\input texinfo   @c -*-texinfo-*-
@c %**start of header
@setfilename sketch.info
@include version.texi
@settitle Sketch

@c an index for sketch commands
@defcodeindex sx
@defcodeindex op
@syncodeindex op sx
@c %**end of header

@copying
Copyright @copyright{} 2005 to 2012 Eugene K. Ressler.

This manual is for @code{sketch}, version @value{VERSION},
@value{UPDATED}, a program that converts descriptions of simple
three-dimensional scenes into line drawings. This version generates
@code{PSTricks} or @code{PGF/TikZ} code suitable for use with the
@TeX{} document processing system.

@code{Sketch} is free software. You can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 3, or (at your option)
any later version.

Sketch is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with @code{sketch}; see the file COPYING.txt.  If not, see
@verb{|http://www.gnu.org/copyleft|}.

@end copying

@dircategory TeX
@direntry
* Sketch: (sketch).             Simple 3D sketching for TeX
@end direntry
 
@titlepage
@title Sketch
@subtitle Simple 3D sketching
@subtitle Version @value{VERSION}, @value{UPDATED}
@author Gene Ressler
@page
@vskip 0pt plus 1fill
@insertcopying
@end titlepage

@c TOC
@contents

@ifnottex
@node Top, About sketch, (dir), (dir)
@top Sketch

@insertcopying
@end ifnottex

@menu
* About sketch::                Why sketch exists and what it does.
* Introduction by example::     Most features shown as working code.
* Input language::              Syntax and semantics of @code{sketch} commands. 
* Building a drawing::          How to use @code{sketch} productively.
* Command line::                Options and their usage.
* Installing sketch::           Building and installing from sources.
* Index of syntax::             
* Index::                       

@detailmenu
 --- The Detailed Node Listing ---

About sketch

* Reporting bugs::              Let use know what's wrong!
* Contributions::               How you can help@dots{}.

Introduction by example

* Hello world::                 Simplest possible @code{sketch} program.
* Drawing options::             Controlling object appearance.
* Drawing a solid::             Drawing an object with 3d appearance.
* Special objects::             Laying @TeX{} over, in, or under drawings.
* Object transforms::           Rotate, translate, scale, and others.
* Repeated objects::            Making transformed copies.
* Swept objects::               Sweeping objects in space to make new shapes.

Swept objects

* Point sweeps::                Swept points make lines and polygons.
* Polyline sweeps::             Swept lines make surfaces.
* Nested sweeps::               Swept sweeps are useful!
* Polygon sweeps::              Swept polygons make solids...
* Polyline sweeps with closure::  and so do closed polyline sweeps.
* Affine arithmetic::           Sketch useful math expression.
* More to learn::               Check out the Mobius strip!

Input language

* Language basics::             Case, space, comments, include files.
* Drawables::                   Things that can be drawn.
* Definitions::                 Giving things names.
* Global environment::          Affect the entire drawing.

Basics 

* Identifiers::                 Names for things.
* Key and reserved words::      Names you shouldn't use.
* Literals::                    Constants and constructors.
* Arithmetic::                  Rules for expressions.
* Options::                     Modifying object appearance.

Literals

* Scalar literals::             Just the numbers.
* Point and vector literals::   3d quantities.
* Transform literals::          Matrix form.

Arithmetic expressions

* Two-operand (binary) forms::  A op B
* Unary forms::                 op A (and others)

Options

* PSTricks options::            Options inherited from @code{PSTricks}.
* TikZ/PGF options::            Options inherited from @code{TikZ/PGF}.
* Dots in TikZ/PGF::            Sketch uses @code{TikZ/PGF} circles for dots.
* TikZ/PGF user-defined styles::  Support for @code{TikZ/PGF} named, user-defined styles.
* Transparency::                See-through polygons.
* Internal options::            Options used by @code{sketch}.

Point lists

* Drawables::                   Things that are drawn.
* Definitions::                 Things with names.

Drawables

* Dots::                        Draw dots.
* Lines::                       Draw polylines.
* Curves::                      Draw curves.
* Polygons::                    Draw polygons.
* Specials::                    Embed raw @LaTeX{} and @code{PSTricks}.
* Sweeps::                      Draw sweeps of dots and polylines.
* Blocks::                      Group other drawables.
* Repeats::                     Draw transformed copies of objects.
* Puts::                        Draw one object transformed.

Sweeps

* Swept points::                Swept points make lines or polygons.
* Swept lines::                 Swept lines make open or closed surfaces.
* Swept polygons::              Swept polygons make closed surfaces.
* Swept blocks::                Swept block @equiv{} block of sweeps.
* Sweep face splitting::        Fixing warped faces with triangles.

Definitions

* Forms of definitions::        Different defs for different purposes.
* Forms of references::         How references denote types.

Global environment

* Global options::              Attributes of the entire drawing.
* Camera::                      A final camera transformation of the scene.
* Picture box::                 Setting the bounding box and 2d clipping.
* Frame::                       Adding a box around the drawing.
* Language::                    Setting the output language.

Building a drawing

* Overview::                    Building a substantial drawing.
* A technical drawing::         An example with fine placement.
* A hierarchical model::        An example with sweeps and puts.
* Caveats::                     Where trouble can occur.

Caveats

* Limits on error detection::   What sketch doesn't do.
* Clipping::                    No clipping at present.
* Hidden surface removal::      Imperfections to fix.

Hidden surface removal and polygon splitting

* Statistics::                  Performance numbers on depth sort.
* Bugs and anomalies::          Imperfections in this implementation.

@end detailmenu
@end menu

@node About sketch, Introduction by example, Top, Top
@comment  node-name,  next,  previous,  up
@chapter About sketch

@menu
* Reporting bugs::              Let use know what's wrong!
* Contributions::               How you can help@dots{}.
@end menu

@code{Sketch} is a small, simple system for producing line drawings of
two- or three-dimensional objects and scenes.  It began as a way to
make illustrations for a textbook after we could find no suitable
tool for this purpose.  Existing scene processors emphasized GUIs
and/or photo-realism, both un-useful to us.  We wanted to produce
finely wrought, mathematically-based illustrations with no extraneous
detail.

@code{Sketch} accepts a tiny scene description language and generates
@code{PSTricks} or @code{TikZ/PGF} code for @LaTeX{}.  The
@code{sketch} language is similar to @code{PSTricks}, making it easy
to learn for current @code{PSTricks} users.  See
@cindex PSTricks
@verb{|www.pstricks.de|} for information on @code{PSTricks}.
@code{TikZ/PGF} is similar.
See
@cindex TikZ/PGF
@verb{|http://sourceforge.net/projects/pgf|}.  One can easily include
arbitrary
@code{PSTricks} or @code{TikZ/PGF} drawings and text 
over, in, or under
@code{sketch} drawings, providing access to the full power of @LaTeX{}
text and mathematics formatting in a three-dimensional setting.

@node Reporting bugs, Contributions, About sketch, About sketch
@comment  node-name,  next,  previous,  up
@section Reporting bugs and recommending improvements.
The group @verb{|http://groups.google.com/group/sketch-users|}
is the best place to report bugs and make improvements.
A second method that will probably produce a slower
response is email to @verb{|sketch@frontiernet.net|}.
We will try to respond, but can't promise.  In any event, don't be
offended if a reply is not forthcoming.  We're just busy and will get
to your suggestion eventually.

For bugs, attach a @code{sketch} input file that causes the bad
behavior.  Embed comments that explain what to look for in
the behavior of @code{sketch} or its output.

A recommendation for improvement from one unknown person counts as one
vote. We use overall vote tallies to decide what to do next as
resources permit.  We reserve the right to assign any number of votes
to suggestions from people who have been helpful and supportive in the
past.

@node Contributions,  , Reporting bugs, About sketch
@comment  node-name,  next,  previous,  up
@section Contributions
If you intend to implement an enhancement of your own, that's
terrific!  Consider collaborating with us first to see if we're
already working on your idea or if we can use your work in the
official release.

@center @image{ex000}
@anchor{Solid coil example}

@node Introduction by example, Input language, About sketch, Top
@comment  node-name,  next,  previous,  up
@chapter Introduction by example
The @code{sketch} input language will seem familiar to users of the
@code{PSTricks} package for @LaTeX{}.  The following program draws a
triangular polygon pierced by a line.
@verbatim
  polygon(0,0,1)(1,0,0)(0,1,0)
  line(-1,-1,-1)(2,2,2)
@end verbatim
@noindent
The coordinate system 
@cindex coordinate system, right-handed
@cindex right-hand coordinate system
is a standard right-handed Cartesian one.

@center @image{ex010}

@menu
* Hello world::                 Simplest possible @code{sketch} program.
* Drawing options::             Controlling object appearance.
* Drawing a solid::             Drawing an object with 3d appearance.
* Special objects::             Laying @TeX{} over, in, or under drawings.
* Object transforms::           Rotate, translate, scale, and others.
* Repeated objects::            Making transformed copies.
* Swept objects::               Sweeping objects in space to make new shapes.
@end menu

@node Hello world, Drawing options, Introduction by example, Introduction by example
@comment  node-name,  next,  previous,  up
@section Hello world
The @code{sketch} program above is nearly the simplest one possible,
the equivalent of a ``hello world''
@cindex hello world
@cindex program, hello world
program you might find at the start of a programming language text.
If it is saved in the file @file{simple.sk}, then the command
@cindex command line, @code{sketch}
@cindex running @code{sketch}
@verbatim
  sketch simple.sk -o simple.tex
@end verbatim
@noindent
creates a file @file{simple.tex} containing @code{PSTricks} commands to
draw these objects on paper.  The contents of @file{simple.tex}
look like this.
@verbatim
  \begin{pspicture}(-1,-1)(2,2)
  \pstVerb{1 setlinejoin}
  \psline(-1,-1)(.333,.333)
  \pspolygon[fillstyle=solid,fillcolor=white](0,0)(1,0)(0,1)
  \psline(.333,.333)(2,2)
  \end{pspicture}
@end verbatim
@noindent
The hidden surface algorithm 
@cindex hidden surface algorithm
of @code{sketch} has split 
@cindex splitting, line and surface
the line into
two pieces and ordered the three resulting objects so that the correct
portion of the line is hidden.

If you've noticed that the projection we are using seems equivalent to
erasing the @math{z}-coordinate of the three-dimensional input points,
pat yourself on the back.  You are correct.  This is called a
@dfn{parallel projection}.
@cindex parallel projection
@cindex projection, parallel
The @math{z}-coordinate axis is pointing straight out of the paper at
us, while the @math{x}- and @math{y}-axes point to the right and up as
usual.

The resulting picture file can be included in a @LaTeX{} document with
@verb{|\input{simple}|}.  Alternately, adding the command line option
@option{-T}@footnote{Or for European users of A4 size paper,
@option{-Te}.} 
@cindex command line option
@cindex option, command line
causes the @code{pspicture} to be wrapped in a short
but complete document, ready to run though @LaTeX{}.
@cindex document template
@cindex template, document
In a finished, typeset document, the picture looks like this.  (The
axes have been added in light gray.)

@center @image{ex020}

@noindent
It is important to know that only the ``outside'' 
@cindex outside of a polygon
@cindex polygon, outside of
of a polygon is
normally drawn.  The @dfn{outside} is where the vertices given in the
@code{polygon} 
@sxindex polygon
command appear in @emph{counter-clockwise} 
@cindex counter-clockwise polygon vertex order
@cindex polygon vertex order
@cindex order, polygon vertex
order.  Thus, if the command above had been
@verbatim
  polygon(0,1,0)(1,0,0)(0,0,1)
@end verbatim
@noindent
the polygon would not appear in the picture at all.  It would have
been @dfn{culled}
@cindex culling
from the scene.  This culling behavior may seem
strange, but stay tuned.

@node Drawing options, Drawing a solid, Hello world, Introduction by example
@comment  node-name,  next,  previous,  up
@section Options
Many @code{PSTricks} and @code{TikZ/PGF} options 
@cindex option
work just fine in @code{sketch}. If generating @code{PSTricks}, the code
@sxindex line
@verbatim
  polygon[fillcolor=lightgray,linewidth=3pt](0,0,1)(1,0,0)(0,1,0)
  line[linestyle=dotted](-1,-1,-1)(2,2,2)
@end verbatim
@noindent
produces

@center @image{ex030}

To produce @code{TikZ/PGF}, the corresponding code is 
@verbatim
  polygon[fill=lightgray,line width=3pt](0,0,1)(1,0,0)(0,1,0)
  line[style=dotted](-1,-1,-1)(2,2,2)
  global { language tikz }
@end verbatim
@noindent
The final @code{global} 
@cindex options, global
@cindex global options
instructs @code{sketch} to produce @code{TikZ/PGF} code as output
rather than the default, @code{PSTricks}.  Note that @code{polygon}
fill color and @code{line} style options both conform to @code{TikZ}
syntax rules.  The resulting @code{TikZ/PGF} output is
@verbatim
  \begin{tikzpicture}[join=round]
  \draw[dotted](-1,-1)--(.333,.333);
  \filldraw[fill=lightgray,line width=3pt](0,0)--(1,0)--(0,1)--cycle;
  \draw[dotted](.333,.333)--(2,2);
  \end{tikzpicture}
@end verbatim
@noindent
The remaining examples of this manual are in PSTricks style.

@node Drawing a solid, Special objects, Drawing options, Introduction by example
@comment  node-name,  next,  previous,  up
@section Drawing a solid
Let's try something more exciting.  @code{Sketch} has no notion of a
solid, 
@cindex solid
but polygonal @dfn{faces} 
@cindex faces
can be used to represent the
boundary of a solid.  To the previous example, let's add three more
triangular polygons to make the faces of an irregular tetrahedron.
@cindex tetrahedron
@sxindex def
@sxindex polygon
@verbatim
  % vertices of the tetrahedron
  def p1 (0,0,1) def p2 (1,0,0)
  def p3 (0,1,0) def p4 (-.3,-.5,-.8)

  % faces of the tetrahedron.
  polygon(p1)(p2)(p3) % original front polygon
  polygon(p1)(p4)(p2) % bottom
  polygon(p1)(p3)(p4) % left
  polygon(p3)(p2)(p4) % rear

  % line to pierce the tetrahedron
  line[linecolor=red](-1,-1,-1)(2,2,2)
@end verbatim
@noindent
This example uses @dfn{definitions}, 
@cindex definition
which begin with 
@code{def}.
@sxindex def
These @dfn{define} or give names to points, 
@cindex definition, point
@cindex point definition
which are then available
as @dfn{references} 
@cindex reference, point
by enclosing the names in parentheses,
e.g.@ @verb{|(foo)|}.
@sxindex (foo)@r{, point reference}
The parentheses denote that the names refer to points; they are
required.  There can be no
@cindex white space
white space between them and the name.

As you can see, comments 
@cindex comments
start with @verb{|%|} as in @TeX{} and extend
to the end of the line (though @verb{|#|} will work as well).  White
space, 
@cindex white space
including spaces, tabs and blank lines, has no effect in the @code{sketch}
language.

@center @image{ex040}

@noindent
If we look inside the @TeX{} file produced by @code{sketch}, there
will be only three polygons.  The fourth has been 
@cindex culling
culled because it is
a ``back face'' 
@cindex back face
of the tetrahedron, invisible to our view.  It is
unnecessary, and so it is removed.

In some drawings, polygons act as zero-thickness solid surfaces with
both sides visible rather than as the faces of solid objects, where
back faces can be culled.  For zero-thickness solids, culling 
@cindex culling
is a
problem.  One solution is to use a pair of @code{sketch} polygons for
each zero-thickness face, identical except with opposite vertex
orders.  This is unwieldy and expensive.  A better way is to
set the @code{sketch} internal option @code{cull} to @code{false} in
the usual @code{PSTricks} manner.
@opindex cull
@verbatim
  polygon[cull=false](p1)(p2)(p3)
@end verbatim
@noindent
The following shows the same helix
@cindex helix
shape drawn first with
@verb{|cull=true|} (the default) and then @verb{|cull=false|}.

@center @image{ex045} @anchor{Helix with cull set false then true} 

@noindent
We'll soon see how to produce these helixes with a few lines 
of @code{sketch} language code.

It may be tempting to turn culling off gratuitously so that vertex order
can be ignored.  This is not a good idea because output file size and
@TeX{} and Postscript processing time both depend on the number of
output polygons.  Culling usually improves performance by a factor of
two.  On the other hand, globally setting @code{cull=false} is 
reasonable while debugging.  See @ref{Global options} and 
@ref{Limits on error detection}.

@node Special objects, Object transforms, Drawing a solid, Introduction by example
@comment  node-name,  next,  previous,  up
@section Special objects
We can add labels 
@cindex labels
to a drawing by using @verb{|special|} 
@sxindex special
@cindex special object
objects, which provide a way to embed raw @LaTeX{} and @code{PSTricks}
code.  Adding this to the tetrahedron does the trick.
@verbatim
  special |\footnotesize
           \uput{2pt}[ur]#1{$P1$}
           \uput[r]#2{$P2$}
           \uput[u]#3{$P3$}
           \uput[d]#4{$P4$}|
    (p1)(p2)(p3)(p4)
@end verbatim
@noindent
Here is the result.

@center @image{ex042}

There are several details to note here.  First, the quoting convention
@cindex quoting, special
for the raw code is similar to the @LaTeX{} @verb{|\verb|} command.  The
first non-white space character following @verb{|special|} is
understood to be the quote character, 
in this case @samp{|}.  The raw
text continues until this character recurs.

Second, the argument references 
@cindex argument, special
@verb{|#1|}, @verb{|#2|}, @verb{|#3|}, and @verb{|#4|} refer to point,
vector, or scalar values in the list that follow.  This is similar to
@TeX{} macro syntax.  The transformed and two-dimensional projections
of these three-dimensional points are substituted
@cindex substitution, special
@cindex special argument substitution
in the final output.  An argument reference of the form @verb{|#1-2|}
is replaced with the angle in degrees of the two-dimensional vector
that connects the projections of the two respective argument points,
here @verb{|#1|} and @verb{|#2|}.  The substituted angle is enclosed
in curly braces @code{@{ @}}.  When @code{TikZ/PGF} output is being
generated, the angle is rounded to the nearest degree because
non-integer angles are not allowed by @code{TikZ/PGF} primitives.

As of Version 0.3 of @code{sketch}, special arguments may be scalars
or vectors in addition to points. References to scalar arguments are
merely replaced with a number formatted just as any point
coordinate. References to vectors become two-dimensional points. The
tick operator that selects individual components of points and vectors
elsewhere in @code{sketch} (see for example @ref{Affine arithmetic})
can also be applied to point and vector argument references.  All
three dimensions of a transformed point or vector can also be
substitued with @code{'3}. See @ref{Specials} for details.

By default, @code{special} objects are printed last, overlaying all
other objects in the scene.  If you specify the internal option
@cindex internal option
@cindex option, internal
@code{lay=in}, the hidden surface algorithm 
@opindex lay
@cindex hidden surface algorithm
considers the entire special object to be the first point
(@verb{|#1|}) in the argument list.  If that point is behind (of
smaller @math{z}-component than) any drawable, then the entire special
object is drawn before that drawable, so the drawable obscures parts of
the special object that overlaps it.  In our example, @verb{|p1|} is
the front-most point in the scene (has the largest
@math{z}-component), so adding @code{lay=in} has no effect.  

With option @code{lay=under}, a special is drawn @emph{before}, hence
appears @emph{under} any of the objects handled by the hidden surface
algorithm.  This is how the light gray axes were added to the ``hello
world'' example @ref{Hello world}.

@verb{|Special|} objects are powerful, with many possible uses.

@node Object transforms, Repeated objects, Special objects, Introduction by example
@comment  node-name,  next,  previous,  up
@section Transforms
@cindex transform
Now let's add a second copy of the pierced tetrahedron.  We'll rotate
the copy 90 degrees about the @math{x}-axis with the origin as
@dfn{center of rotation}
@cindex center of rotation
@cindex rotation, center of
so we can see the back,
then translate it to the right---in the positive
@math{x}-direction---so it doesn't collide with the original. To help
us see what's going on, make the back side gray.
@sxindex def
@sxindex put
@sxindex line
@sxindex polygon
@opindex linecolor
@opindex fillcolor
@sxindex rotate
@sxindex translate
@sxindex then
@verbatim
  def pierced_tetrahedron {
    def p1 (0,0,1) def p2 (1,0,0)
    def p3 (0,1,0) def p4 (-.3,-.5,-.8)
    polygon(p1)(p2)(p3)                      % original
    polygon(p1)(p4)(p2)                      % bottom
    polygon(p1)(p3)(p4)                      % left
    polygon[fillcolor=lightgray](p3)(p2)(p4) % rear
    line[linecolor=red](-1,-1,-1)(2,2,2)
  }
  {pierced_tetrahedron}  % tetrahedron in original position
  put { rotate(90, (0,0,0), [1,0,0]) % copy in new position
        then translate([2.5,0,0]) } {pierced_tetrahedron}
@end verbatim
@noindent
Here the entire code of the previous example has been wrapped in a
definition by forming a @dfn{block} 
@cindex block
@sxindex @{ @}@r{, block drawable}
with braces (a single item would not need them).  The point
definitions nested inside the braces are @dfn{lexically scoped}.
@cindex lexical scope
@cindex scope, identifier
Their meaning extends only to the end of the block.  The outer
@verb{|def|} is called a @dfn{drawable}
@cindex drawable
definition 
@cindex definition, drawable
@cindex drawable definition
because it describes something that can be drawn.

A drawable definition by itself causes nothing to happen until its
name is referenced.  Drawable references must be enclosed in curly
braces, e.g.@ @verb{|{foo}|}, with no intervening
@cindex white space
white space.  In the code
above, the first reference 
@cindex reference, drawable
@verb{|{pierced_tetrahedron}|} 
@sxindex @{foo@}@r{, drawable reference}
is a plain
one. Its effect is merely to duplicate the earlier drawing.  Almost
any series of @code{sketch} commands @verb{|stuff|} may be replaced
with @verb{|def foo { stuff } {foo}|} without changing its meaning.

The @verb{|put|} command supplies a second reference, this time with
a @dfn{transform} applied first.  The @verb{|rotate|}
@sxindex rotate
@cindex rotation
transform turns the tetrahedron 90 degrees about the origin.  The
axis of rotation 
@cindex axis, rotation
is the vector @math{[1,0,0]}.  By the @dfn{right
hand rule}, 
@cindex right hand rule
this causes the top of the tetrahedron to rotate toward
the viewer and the bottom away.  The rule receives its name from the
following definition:
@quotation
@anchor{Right hand rule}
@strong{Right hand rule.}  If the right hand is wrapped around any
axis with the thumb pointing in the axis direction, then the fingers
curl in the direction of positive rotation about that axis.
@end quotation
The @verb{|translate|} 
@sxindex translate
@cindex translation transform
@cindex transform, translation
transform moves the pyramid laterally to
the right by adding the vector 
@cindex vector
@math{[2.5,0,0]} to each vertex
coordinate.  The result is shown here.

@center @image{ex050}

@node  Repeated objects, Swept objects, Object transforms, Introduction by example
@comment  node-name,  next,  previous,  up
@section Repeated objects
To draw seven instances of the tetrahedron, each differing from the
last by the same transform, replace the last two commands of the
previous example with
@sxindex repeat
@sxindex rotate
@sxindex translate
@verbatim
  repeat { 7, rotate(15, (0,0,0), [1,0,0]) % copy in new position
              then translate([2,0,0]) } {pierced_tetrahedron}
@end verbatim
@noindent
And the result@enddots{}

@center @image{ex060}

@node  Swept objects,  , Repeated objects, Introduction by example
@comment  node-name,  next,  previous,  up
@section Swept objects
@cindex swept object
@cindex sweep
Many familiar shapes can be generated by sweeping simpler ones through
space and considering the resulting path, surface, or volume.
@code{Sketch} implements this idea in the @verb{|sweep|} command.
@sxindex sweep
@sxindex rotate
@verbatim
  def n_segs 8
  sweep { n_segs, rotate(180 / n_segs, (0,0,0), [0,0,1]) } (1,0,0)
@end verbatim
@noindent
This code sweeps the point @math{(1,0,0)} 
@cindex point sweep
@cindex swept point
eight times by rotating it
@math{180/8 = 22.5} degrees each time and connecting the resulting
points with line segments.  The @verb{|def|} used here is a
@dfn{scalar} definition.
@cindex definition, scalar
@cindex scalar definition
References to
@cindex reference, scalar
scalars have no enclosing brackets at all.

@menu
* Point sweeps::                Swept points make lines and polygons.
* Polyline sweeps::             Swept lines make surfaces.
* Nested sweeps::               Swept sweeps are useful!
* Polygon sweeps::              Swept polygons make solids...
* Polyline sweeps with closure::  and so do closed polyline sweeps.
* Affine arithmetic::           Sketch useful math expression.
* More to learn::               Check out the Mobius strip!
@end menu

@node Point sweeps, Polyline sweeps, Swept objects, Swept objects
@comment  node-name,  next,  previous,  up
@subsection Point sweeps
Sweeping a point makes a one-dimensional path, which is a polyline.
Since we have swept with a rotation, the result is a circular arc.
Here is what it looks like.

@center @image{ex070}

This is the first example we have seen of @code{sketch} arithmetic.
The expression @verb{|180 / n_segs|} causes the eight rotations to add
to 180.  If you're paying attention, you'll have already noted that
there are @emph{nine} points, producing eight line segments.

You can cause the swept point to generate a single polygon rather than
a polyline by using the @dfn{closure tag} @verb{|<>|} 
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
after the number
of swept objects. Code and result follow
@sxindex def
@sxindex rotate
@sxindex sweep
@verbatim
  def n_segs 8
  sweep { n_segs<>, rotate(180 / n_segs, (0,0,0), [0,0,1]) } (1,0,0)
@end verbatim
@center @image{ex080}

@node Polyline sweeps, Nested sweeps, Point sweeps, Swept objects
@comment  node-name,  next,  previous,  up
@subsection Polyline sweeps
Sweeping a polyline produces a 
@cindex line sweep
@cindex swept line
@cindex surface
surface composed of many faces.  
@cindex faces
The unbroken helix in the
example @ref{Helix with cull set false then true} is produced by this
code (plus a surrounding @verb{|put|} rotation to make an interesting
view; this has been omitted).
@sxindex def
@sxindex sweep
@sxindex rotate
@sxindex translate
@opindex cull
@opindex linewidth
@verbatim
  def K [0,0,1]
  sweep[cull=false] {
    60, 
    rotate(10, (0,0,0), [K]) then translate(1/6 * [K]) 
  } line[linewidth=2pt](-1,0)(1,0)
@end verbatim
@noindent
Again, 60 segments of the helix 
@cindex helix
are produced by connecting 61
instances of the swept line.  Options 
@cindex options, sweep
applied to the sweep, here
@verb{|cull=false|}, are treated as options for the generated polygon
or polyline.  Options of the swept line itself, here
@verb{|linewidth=2pt|}, are ignored, though with a warning. This
@verb{|def|} is a @dfn{vector} definition, 
@cindex definition, vector
@cindex vector definition
which must be referenced
with square brackets, e.g.@ @verb{|[foo]|}.
@cindex reference, vector
@sxindex [foo]@r{, vector reference}

@node Nested sweeps, Polygon sweeps, Polyline sweeps, Swept objects
@comment  node-name,  next,  previous,  up
@subsection Nested sweeps
When the center point of rotation is omitted, 
@cindex center of rotation
@cindex rotation, center of
the origin is assumed.
When a point has only two coordinates, they are taken as
@math{x}@tie{}and @math{y},@tie{}with @math{z=0} assumed.  A toroid 
@cindex toroid
is therefore obtained with this code.
@sxindex def
@sxindex sweep
@sxindex rotate
@verbatim
  def n_toroid_segs 20   def n_circle_segs 16
  def r_minor 1          def r_major 1.5
  sweep { n_toroid_segs, rotate(360 / n_toroid_segs, [0,1,0]) }
    sweep { n_circle_segs, rotate(360 / n_circle_segs, (r_major,0,0)) } 
      (r_major + r_minor, 0)
@end verbatim
@indent
For intuition, the idea of the code is to sketch a circle to the right
of the origin in the @math{xy}-plane, then rotate that circle ``out of
the plane'' about the @math{y}-axis to make the final figure.  This
produces the following.  (A view rotation and some axes have been
added.)

@center @image{ex090}

This example also shows that the swept object may itself be another
@code{sweep}.
@sxindex sweep
@cindex nesting, swept object
In fact, it may be @emph{any} @code{sketch} expression that results in
a list of one or more points or, alternately, a list of one or more
polylines and polygons.  The latter kind of list can be created with a
@verb{|{ }|}-enclosed block, perhaps following a 
@sxindex put
@verb{|put|} or
@sxindex repeat
@verb{|repeat|}.
@sxindex @{ @}@r{, block drawable}

@node Polygon sweeps, Polyline sweeps with closure, Nested sweeps, Swept objects
@comment  node-name,  next,  previous,  up
@subsection Polygon sweeps
Sweeping a polygon 
@cindex polygon sweep
@cindex swept polygon
creates a closed surface with polygons at the ends,
which are just copies of the original, appropriately
positioned.  @xref{Solid coil example}. 
@cindex options, swept object
Options on the swept polygon,
if they exist, are applied to the ends.  Otherwise the sweep options
@cindex options, sweep
are used throughout.

@node Polyline sweeps with closure, Affine arithmetic, Polygon sweeps, Swept objects
@comment  node-name,  next,  previous,  up
@subsection Polyline sweeps with closure
A polyline sweep with a closure tag 
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
creates another kind of closed
surface.  First, the polyline segments are connected by faces, just as
without the closure tag.  Then, each set of end points is joined to
make a polygon, one for each end.  A code for several views of a
cylindrical prism follows.
@sxindex def
@sxindex repeat
@sxindex rotate
@sxindex then
@sxindex translate
@sxindex sweep
@sxindex line
@opindex fillcolor
@verbatim
  def n_cyl_segs 20  def n_views 5  def I [1,0,0]
  def endopts [fillcolor=lightgray]
  repeat { n_views, rotate(180/n_views, [I]) then translate([I] * 2.1) } 
    sweep[endopts]{ n_cyl_segs<>, rotate(360/n_cyl_segs, [0,1,0]) } 
      line[fillcolor=white](1,-1)(1,1)
@end verbatim
@noindent
It produces this drawing.

@center @image{ex110}

@noindent
The options of the swept line, if any, are applied to the faces
produced by sweeping the line, but not the end polygons.  Otherwise,
the sweep options are applied throughout.
@cindex options, swept object
The @verb{|def|} in this example is an @dfn{option} definition.
@cindex definition, options
@cindex options definition
References to options must be enclosed in square brackets, e.g.@tie{}
@verb{|[foo]|}.
@cindex reference, options
@sxindex [foo]@r{, options reference}
You can concatenate several sets of options with a single
reference, e.g.@tie{}
@verb{|[color, size, style]|}
would cause the option definitions for @code{color}, @code{size},
and @code{style} to appear in sequence in the output 
created by the @code{sketch} command containing the reference.
Happily, the syntax of @code{sketch} is such that
options references can never be confused with vector references.  While
not apparent in this example, options references are useful when
defining many objects with a similar appearance.

@node Affine arithmetic, More to learn, Polyline sweeps with closure, Swept objects
@comment  node-name,  next,  previous,  up
@subsection Affine arithmetic
The arithmetic @verb{|[I] * 2.1|} above hints at a larger truth.
@code{Sketch} operators work on scalars, vectors, points, and
transforms according to the general rules of @dfn{affine algebra}.
@cindex affine arithmetic
This can be helpful for setting up diagrams with computed geometry.
For example, if you have triangle vertices @verb{|(p1)|} through
@verb{|(p3)|} and need to draw a unit normal vector pointing out of
the center of the triangle, this code does the trick.
@sxindex def
@sxindex polygon
@sxindex line
@opindex arrows
@verbatim
  def p1 (1,0,0)  def p2 (0,0.5,0)  def p3 (-0.5,-1,2)
  def O (0,0,0)
  def N unit( ((p3) - (p2)) * ((p1) - (p2)) )
  def n1 ((p1)-(O) + (p2)-(O) + (p3)-(O)) / 3 + (O)
  def n2 (n1)+[N]
  polygon(p1)(p2)(p3)
  line[arrows=*->](n1)(n2)
@end verbatim
@noindent
The first line computes the cross product of two edge vectors of the
triangle and scales it to unit length.  The second computes the
average of the vertices.  Note that subtraction and addition of the
origin effectively convert vectors to points and @emph{vice versa}.
The line command draws the normal at the correct spot.

@center @image{ex100}

Two caveats regarding this example remain.  First, the only way to use
@code{PSTricks}-style arrows is with @verb{|arrows=|}.  
@opindex arrows
The alternative syntax for @code{PSTricks} arrows is not allowed in
@verb{|sketch|}. Second, you might like to eliminate the third
@verb{|def|} and write instead the following.
@verbatim
  line[arrows=*->](n1) (n1)+[N]
@end verbatim
@noindent
This is not allowed.  The point lists in drawables may consist only of
explicit points or point references.  You may, however, use arithmetic
to calculate point components.  The following works, though it's
a little cumbersome.
@verbatim
  line[arrows=*->](n1)((n1)'x+(N)'x, (n1)'y+(N)'y, (n1)'z+(N)'z)
@end verbatim
@noindent
Obviously, the @dfn{tick operator} 
@cindex tick operator (@code{'})
@sxindex 'x@r{,} 'y@r{, and }'z
@samp{'x} extracts components of points and
vectors. 

@node More to learn,  , Affine arithmetic, Swept objects
@comment  node-name,  next,  previous,  up
@subsection More to learn
This is not the end of the story on sweeps!  We invite the reader into
the main body of this documentation @ref{Sweeps} to learn more.

@center @image{ex120}

@noindent
Who knows where you'll finish?

@node Input language, Building a drawing, Introduction by example, Top
@comment  node-name,  next,  previous,  up
@chapter Input language
This chapter describes the @code{sketch} input language in detail.

@menu
* Language basics::             Case, space, comments, include files.
* Drawables::                   Things that can be drawn.
* Definitions::                 Giving things names.
* Global environment::          Affect the entire drawing.
@end menu

@node Language basics, Drawables, Input language, Input language
@comment  node-name,  next,  previous,  up
@section Basics 
@code{Sketch} input is plain ASCII text, usually stored in an input
file.  
@cindex input file
@cindex file, input
It describes a @dfn{scene},
so the sketch language is a @dfn{scene description
language}.
@cindex scene description language
@cindex language, scene description
@code{Sketch} input is also @dfn{declarative}.
@cindex declarative language
@cindex language, declarative
It merely
declares what the scene ought to look like when drawing is complete
and says very little about how @code{sketch} should do its work.
@code{Sketch} commands are not executed sequentially as in the usual
programming language.  They merely contribute to that declaration.

A few syntactic details are important.  Case is significant in the
@code{sketch} language.  With a few exceptions, white space is not.
This includes line breaks.
@cindex white space
Comments begin with @code{%} or @code{#} and extend to the end of the
line.  You can disable a chunk of syntactically correct @code{sketch}
code by enclosing it in a @code{def}.
@cindex comments
There is a simple ``include file'' mechanism. 
@cindex include file
@cindex file, include
The command
@sxindex input
@verbatim
  input{otherfile.sk}
@end verbatim
@noindent
causes the contents of @file{otherfile.sk} to be inserted as though
they were part of the current file.

@menu
* Identifiers::                 Names for things.
* Key and reserved words::      Names you shouldn't use.
* Literals::                    Constants and constructors.
* Arithmetic::                  Rules for expressions.
* Options::                     Modifying object appearance.
@end menu

@node Identifiers, Key and reserved words, Language basics, Language basics
@comment  node-name,  next,  previous,  up
@subsection Identifiers
Identifiers in @code{sketch} are references to earlier-defined
options, scalars, points, vectors, transforms, drawables, and tags.
@cindex identifiers
Definitions are explained in @ref{Definitions}.

An identifier consists of a leading letter followed by letters,
numbers and underscores.  The last character may @emph{not} be an
underscore.  Keywords cannot be used as identifiers, and reserved
words ought to be avoided. @xref{Key and reserved words}.

@node Key and reserved words, Literals, Identifiers, Language basics
@comment  node-name,  next,  previous,  up
@subsection Key and reserved words
@cindex keywords
The keywords of @code{sketch} are @code{picturebox} @code{curve}
@code{def} @code{dots} @code{frame} @code{global} @code{input}
@code{line} @code{polygon} @code{put} @code{repeat} @code{set}
@code{sweep} and @code{then}.  The @code{sketch} parser will note a
syntax error if any of these are used in place of a proper identifier.

In addition, there are reserved words 
@cindex reserved words
that can currently be defined by the user, but with the risk that
future versions of @code{sketch} will reject those definitions.  The
reserved words are @code{acos} @code{asin} @code{atan2} @code{cos}
@code{inverse} @code{perspective} @code{project} @code{rotate}
@code{scale} @code{sin} @code{special} @code{sqrt} @code{translate}
@code{unit} and @code{view}.

@node Literals, Arithmetic, Key and reserved words, Language basics
@comment  node-name,  next,  previous,  up
@subsection Literals
Literals in @code{sketch} include scalars, points, vectors, and
transforms.  Literals, along with defined object references,
are used in arithmetic expressions.  @xref{Arithmetic}.

@menu
* Scalar literals::             Just the numbers.
* Point and vector literals::   3d quantities.
* Transform literals::          Matrix form.
@end menu

@node Scalar literals, Point and vector literals, Literals, Literals
@comment  node-name,  next,  previous,  up
@subsubsection Scalar literals
@cindex scalar literal
@cindex literal, scalar
Scalar literals are positive floating point numbers with syntax
according to C conventions.  The following are some examples.
@example
0 1004 .001 8.3143 3. 1.60E-19 6.02e+23 
@end example
@noindent
Scalar literals may not contain embedded spaces.

@node Point and vector literals, Transform literals, Scalar literals, Literals
@comment  node-name,  next,  previous,  up
@subsubsection Point and vector literals
@cindex point literal
@cindex literal, point
@cindex vector literal
@cindex literal, vector
Points and vector literals have these forms respectively.
@example
(@i{X},@i{Y},@i{Z})  [@i{X},@i{Y},@i{Z}]
@end example
@noindent
Each of the components is itself a scalar expression.  The
@math{z}-components are optional and default to zero.

@node Transform literals,  , Point and vector literals, Literals
@comment  node-name,  next,  previous,  up
@subsubsection Transform literals
@cindex transform literal
@cindex literal, transform
Most transform literals are formed by @dfn{constructors}.
@cindex constructor
These are summarized in the following table.
@multitable {@code{[[@math{a_{11}},@math{a_{12}},@math{a_{13}},@math{a_{14}}],}}{point,vector,vector}{Long column meant to wrap but does it?  Maybe.}
@headitem Constructor @tab Param types @tab Description
@item @code{rotate(A,P,X)} 
 @sxindex rotate
 @cindex rotation transform
 @cindex transform, rotation
 @tab scalar,point,vector 
 @tab Rotate @code{A} degrees about point @code{P} with axis @code{X}
   according to the right hand rule. @xref{Right hand rule}.
   @code{P} and @code{X} are both optional and default to the origin and
   the @math{z}-axis respectively.
@item @code{translate(X)} 
 @sxindex translate
 @cindex translation transform
 @cindex transform, translation
 @tab vector 
 @tab Translate by @code{X}.
@item @code{scale(S)} 
 @sxindex scale
 @cindex scale transform
 @cindex transform, scale
 @tab scalar
 @tab Scale uniformly by factor @code{S}.
@item @code{scale(V)} 
 @sxindex scale
 @cindex scale transform
 @cindex transform, scale
 @tab vector
 @tab Scale along each axis by components of @code{V}.
@item @code{project()} 
 @sxindex project
 @cindex parallel projection
 @cindex projection, parallel
 @tab ---
 @tab Same as @code{scale([1,1,0])}.
@item @code{project(S)} 
 @sxindex project
 @cindex perspective projection
 @cindex projection, perspective
 @tab scalar
 @tab Perspective projection with view center at origin and projection
 plane @math{z=-@code{S}}.
@item @code{perspective(S)} 
 @sxindex perspective
 @cindex perspective projection
 @cindex projection, perspective
 @tab scalar
 @tab Perspective @emph{transform} identical to @code{project(S)}
 except that the @math{z}-coordinate of the transformed result is
 @dfn{pseudodepth}, usable by the hidden surface algorithm. 
 @cindex hidden surface algorithm
@item @code{view(E,D,U)} 
 @sxindex view
 @cindex view transform
 @cindex transform, view
 @tab point,vector,vector
 @tab View transform similar to that of @code{OpenGL}'s.  The
@emph{eye point} @code{E} is translated to the origin while a rotation
is also applied that makes the @emph{view direction vector} @code{D}
and the @emph{view ``up'' vector} @code{U} point in the negative
@math{z}- and the @math{y}-directions respectively.  If @code{U} is
omitted, it defaults to @math{[0,1,0]}. When @code{U} is omitted,
@code{D} may be also; it defaults to @code{(0,0,0)-(E)}, a vector
pointing from the eye toward the origin.
@item @code{view(E,L,U)} 
 @sxindex view
 @cindex view transform
 @cindex transform, view
 @tab point,point,vector
 @tab An alternate form of @code{view(E,D,U)} above where 
 the view direction parameter @code{D} is replaced with a
``look at'' point @code{L}, i.e., a point where the viewer is focusing
her attention.  This form of view is equivalent to
@code{view(E, (L)-(E), U)}, where @code{(L)-(E)} is a direction
vector. @code{U} is optional and defaults to @math{[0,1,0]}.
@item
@code{[[@math{a_{11}},@math{a_{12}},@math{a_{13}},@math{a_{14}}]}@*
@code{@w{ }[@math{a_{21}},@math{a_{22}},@math{a_{23}},@math{a_{24}}]}@*
@code{@w{ }[@math{a_{31}},@math{a_{32}},@math{a_{33}},@math{a_{34}}]}@*
@code{@w{ }[@math{a_{41}},@math{a_{42}},@math{a_{43}},@math{a_{44}}]]}@*
  @sxindex [[ ][ ][ ][ ]]@r{, transform literal}
  @cindex direct transform
  @cindex transform, direct
  @tab 16 scalars 
  @tab Direct transform matrix definition. Each
  of the @math{a_{ij}} is a scalar expression. See text for details.
@end multitable
@noindent
The @code{project} 
@sxindex project
constructor is not generally useful because it
defeats hidden surface removal by collapsing the scene onto a single
plane.  It is a special purpose transform for drawing pictures of
scenes where three-dimensional objects are being projected onto
planes.  See, for example, @ref{Overview}.

The direct transform constructor 
@sxindex [[ ][ ][ ][ ]]@r{, transform literal}
@cindex direct transform
@cindex transform, direct
allows direct entry of a 4x4 transformation matrix. Most 3d graphics
systems, including @code{sketch}, use 4-dimensional homogeneous
coordinates internally. These are transformed by multiplication with
4x4 matrices.  The built-in constructors (@code{rotate}, @code{scale},
etc.) are all translated internally to such 4x4 matrices. For more
information on homogeneous coordinate transformations, see any good
computer graphics textbook. The direct transform feature of
@code{sketch} allows you to enter a matrix of your own choice. There
are not many cases where this is useful, but if you do need it, you
need it badly!

@node Arithmetic, Options, Literals, Language basics
@subsection Arithmetic expressions
Arithmetic expressions over @code{sketch} literals and
defined identifiers are summarized in the following tables.

@menu
* Two-operand (binary) forms::  A op B
* Unary forms::                 op A (and others)
@end menu

@node Two-operand (binary) forms, Unary forms, Arithmetic, Arithmetic
@comment  node-name,  next,  previous,  up
@subsubsection Two-operand (binary) forms and precedence
Most two-operand binary 
@cindex binary form
@cindex two-operand form
forms have meanings dependent on the types of
their arguments.  An exhaustive summary of the possibilities is given
in the following table.
@multitable {transform}{@code{then}}{transform}{transform}{a long description that really ought to wrap but does it I do not know}
@headitem Left @tab Op @tab Right @tab Result @tab Description
@item scalar @tab @code{+} @tab scalar 
@sxindex +@r{, plus operator}
@tab scalar @tab Scalar sum.
@item vector @tab @code{+} @tab vector
@tab  vector @tab Vector sum.
@item point  @tab @code{+} @tab vector
@tab  point  @tab Point-vector affine sum.
@item vector @tab @code{+} @tab point
@tab  "         @tab " 
@item scalar    @tab @code{-} @tab scalar
@sxindex -@r{, minus operator}
@tab  scalar    @tab Scalar difference.
@item vector    @tab @code{-} @tab vector
@tab  vector    @tab Vector difference.
@item point     @tab @code{-} @tab point
@tab  vector    @tab Point-point affine difference.
@item point     @tab @code{-} @tab vector
@tab  point     @tab Point-vector affine difference.
@item scalar    @tab @code{*} or 
                     @code{.} @tab scalar
@sxindex *@r{, multiplication operator}
@sxindex .@r{, dot operator}
@tab  scalar    @tab Scalar product.
@item scalar    @tab @code{*} or 
                     @code{.} @tab vector
@tab  vector    @tab Scalar-vector product.
@item vector    @tab @code{*} or 
                     @code{.} @tab scalar
@tab  "         @tab " 
@item vector    @tab @code{*} @tab vector
@tab  vector    @tab Vector cross-product.
@item vector    @tab @code{.} @tab vector
@tab  scalar    @tab Vector dot product.
@item scalar    @tab @code{^} @tab scalar
@sxindex ^@r{, exponentiation operator}
@tab  scalar    @tab Raise scalar to scalar power.
@item transform @tab @code{^} @tab integer
@tab  transform @tab Raise transform to integer power.
@cindex transform
@item transform @tab @code{*} or 
                     @code{.} @tab point
@tab  point     @tab Affine point transform (right-to-left).
@item transform @tab @code{*} or 
                     @code{.} @tab vector
@tab  vector    @tab Affine vector transform (right-to-left).
@item transform @tab @code{*} or 
                     @code{.} @tab transform
@tab  transform @tab Transform composition (right-to-left).
@item point     @tab @code{then} @tab transform
@sxindex then
@tab  point     @tab Affine point transform (left-to-right).
@item vector    @tab @code{then} @tab transform
@tab  vector    @tab Affine vector transform (left-to-right).
@item transform @tab @code{then} @tab transform
@tab  transform @tab Transform composition (left-to-right).
@item scalar    @tab @code{/} @tab scalar
@sxindex /@r{, division operator}
@tab  scalar    @tab Scalar division.
@item vector    @tab @code{/} @tab scalar
@tab  vector    @tab Vector component-wise division by scalar.
@item point     @tab @code{'} @tab @code{x}, @code{y}, or @code{z}
@cindex tick operator (@code{'})
@sxindex 'x@r{,} 'y@r{, and }'z
@tab  scalar    @tab Point component extraction.
@item vector    @tab @code{'} @tab @code{x}, @code{y}, or @code{z}
@tab  scalar    @tab Vector component extraction.
@end multitable
@sp 1
@noindent
Operator precedence 
@cindex precedence, operator
@cindex operator precedence
is shown in this table.
@multitable {@code{then}} {highest (most tightly binding)}
@headitem Op        @tab Precedence
@item @code{'}      @tab highest (most tightly binding)
@item @code{^}      @tab 
@item @code{-}      @tab (unary negation)
@item @code{*}
      @code{.}
      @code{/}      @tab 
@item @code{+}
      @code{-}      @tab
@item @code{then}   @tab lowest (least tightly binding)
@end multitable
@sp 1
@noindent
All operations are left-associative 
@cindex operator associativity
@cindex associativity, operator
except for @samp{^}.
Parentheses @samp{( )} 
@cindex parentheses
@sxindex ( )@r{, grouping}
are used for grouping to override precedence in the usual way.

As you can see, the dot operator @samp{.} 
@sxindex .@r{, dot operator}
is usually a synonym for run-of-the-mill multiplication, @samp{*}.
The meanings differ only for vector operands.  The @code{then}
operator
@sxindex then
merely reverses the operand
order with respect to normal multiplication @samp{*}.  The intent
here is to make compositions read more naturally. The code
@example
(1,2,3) then scale(2) then rotate(30) then translate([1,3,0]) 
@end example
@noindent
expresses a series of successive modifications to the point,
whereas the equivalent form
@sxindex *@r{, multiplication operator}
@example
translate([1,3,0]) * rotate(30) * scale(2) * (1,2,3)
@end example
@noindent
will be intuitive only to mathematicians (and perhaps Arabic
language readers). 

@node Unary forms,  , Two-operand (binary) forms, Arithmetic
@comment  node-name,  next,  previous,  up
@subsubsection Unary forms
Unary or one-operand forms
@cindex unary form
@cindex one-operand form
are summarized in the following table, where @code{X}
stands for the operand.
@multitable{@code{inverse(X)}}{transform}{transform}{Long description meant to wrap eventually.}
@headitem Op         @tab Operand 
 @tab Result @tab Description
@item @code{-X}      @tab scalar
@sxindex -@r{, unary minus operator}
 @tab scalar @tab Unary scalar negation.
@item @code{-X}      @tab vector
 @tab vector @tab Unary vector negation.
@item @code{|X|}     @tab vector
 @sxindex |@math{X}|@r{, magnitude operator}
 @tab scalar @tab Vector length.
@item @code{unit(X)} @tab vector
 @sxindex unit
 @tab vector @tab Unit vector with same direction.
@item @code{sqrt(X)} @tab scalar
 @sxindex sqrt
 @tab scalar @tab Scalar square root.
@item @code{sin(X)} @tab scalar
 @sxindex sin
 @tab scalar @tab Trigonometric sine (@code{X} in degrees).
@item @code{cos(X)} @tab scalar
 @sxindex cos
 @tab scalar @tab Trigonometric cosine (@code{X} in degrees).
@item @code{asin(X)} @tab scalar
 @sxindex asin
 @tab scalar @tab Inverse sine (@code{X} in degrees).
@item @code{acos(X)} @tab scalar
 @sxindex acos
 @tab scalar @tab Inverse cosine (@code{X} in degrees).
@item @code{atan2(X,Y)} @tab scalar
 @sxindex atan2
 @tab scalar @tab Polar angle in degrees of vector @math{[X,Y]}.
@item @code{inverse(X)} @tab transform
 @sxindex inverse
 @tab transform @tab Inverse transform.
@end multitable
@sp 1
@noindent
Errors are reported when @code{|X|}, @code{unit}, @code{asin},
@code{acos}, @code{sqrt}, @code{atan2}, and @code{inverse} fail due to
operands outside their respective domains.

@node Options,  , Arithmetic, Language basics
@comment  node-name,  next,  previous,  up
@subsection Options
@cindex options
@strong{Syntax:}
@example
[@var{key1}=@var{val1},@var{key2}=@var{val2},@dots{}]
@end example
@noindent
Options are used to specify details of the appearance of drawables.
As shown above, they are given as comma-separated key-value
pairs. 

@menu
* PSTricks options::            Options inherited from @code{PSTricks}.
* TikZ/PGF options::            Options inherited from @code{TikZ/PGF}.
* Dots in TikZ/PGF::            Sketch uses @code{TikZ/PGF} circles for dots.
* TikZ/PGF user-defined styles::  Support for @code{TikZ/PGF} named, user-defined styles.
* Transparency::                See-through polygons.
* Internal options::            Options used by @code{sketch}.
@end menu

@node PSTricks options, TikZ/PGF options, Options, Options
@comment  node-name,  next,  previous,  up
@subsubsection @code{PSTricks} options
When @code{language pstricks} is selected (the default), permissible
key-value pairs include all those for similar @code{PSTricks} objects.
For example, a polygon might have the options
@verbatim
  [linewidth=1pt,linecolor=blue,fillcolor=cyan]
@end verbatim
@noindent
@code{Sketch} merely passes these on to @code{PSTricks} without
checking or modification.  Option lists are always optional.  A
missing options list is equivalent to an empty one @samp{[]}.

When a @code{polygon} has options for both its face and its edges, and
the polygon is split by the hidden surface algorithm, @code{sketch}
must copy the edge options to @code{psline}s for the edge segments and
the face options to @code{pspolygon}s.  Options known to @code{sketch}
for purposes of this splitting operation include @code{arrows},
@code{dash}, @code{dotsep}, @code{fillcolor}, @code{fillstyle},
@code{linecolor}, @code{linestyle}, @code{linewidth}, @code{opacity}, 
@code{showpoints}, @code{strokeopacity}, and @code{transpalpha}.

@node TikZ/PGF options, Dots in TikZ/PGF, PSTricks options, Options
@comment  node-name,  next,  previous,  up
@subsubsection @code{TikZ/PGF} options
@code{TikZ/PGF} options are handled much as for @code{PSTricks}.
Though @code{TikZ/PGF} often allows colors and styles to be given
without corresponding keys, for example,
@verbatim
  \draw[red,ultra thick](0,0)--(1,1);
@end verbatim
@noindent
this is not permitted in @code{sketch}.  To draw a red, ultra-thick
line in @code{sketch}, the form is
@verbatim
  line[draw=red,style=ultra thick](0,0)(1,1)
@end verbatim

Just as for @code{PSTricks}, when a @code{polygon} has options for
both its face and its edges, and the polygon is split by the hidden
surface algorithm, @code{sketch} must copy the edge options to
@code{psline}s for the edge segments and the face options to
@code{pspolygon}s.  @code{TikZ/PGF} options known to @code{sketch} for
purposes of this splitting operation include @code{arrows},
@code{cap}, @code{color}, @code{dash pattern}, @code{dash phase},
@code{double},
@code{double distance}, @code{draw}, @code{draw opacity}, @code{fill},
@code{fill opacity}, @code{join}, @code{line width}, @code{miter
limit}, @code{pattern}, @code{pattern color}, and @code{style}.

The @code{style} option can contain both face and edge information, so
@code{sketch} must check the style value.  Values known to
@code{sketch} include @code{dashed}, @code{densely dashed},
@code{densely dotted}, @code{dotted}, @code{double}, @code{loosely
dashed}, @code{loosely dotted}, @code{nearly opaque}, @code{nearly
transparent}, @code{semithick}, @code{semitransparent}, @code{solid},
@code{thick}, @code{thin}, @code{transparent}, 
@code{ultra nearly transparent}, @code{ultra thick}, @code{ultra thin}, 
@code{very nearly transparent}, @code{very thick}, and @code{very thin}.

@node  Dots in TikZ/PGF, TikZ/PGF user-defined styles, TikZ/PGF options, Options
@comment  node-name,  next,  previous,  up
@subsubsection Dots in @code{TikZ/PGF}
@code{TikZ/PGF} does not have a @code{dots} command as does PSTricks.
Instead, @code{Sketch} emits dots as @code{filldraw} circles.  The
diameter may be set using the option @code{dotsize} borrowed from
PSTricks.  The @code{dotsize} option will be removed from the option
list in the output @code{filldraw} command.  Other options work in the
expected way.  For example, @code{fill} sets fill color and
@code{color} sets line color of the circles.

@node  TikZ/PGF user-defined styles, Transparency, Dots in TikZ/PGF, Options
@comment  node-name,  next,  previous,  up
@subsubsection @code{TikZ/PGF} user-defined styles
@code{TikZ/PGF} allows named styles defined by the user, for
example
@verbatim
  \tikzstyle{mypolygonstyle} = [fill=blue!20,fill opacity=0.8]  
  \tikzstyle{mylinestyle} = [red!20,dashed]
@end verbatim
@noindent
Since @code{sketch} has no information on the contents of such styles,
it omits them entirely from lines, polygons, and their edges during
option splitting.  For example,
@verbatim
  polygon[style=mypolygonstyle,style=thick](0,0,1)(1,0,0)(0,1,0)
  line[style=mylinestyle](-1,-1,-1)(2,2,2)
@end verbatim
@noindent
produces the @code{TikZ} output
@verbatim
  \draw(-1,-1)--(.333,.333);
  \filldraw[thick,fill=white](0,0)--(1,0)--(0,1)--cycle;
  \draw(.333,.333)--(2,2);
@end verbatim
@noindent
Note that the user-defined styles are not present.  Sketch also issues
warnings:
@verbatim
  warning, unknown polygon option style=mypolygonstyle will be ignored
  warning, unknown line option style=mylinestyle will be ignored
@end verbatim

The remedy is to state explicitly whether a user-defined style should
be attched to polygons or lines in the @code{TikZ} output using
@emph{pseudo-options} @code{fill style} and @code{line style},
@cindex pseudo-options
@sxindex fill style
@sxindex line style
@verbatim
  polygon[fill style=mypolygonstyle,style=thick](0,0,1)(1,0,0)(0,1,0)
  line[line style=mylinestyle](-1,-1,-1)(2,2,2)
@end verbatim
@noindent
Now, the output is
@verbatim
  \draw[mylinestyle](-1,-1)--(.333,.333);
  \filldraw[mypolygonstyle,thick](0,0)--(1,0)--(0,1)--cycle;
  \draw[mylinestyle](.333,.333)--(2,2);
@end verbatim

A useful technique is to include user-defined style definitions in
@code{sketch} code as @code{special}s with option @code{[lay=under]}
to ensure that the styles are emitted first in the output, before
any uses of the style names.
@opindex lay
@footnote{This clever trick is due to Kjell Magne Fauske.}  For
example,
@verbatim
  special|\tikzstyle{mypolygonstyle} = [fill=blue!20,fill opacity=0.8]|[lay=under]
  special|\tikzstyle{mylinestyle} = [red!20,dashed]|[lay=under]
@end verbatim
@noindent
The author is responsible for using the key, @code{line style}
or @code{fill style}, that matches the content of the style
definition.

@node  Transparency, Internal options, TikZ/PGF user-defined styles, Options
@comment  node-name,  next,  previous,  up
@subsubsection Transparency
@cindex transparency
Both @code{PSTricks} and @code{TikZ/PGF} support polygon options that
have the effect of making the polygon appear transparent.  For
@code{PSTricks}, keyword @code{transpalpha} was used during initial
development of transparency features, and @code{opacity} was adopted 
later. @code{Sketch} honors both. @code{TikZ/PGF} uses @code{opacity} only.
@opindex transpalpha
@opindex opacity
@opindex fill opacity
When transparent polygons are in the foreground, objects behind them
(drawn earlier) are visible with color subdued and tinted.  The hidden
surface algorithm of @code{sketch} works well with such transparent
polygons.

Note that @verb{|cull=false|}
@opindex cull
must be used for rear-facing polygons to be visible when positioned
behind other transparent surfaces.

@node Internal options,  , Transparency, Options
@comment  node-name,  next,  previous,  up
@subsubsection Internal options
There are also @emph{internal} options
@cindex options, internal
@cindex internal options
used only by @code{sketch} and not
passed on to @code{PSTricks}.  These are summarized in the following
table.
@sxindex cull
@sxindex lay
@sxindex split
@multitable {@code{split}}{@code{over}, @code{in}, @code{under}}{A column that ought to wrap.  Will it wrap?  I do not know.}
@headitem Key @tab Possible values @tab Description
@item @code{cull}
@tab  @code{true}, @code{false}
@tab  Turn culling of backfaces on and off respectively for this object.
The default value is @code{true}.
@item @code{lay}
@tab  @code{over}, @code{in}, @code{under}
@tab  Force this object to be @code{under} or
@code{over} all other objects in the depth sort
@cindex depth sort
order created by the hidden surface algorithm.  The default value
@code{over} guarantees that output due to the @code{special} will be
visible.
@item @code{split}
@tab  @code{true}, @code{false}
@tab  Turn splitting of sweep-generated body polygons 
@cindex body polygon
@cindex polygon, body
on and off respectively. @xref{Sweeps}.  The default value @code{true}
causes ``warped'' polygons to be split into triangles, which avoids
mistakes by the hidden surface algorithm.
@end multitable

@subsection Point lists
@sxindex point list
@strong{Syntax:}
@example
(@var{x1},@var{y1},@var{z1})(@var{x2},@var{y2},@var{z2})@dots{}
@end example
@noindent
A sequence of one or more points makes a point list, a feature
common to all drawables.  Each of the point components is a scalar
arithmetic expression.  Any point may have the @math{z}-component
omitted; it will default to @math{z=0}.

@menu
* Drawables::                   Things that are drawn.
* Definitions::                 Things with names.
@end menu

@node Drawables, Definitions, Language basics, Input language
@comment  node-name,  next,  previous,  up
@section Drawables
@cindex drawable
Drawables are simply @code{sketch} objects that might appear in the
drawing.  They include dots, polylines, curves, polygons, and more
complex objects that are built up from simpler ones in various ways.
Finally, @dfn{special} objects are those composed of @LaTeX{} or
@code{PSTricks} code, perhaps including coordinates and angles
computed by @code{sketch}.

@menu
* Dots::                        Draw dots.
* Lines::                       Draw polylines.
* Curves::                      Draw curves.
* Polygons::                    Draw polygons.
* Specials::                    Embed raw @LaTeX{} and @code{PSTricks}.
* Sweeps::                      Draw sweeps of dots and polylines.
* Blocks::                      Group other drawables.
* Repeats::                     Draw transformed copies of objects.
* Puts::                        Draw one object transformed.
@end menu

@node Dots, Lines, Drawables, Drawables
@comment  node-name,  next,  previous,  up
@subsection Dots
@sxindex dots
@strong{Syntax:}
@example
dots[@var{options}] @var{point_list}
@end example
@noindent
This command is the three-dimensional equivalent of the
@code{PSTricks} command @code{\psdots}.

@node Lines, Curves, Dots, Drawables
@comment  node-name,  next,  previous,  up
@subsection Lines
@sxindex line
@strong{Syntax:}
@example
line[@var{options}] @var{point_list}
@end example
@noindent
This command is the three-dimensional equivalent of the 
@code{PSTricks} command @code{\psline}.

@node Curves, Polygons, Lines, Drawables
@comment  node-name,  next,  previous,  up
@subsection Curves
@sxindex curve
@strong{Syntax:}
@example
curve[@var{options}] @var{point_list}
@end example
@noindent
This command is the three-dimensional equivalent of the 
@code{PSTricks} command @code{\pscurve}. @b{It is not
implemented in the current version of @code{sketch}}.

@node Polygons, Specials, Curves, Drawables
@comment  node-name,  next,  previous,  up
@subsection Polygons
@sxindex polygon
@strong{Syntax:}
@example
polygon[@var{options}] @var{point_list}
@end example
@noindent
@noindent
This command is the three-dimensional equivalent of the
@code{PSTricks} command @code{\pspolygon}.  The @code{sketch} hidden
surface algorithm assumes that polygons are convex and planar.  
@cindex polygon, planar
@cindex planarity of polygons
In practice, drawings may well turn out correctly even if these
assumptions are violated.

@node Specials, Sweeps, Polygons, Drawables
@comment  node-name,  next,  previous,  up
@subsection Specials
@sxindex special
@strong{Syntax:}
@example
special $@var{raw_text}$[@var{options}] @var{arg_list}
@end example
@noindent
Here @code{$} 
@cindex quoting, special
can be any character and is used to delimit the start
and end of @var{raw_text}.  The command embeds @var{raw_text} in the
@code{sketch} output after performing substitutions as follows.
@cindex special argument substitution
@cindex argument, special
@itemize
@item 
@code{#@var{i}} where @var{i} is a positive integer is replaced by the
@var{i}'th value in @var{arg_list}.  Point and vector arguments become
two-dimensional points, which are the transformed 3d arguments
projected onto the @var{x}-@var{y} plane. This allows
two-dimentional output elements such as labels to be easily positioned
with respect to three-dimensional features in the drawing. Scalar 
arguments are subsituted directly. No transformation is applied.
@item
@code{#@{@var{i}@}} is also replaced as above.
@item
@code{#@var{i}'2} is replaced as above for points or vectors.  It is
an error for the @var{i}'th argument to be a scalar.
@item
@code{#@var{i}'x}, @code{#@var{i}'y}, or @code{#@var{i}'z} is
replaced, respectively, by the scalar @var{x}, @var{y}, or
@var{z}-coordinate of the argument point or vector. It is an error for
the @var{i}'th argument to be a scalar.
@item
@code{#@var{i}'3} is replaced by the three-dimensional
transformed argument. Note that if a perspective transformation has
been applied, the @var{z}-coordinate has little geometric
significance, though it accurately indicates relative depth.
@item
@code{#@var{i}-@var{j}} where @var{i} and @var{j} are positive
integers is replaced by a string @code{@{@var{angle}@}} where
@var{angle} is the polar angle of a vector from the @var{i}'th point
in @var{point_list} to the @var{j}'th point projected into the
@math{x}-@math{y} plane.  It is an error for the @var{i}'th or
@var{j}'th argument to be a scalar or a vector.
@item
@code{#@{@var{i}-@var{j}@}} is also replaced as above.
@item 
@code{##} is replaced with @code{#}.
@end itemize
@noindent
The forms with braces @samp{@{ @}} are useful when the argument is
immediately followed by a digit or a tick character that is part of
the @TeX{} code.

The only useful option of @code{special} is @code{lay}, which
determines if the substitued raw text is emitted before, after,
or using the first point in @var{point_list} as an indicator
of depth.  These occur, respectively, with @code{lay=under},
@code{lay=over}, and @code{lay=in}.  See @ref{Special objects}
and @ref{TikZ/PGF user-defined styles} for examples.
@sxindex lay
@xref{Internal options}.

@node Sweeps, Blocks, Specials, Drawables
@comment  node-name,  next,  previous,  up
@subsection Sweeps
@sxindex sweep
@strong{Syntax:}
@example
sweep @{ @var{n}, @var{T_1}, @var{T_2}, @dots{}, @var{T_@math{r}} @}[@var{options}] @var{swept_object}
sweep @{ @var{n}<>, @var{T_1}, @var{T_2}, @dots{}, @var{T_@math{r}} @}[@var{options}] @var{swept_object}
@end example
@noindent
The sweep connects @var{n} (or perhaps @math{@var{n}+1}) copies of
@var{swept_object} 
@cindex swept object
in order to create a new object of higher
dimension.  The @var{T_@math{i}} (for @math{i} between @math{1} and
@math{r}) are transforms.  
@cindex transform
The @math{k}'th copy of @var{swept_object} is produced by applying the
following transform to the original.
@example
@math{@var{T_1}^k} then @math{@var{T_2}^k} then @dots{} then @math{@var{T_r}^k}
@end example
@noindent
Here @math{@var{T}^k} means ``transform @var{T} applied @math{k}
times.''  The original object is the zero'th copy, with @math{k=0} and
effectively no transform applied (@math{T^0=I}, the identity
transform).

The method of connecting the copies depends on the type of
@var{swept_object} and on whether the closure tag
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
@samp{<>} is present
or not.

An example of a sweep where @math{r=2} is the Mobius figure at
@ref{More to learn}.

@menu
* Swept points::                Swept points make lines or polygons.
* Swept lines::                 Swept lines make open or closed surfaces.
* Swept polygons::              Swept polygons make closed surfaces.
* Swept blocks::                Swept block @equiv{} block of sweeps.
* Sweep face splitting::        Fixing warped faces with triangles.
@end menu

@node Swept points, Swept lines, Sweeps, Sweeps
@comment  node-name,  next,  previous,  up
@subsubsection Swept points
@cindex swept point
@cindex point sweep
If @var{swept_object} is a point list and there is no closure tag,
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
then @code{sweep} connects @math{@var{n}+1} successive copies of each
point (including the original) with straight line segments to form a
polyline.  If there are @math{m} points in the original point list,
@cindex point list
then @math{m} polylines with @var{n} segments each are formed by the
sweep.  In this manner, @code{sweep} forms a set of one-dimensional
objects (polylines) from zero-dimensional ones (points).

When there @emph{is} a closure tag, 
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
@code{sweep} connects @var{n}
successive copies of each point (including the original) with straight
line segments and finally connects the last copy back to the original
to form a polygon with @var{n} sides.  If there are @math{m} points in
the original point list, then @math{m} polygons with @var{n} sides
each are formed by the sweep.  In this manner, @code{sweep} forms a
set of two-dimensional objects (polygons) from zero-dimensional ones
(points).

Options 
@cindex options, sweep
of the @code{sweep} are copied directly to the resulting
polyline(s).

@node Swept lines, Swept polygons, Swept points, Sweeps
@comment  node-name,  next,  previous,  up
@subsubsection Swept lines
@cindex swept line
@cindex line sweep
If @var{swept_object} is a polyline and there is no closure tag, 
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
then
@code{sweep} connects @math{@var{n}+1} successive copies of the
polyline (including the original) with four-sided polygons, each pair
of copies giving rise to a ``polygon strip.''  If there are @math{m}
points in the original polyline, then @math{(m-1)@var{n}} polygons are
formed by the sweep.  We call these @dfn{body polygons}.
@cindex body polygon
@cindex polygon, body
In this manner, @code{sweep} forms a
two-dimensional surface from from a one-dimensional polyline.

The order of vertices 
@cindex polygon vertex order
@cindex order, polygon vertex
produced by @code{sweep} is important.  If a
polygon's vertices do not appear in counter-clockwise order in the
final image, the polygon will be culled 
@cindex culling
(unless @code{cull=false} is
set).  If the points in the @math{k}'th copy of the polyline are
@math{P_1}, @math{P_2}, @dots{}, @math{P_m}, and the points in the
next copy, the @math{(k+1)}st, are @math{P_1'}, @math{P_2'}, @dots{},
@math{P_m'}, then the vertex order of the generated polygons is
@display
Body polygon 1: @math{P_2} @math{P_1} @math{P_1'} @math{P_2'} 
Body polygon 2: @math{P_3} @math{P_2} @math{P_2'} @math{P_3'} 
@dots{}
Body polygon @math{m-1}: @math{P_m} @math{P_{m-1}} @math{P_{m-1}'} @math{P_m'} 
@end display

Options of unclosed line sweeps 
@cindex options, sweep
are copied to each output polygon.
Options of the swept line are ignored.
@cindex options, swept object

When there @emph{is} a closure tag, 
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
then @code{sweep} connects @var{n}
successive copies of the polyline (including the original) with
four-sided body polygons just as the case with no closure tag.  It then
connects the last copy back to the original to form a ribbon-shaped
surface that closes on itself with two holes remaining.

Finally, the sweep adds two more polygons to seal the holes and form a
closed surface that, depending on the sweep transforms, may
represent the boundary of a solid.  In this manner, @code{sweep} forms
the boundary of a three-dimensional object from a one-dimensional
polyline.  We call these hole-filling polygons @dfn{ends}.
@cindex end polygon
@cindex polygon, end

The order of vertices of end polygons 
@cindex polygon vertex order
@cindex order, polygon vertex
is important for correct culling
as described above.  If @math{P_1^1}, @math{P_1^2}, @dots{},
@math{P_1^n} are the @var{n} copies of the first polyline point and
@math{P_m^1}, @math{P_m^2}, @dots{} ,@math{P_m^n} are the @var{n}
copies of the last polyline point, then the end polygon vertex order
is
@display
End polygon 1: @math{P_1^n}, @math{P_1^{n-1}}, @dots{} ,@math{P_1^1}
End polygon 2: @math{P_m^1}, @math{P_m^2}, @dots{} ,@math{P_m^n}
@end display

If there are no options on the swept line, 
@cindex options, swept object
then the @samp{sweep}
options 
@cindex options, sweep
are copied to each output polygon.  If the swept line does
have options, these are copied to corresponding body polygons; the
sweep options are copied to the end polygons.  In this manner, body
and ends may be drawn with different characteristics such as
@code{fillcolor}.

@node Swept polygons, Swept blocks, Swept lines, Sweeps
@comment  node-name,  next,  previous,  up
@subsubsection Swept polygons
@cindex polygon sweep
If @var{swept_object} is a polygon, the @code{sweep} connects
@math{@var{n}+1} successive copies of the closed polyline border of
the polygon to form body polygons exactly as though the border were a
swept polyline as described in @ref{Swept lines}.
@cindex body polygon
@cindex polygon, body
If there are @math{m} points in the
original polygon, then @math{m@var{n}} body polygons are formed by
this sweep.  The body polygons form an @dfn{extrusion} of the boundary of the
original polygon with two holes at the open ends.  

Finally, the sweep adds two copies of the original polygon to cover
the holes.  We call these hole-filling polygons @dfn{ends}.
@cindex end polygon
@cindex polygon, end
In this manner, @code{sweep} forms the boundary of a three-dimensional
object from a two-dimensional polygon.

The order of vertices 
@cindex polygon vertex order
@cindex order, polygon vertex
of end polygons is important for correct culling as described above.
An exact copy of the original polygon with vertex order intact forms
the first end polygon.  The other end polygon results from
transforming and the reversing the order of vertices in the original.
The transform places the original polygon at the uncovered hole;
it is
@example
@math{@var{T_1}^n} then @math{@var{T_2}^n} then @dots{} then @math{@var{T_r}^n}.
@end example
@noindent
If there are no options on the swept polygon, then the @samp{sweep}
options are copied to each output polygon.  If the swept polygon does
have options, these are copied to the ends; the sweep options are
copied to the body polygons.  In this manner, body and ends may be
drawn with different characteristics such as @code{fillcolor}.

@node Swept blocks, Sweep face splitting, Swept polygons, Sweeps
@comment  node-name,  next,  previous,  up
@subsubsection Swept blocks
@cindex swept bock
@cindex block sweep
The swept object @var{swept_object} may also be any collection of
polylines and polygons.  This may be a block 
@cindex block
@sxindex @{ @}@r{, block drawable}
composed of @code{line} 
@sxindex line
and/or @code{polygon} 
@sxindex polygon
commands in braces
@samp{@{ @}}, or it may be the result of a @code{repeat}, another
@code{sweep}, etc.  The sweep acts independently on each object in the
block exactly as if it were a single swept object described above in
@ref{Swept lines} and @ref{Swept polygons}.

@node Sweep face splitting,  , Swept blocks, Sweeps
@comment  node-name,  next,  previous,  up
@subsubsection Sweep face splitting
Before sending each four-sided body polygon of a @code{sweep}
to the output, @code{sketch} tests to see if it is roughly planar.
@cindex polygon, planar
@cindex planarity of polygons
Since planarity is necessary for proper functioning of the hidden
surface algorithm, ``warped'' polygons are automatically split into
two triangles.  

Hole-filling polygons produced by closure-tagged 
@sxindex <>@r{, closure tag}
@cindex closure tag, @code{<>}
line sweeps are not
split.  Nor are original polygons in polygon sweeps.  It is the user's
responsibility to ensure these are planar.

@node Blocks, Repeats, Sweeps, Drawables
@comment  node-name,  next,  previous,  up
@subsection Blocks
@cindex block
@sxindex @{ @}@r{, block drawable}
Any sequence of drawables may be grouped in a @dfn{block} merely by
enclosing them in braces @samp{@{ @}}.  A block is itself drawable.  A
key use of blocks is to extend the effect of a single @code{def},
@ref{Definitions}, @code{put} @ref{Puts}, @code{sweep} @ref{Sweeps},
or @code{repeat} @ref{Repeats} to include several objects rather than
one.

Definitions (@xref{Definitions}.) inside a block have @dfn{lexical
scope} 
@cindex lexical scope
@cindex scope, identifier
extending from the place of definition to the end of the block.

@node Repeats, Puts, Blocks, Drawables
@comment  node-name,  next,  previous,  up
@subsection Repeats
@sxindex repeat
@cindex repeated object
@strong{Syntax:}
@example
repeat @{ @var{n}, @var{T_1}, @var{T_2}, @dots{}, @var{T_r} @} @var{repeated_object}
@end example
@noindent
The repeat makes @var{n} transformed copies of @var{repeated_object}
(including the original).  The @var{T_@math{i}} are transforms.
@cindex transform
The @math{k}'th copy of the @var{repeated_object} (for
@math{k=0,1,...,n-1}) is produced in the
same manner as for @code{sweep}s described in @ref{Sweeps}.  This is
repeated here (no pun intended) for convenience.  To make the
@math{k}'th copy, the following transform is applied to the
original object.
@example
@math{@var{T_1}^k} then @math{@var{T_2}^k} then @dots{} then @math{@var{T_r}^k}
@end example
@noindent
Here @math{@var{T}^k} means ``transform @var{T} applied @math{k}
times.'' 

@node Puts,  , Repeats, Drawables
@comment  node-name,  next,  previous,  up
@subsection Puts
@sxindex put
@strong{Syntax:}
@example
put @{ @var{T} @} @var{put_object}
@end example
@noindent
Put merely applies transform @var{T} to the drawable @var{put_object}.

@node Definitions, Global environment, Drawables, Input language
@comment  node-name,  next,  previous,  up
@section Definitions
@cindex definition
Definitions give names to @code{sketch} objects.  Definitions alone
are benign.  A @code{sketch} input file consisting entirely of
definitions will generate no drawing.  Only when definitions are
@dfn{referenced} do they potentially lead to ink on the drawing.

The intent of definitions is to make @code{sketch} code more concise
and readable.  There is no input file employing definitions
that could not be re-written without them.

Definable objects include any result of an affine arithmetic
expression (scalar, point, vector, or transform), any drawable
object (dots, line, curve, polygon, block, sweep, put, repeat, or
special), and option strings.  In addition, @dfn{tag definitions},
@cindex definition, tag
@cindex tag definition
which have no associated object at all, allow the meaning of other
definitions to be selected from a set of alternatives. Since tags may
be defined (and undefined) in the command line of @code{sketch}, they
can be an aid in the script-driven preparation of documents.

@menu
* Forms of definitions::        Different defs for different purposes.
* Forms of references::         How references denote types.
@end menu

@node Forms of definitions, Forms of references, Definitions, Definitions
@comment  node-name,  next,  previous,  up
@subsection Forms of definitions
Definitions have three possible forms, @dfn{simple}, 
@cindex simple definition
@cindex definition, simple
@dfn{with alternatives}, 
@cindex definition with alternatives
@cindex alternatives, definition
and @dfn{tag} 
@cindex tag definition
@cindex definition, tag
as shown here in order.

@noindent
@strong{Syntax:}
@example
def @var{id} @var{object}  % simple def
def @var{id} <@var{tag_1}> @var{object_1}  % def with alternatives
       <@var{tag_2}> @var{object_2}
       @dots{}
       <> @var{default_object}
def @var{id} <>  % tag def
@end example
@noindent
The simple definition merely associates @var{object} with the
identifier @var{id}.  

The definition with alternatives associates
@var{object_i} with @var{id}, where @var{tag_i} is the
first defined tag in the list of alternative tag references.
@cindex tag reference
@cindex reference, tag
@sxindex <foo>@r{, tag reference}
If no tag in the list is defined, then @var{default_object} is
associated with identifier @var{id}.

The final form defines @var{id} as a tag.  Another way to define a tag
is with the @option{-D} command line option. @xref{Command line}.

@node Forms of references,  , Forms of definitions, Definitions
@subsection Forms of references
References to defined names are enclosed in bracketing delimiters.
The delimiter characters imply the type of the associated value as
shown in the table below.  A type error is raised if the type of a
reference does not match the type of the defined value.  The intent of
this mechanism is, again, to make @code{sketch} input files more
readable.
@multitable {transform}{@code{[@var{id}]} or @code{[@var{id1}, ..., @var{idN}]}}
@headitem Type  @tab Reference
@item scalar    @tab @code{@var{id}}
@cindex scalar reference
@cindex reference, scalar
@item point     @tab @code{(@var{id})}
@cindex point reference
@cindex reference, point
@sxindex (foo)@r{, point reference}
@item vector    @tab @code{[@var{id}]}
@cindex vector reference
@cindex reference, vector
@sxindex [foo]@r{, vector reference}
@item transform @tab @code{[[@var{id}]]}
@cindex transform reference
@cindex reference, transform
@sxindex [[foo]]@r{, transform reference}
@item drawable  @tab @code{@{@var{id}@}}
@cindex drawable reference
@cindex reference, drawable
@sxindex @{foo@}@r{, drawable reference}
@item options   @tab @code{[@var{id}]} or @code{[@var{id1},...,@var{idN}]}
@cindex options reference
@cindex reference, options
@sxindex [foo]@r{, options reference}
@cindex options multiple reference
@cindex reference, multiple options
@sxindex [foo,...,bar]@r{, multiple options reference}
@item tag       @tab @code{<@var{id}>}
@cindex tag reference
@cindex reference, tag
@sxindex <foo>@r{, tag reference}
@end multitable
@sp 1
@noindent
Note that square brackets @samp{[ ]} are used both for vector and for
options references.  Details of @code{sketch} syntax make it
impossible for these two reference types to be confused.  The
special multiple reference @code{[@var{id1},@var{id2},...,@var{idN}]}
acts as if the respective lists of options were concatenated.

@node Global environment,  , Definitions, Input language
@comment  node-name,  next,  previous,  up
@section Global environment
An optional global environment block provides a few ways to affect the
entire scene.  The block must appear as the last text in the
@code{sketch} input file.  It may include definitions, but note
that previous definitions at the top level (not nested inside
blocks) are also available.

@noindent
@strong{Syntax:}
@sxindex global
@example
global @{ @var{environment_settings} @}
@end example
@noindent
The contents of @var{environment_settings} are discussed in the
sections that follow.

@menu
* Global options::              Attributes of the entire drawing.
* Camera::                      A final camera transformation of the scene.
* Picture box::                 Setting the bounding box and 2d clipping.
* Frame::                       Adding a box around the drawing.
* Language::                    Setting the output language.
@end menu

@node Global options, Camera, Global environment, Global environment
@comment  node-name,  next,  previous,  up
@subsection Global options
@cindex options, global
@cindex global options
@sxindex set
@strong{Syntax:}
@example
set [ @var{options} ]
@end example
@noindent
The contents of @var{options}, except for @code{sketch} internal
options, are copied as-is to a @verb{|\psset|} that appears before
anything else in the output file.  This is a good place to set
@code{unit}, a default @code{linewidth}, etc.

Internal options 
@cindex options, internal
@cindex internal options
work on all objects where they make sense.
This includes 
@sxindex cull
@sxindex split
@sxindex lay
@code{cull} and @code{split} (but not @code{lay}).
@xref{Internal options}. 

@node Camera, Picture box, Global options, Global environment
@comment  node-name,  next,  previous,  up
@subsection Camera
@cindex camera
@strong{Syntax:}
@sxindex camera
@example
camera @var{transform_expression}
@end example
The @var{transform_expression} is applied after all other
transformations of the scene.  This is currently only useful for
transforming the bounding box.  @xref{Picture box}.  It will play a
role in any future implementation of clipping.

@node Picture box, Frame, Camera, Global environment
@comment  node-name,  next,  previous,  up
@subsection Picture box
@cindex picture box
@strong{Syntax:}
@sxindex picturebox
@example
picturebox[@var{baseline}]
picturebox[@var{baseline}] (@var{p1})(@var{p2})
@end example
@noindent
The first form of @code{picturebox} causes a scalar @var{baseline}
fraction to be emitted in the @code{pspicture}
@cindex @code{pspicture}
@cindex baseline fraction
environment of the output.  See
@code{PSTricks} documentation for @code{pspicture}.

In the second form, the @var{baseline} fraction is optional, and the
two points that follow define the diagonal of a three-dimensional
bounding box
@cindex bounding box
for the completed scene.  The parallel projection 
@cindex parallel projection
@cindex projection, parallel
of the bounding box
determines the corners of the drawing's @code{pspicture*} environment,
which is used in place of @code{pspicture}.  This causes PostScript to
clip
@cindex clipping
the final drawing to the bounding box in 2d.  If there is a
@code{camera} specified, the camera tranformation is applied to the
bounding box, and the @code{pspicture} is set just large
enough to include the transformed box.

When no bounding box is given, @code{sketch} computes one
automatically.

@node Frame, Language, Picture box, Global environment
@comment  node-name,  next,  previous,  up
@subsection Frame
@cindex frame box
@strong{Syntax:}
@sxindex frame
@example
frame [@var{options}]
@end example
@noindent
Causes a @verb{|\psframebox|} 
@cindex @code{psframebox}
to surround the @code{pspicture}
environment in the output.  If @var{options} are present, they are
copied as-is.  Normally one would want to set 
@opindex linewidth
@code{linewidth},
@opindex linestyle
@code{linestyle}, 
@opindex linecolor
@code{linecolor}, etc. 
If omitted, then 
@opindex framesep
@code{framesep=0pt} is
added so that the frame tightly hugs the @code{pspicture}.

@node Language,  , Frame, Global environment
@comment  node-name,  next,  previous,  up
@subsection Language
@cindex language, output
@cindex output language
@sxindex language
@example
language tikz
language tikz, context
language pstricks
language pstricks, latex
@end example
@noindent
Sets the output language generated by @code{sketch}.  
@sxindex pstricks
@sxindex tikz
The set of options understood by sketch also changes.  For example,
the @code{PSTricks} option @code{linewidth} will not be properly
handled if @code{language} is set to @code{tikz}.  Similarly, the
@code{TikZ} option @code{line style} (note the space) will not be
properly handled if @code{language} is set to @code{pstricks}.  If no
language is specified, the default @code{pstricks} is used.

An optional comma followed by 
@code{latex} 
@sxindex latex
or 
@code{context} 
@sxindex context
specifies the macro package that the output should assume.  This
affects the @code{picture} environment commands emitted and the
document template used with the @option{-T} option. @xref{Command
line}.  Note that at the time this manual was generated,
@code{PSTricks} was not supported by @LaTeX{} or by @code{ConTeXt}.

@node Building a drawing, Command line, Input language, Top
@comment  node-name,  next,  previous,  up
@chapter Building a drawing
Successful drawings with @code{sketch} and with any scene description
language
@cindex scene description language
require that the user develop an accurate mental picture of her code
and its meaning.  This image is best built in small pieces.
Therefore, @code{sketch} inputs are best created in small increments
with frequent pauses to compile and view the results.  Careful
comments in the input often help as a scene grows in complexity.

@menu
* Overview::                    Building a substantial drawing.
* A technical drawing::         An example with fine placement.
* A hierarchical model::        An example with sweeps and puts.
* Caveats::                     Where trouble can occur.
@end menu

@node Overview, A technical drawing, Building a drawing, Building a drawing
@comment  node-name,  next,  previous,  up
@section Overview
As an overview, let's develop a diagram that shows how a perspective
projection transform
@cindex perspective projection
@cindex transform
@cindex projection, perspective
works.  We'll start with the traditional reference object
used in computer graphics textbooks, a house-shaped prism.  Begin
by defining the points of the house.  Rather than defining the faces
of the house as polygons and transforming those, we are going to
transform the points themselves with @code{sketch} arithmetic so that
we have names for the transformed points later.
@sxindex def
@verbatim
  % right side (outside to right)
  def R1 (1,1,1) def R2 (1,-1,1) def R3 (1,-1,-1) def R4 (1,1,-1)
  def R5 (1,1.5,0)

  % left side (outside to right--backward)
  def W  [2,0,0]
  def L1 (R1)-[W] def L2 (R2)-[W] def L3 (R3)-[W] def L4 (R4)-[W]
  def L5 (R5)-[W]
@end verbatim
@noindent
To add a door to the house, we use a polygon slightly in
front of the foremost face of the house.
@verbatim
  % door
  def e .01
  def D1 (0,-1,1+e) def D2 (.5,-1,1+e) def D3 (.5,0,1+e) def D4 (0,0,1+e)
@end verbatim
@noindent
Now let's create a new set of points that are a to-be-determined
transform of the originals.
@verbatim
  def hp scale(1) % house positioner
  def pR1 [[hp]]*(R1) def pR2 [[hp]]*(R2) def pR3 [[hp]]*(R3)
  def pR4 [[hp]]*(R4) def pR5 [[hp]]*(R5)
  def pL1 [[hp]]*(L1) def pL2 [[hp]]*(L2) def pL3 [[hp]]*(L3)
  def pL4 [[hp]]*(L4) def pL5 [[hp]]*(L5)
  def pD1 [[hp]]*(D1) def pD2 [[hp]]*(D2) def pD3 [[hp]]*(D3)
  def pD4 [[hp]]*(D4) 
@end verbatim
@noindent
Note the use of a @dfn{transform definition} 
@cindex transform definition
@cindex definition, transform
@sxindex [[foo]]@r{, transform reference}
and
@dfn{transform references}.
@cindex transform reference
@cindex reference, transform
@sxindex [[foo]]@r{, transform reference}
Now define the seven polygonal faces of the house and the door using
the transformed points as vertices.  Be careful with vertex order!
@cindex polygon vertex order
@cindex order, polygon vertex
@opindex fillcolor
@sxindex def
@sxindex polygon
@sxindex @{ @}@r{, block drawable}
@verbatim
  def rgt polygon (pR1)(pR2)(pR3)(pR4)(pR5)
  def lft polygon (pL5)(pL4)(pL3)(pL2)(pL1)
  def frt polygon (pR2)(pR1)(pL1)(pL2)
  def bck polygon (pR4)(pR3)(pL3)(pL4)
  def tfr polygon (pR1)(pR5)(pL5)(pL1)
  def tbk polygon (pR5)(pR4)(pL4)(pL5)
  def bot polygon (pR2)(pL2)(pL3)(pR3)
  def door polygon[fillcolor=brown] (pD1)(pD2)(pD3)(pD4)
  def house { {rgt}{lft}{frt}{bck}{tfr}{tbk}{bot}{door} }
@end verbatim
Time for a sanity check.  Add the line
@sxindex @{foo@}@r{, drawable reference}
@cindex reference, drawable
@verbatim
  {house}
@end verbatim
@noindent
and this is what we get.

@center @image{ex130}

@noindent
This is correct, but does not reveal very much.  Common errors are
misplaced vertices and polygons missing entirely due to incorrect
vertex order.  
@cindex polygon vertex order
@cindex order, polygon vertex
To rule these out, let's inspect all sides of the
house. This is not hard. Merely replace the reference
@verb{|{house}|} with a @code{repeat}. @xref{Repeats}.
@sxindex @{foo@}@r{, drawable reference}
@cindex reference, drawable
@sxindex repeat
@sxindex rotate
@sxindex translate
@verbatim
  repeat { 13, rotate(30, [1,2,3]), translate([3,0,0]) } {house}
@end verbatim
@center @image{ex140}

@noindent
Again things look correct.  Note that the hidden surface algorithm
handles intersecting polygons correctly where some copies of the house
overlap. 

Let's lay out the geometry of perspective projection of the house onto
a plane with rays passing through the origin.  Begin by positioning the
house twelve units back on the negative @math{z}-axis and adding a set
of coordinate axes.  To move the house we need only change the ``house
positioning'' transform defined earlier.
@sxindex def
@sxindex rotate
@sxindex translate
@opindex arrows
@opindex linewidth
@opindex linecolor
@opindex linestyle
@sxindex special
@sxindex line
@verbatim
  def hp rotate(-40, [0,1,0]) then translate([0,0,-12])
  def axes {
    def sz 1
    line [arrows=<->] (sz,0,0)(O)(0,sz,0)
    line [arrows=->]  (O)(0,0,sz)
    line [linewidth=.2pt,linecolor=blue,linestyle=dashed] (O)(0,0,-10)
    special |\uput[r]#1{$x$}\uput[u]#2{$y$}\uput[l]#3{$z$}|
      (sz,0,0)(0,sz,0)(0,0,sz)
  }
@end verbatim

Time for another test.  Let's build a real view transform,
creating a @dfn{virtual camera}
@cindex virtual camera
to look at the scene we are constructing.  Replace the @code{repeat}
with
@verbatim
  def eye (10,4,10)
  def look_at (0,0,-5)
  put { view((eye), (look_at)) } { {house}{axes} }
@end verbatim
The @dfn{view transform} repositions the scene so that the point
@code{eye} is at the origin and the direction from @code{eye} to
@code{look_at} is the negative @math{z}-axis.  This requires a
rotation and a translation that are all packed into the constructor
@code{view}.

@center @image{ex150}

@noindent
This is starting to look good!  Add the projection plane half way
between the origin and the house at @math{z=-5}.  We'll try
the angle argument feature of @code{special} to position a label.
@verbatim
  def p 5 % projection distance (plane at z=-p)
  def projection_plane {
    def sz 1.5
    polygon (-sz,-sz,-p)(sz,-sz,-p)(sz,sz,-p)(-sz,sz,-p)
    special |\rput[b]#1-2#3{\footnotesize\sf projection plane}| 
      (-sz,-sz,-p)(sz,-sz,-p)(0,-sz+.1,-p)
  }
@end verbatim
@noindent
Add @verb{|{projection_plane}|} to the list of objects in the
@code{put} above.

@center @image{ex160}

@indent
The way we constructed the points of the house now makes it easy to
draw rays of projection.  We'll cast one ray from every visible vertex
of the house and define options so the appearance of
all rays can be changed at the same time.
@verbatim
  def projection_rays {
    def rayopt [linewidth=.3pt,linecolor=lightgray]
    line [rayopt](O)(pR1) line [rayopt](O)(pR2) line[rayopt](O)(pR3)
    line [rayopt](O)(pR4) line [rayopt](O)(pR5)
    line [rayopt](O)(pL1) line [rayopt](O)(pL2) line[rayopt](O)(pL5)
    line [rayopt](O)(pD1) line [rayopt](O)(pD2) 
    line [rayopt](O)(pD3) line [rayopt](O)(pD4) 
  }
@end verbatim
@noindent
The result is shown here.  

@center @image{ex170}

@noindent
The rays pierce the projection plane at the corresponding points on
the perspective image we are trying to draw.  Albrecht D@"urer and his
Renaissance contemporaries had the same idea in the early 1500's.

@center @image{duerer,,1.5in}

All that's left is to find a way to connect the points of the house
on the projection plane.  We could pull out a good computer graphics
text, find the necessary matrix, and enter it ourselves as a
transform literal.  @xref{Transform literals}. That work is
already done, however.  We can use the @code{project(p)} constructor.

There are still some details that require care.  Projection will
flatten whatever is transformed onto the plane @math{z=-p}.  Therefore
any part of the house could disappear behind the projection plane (the
hidden surface algorithm orders objects at the same depth
arbitrarily).  The door may also disappear behind the front of the
house.  To make sure everything remains visible, we'll place the house
a tiny bit in front of the projection plane and a second copy of the
door in front of the house.
@verbatim
  def projection {
    % e is a small number defined above
    put { project(p) then translate([0,0,1*e]) } {house}
    put { project(p) then translate([0,0,2*e]) } {door}
  }
@end verbatim

@center @image{ex180}

If you have studied and understand all this, you are well on the way
to success with @code{sketch}.  Not shown are the 20 or so iterations
that were required to find a reasonable viewing angle and house
position, etc.  Nonetheless, this drawing was completed in about an
hour.  While a GUI tool may have been a little faster, it is unlikely
that a new drawing, itself a perspective projection of the scene,
could be generated with two more minutes' work!  Just change the view
transform to
@verbatim
  put { view((eye), (look_at)) then perspective(9) } { ...
@end verbatim
@noindent 
and produce this.

@center @image{ex190}

@node  A technical drawing, A hierarchical model, Overview, Building a drawing
@comment  node-name,  next,  previous,  up
@section A technical drawing
Let's look at a drawing that represents the kind of problem
@code{sketch} was meant to solve---a pair of textbook figures
regarding a polygonal approximation of a truncated cone.  Here are the
pictures we will produce.

@center @image{ex250}@ @ @ @ @ @image{ex260}

The cone shape is just a swept line with no closure tag and culling
turned off.  Begin by setting up some useful constants.
@sxindex def
@sxindex rotate
@verbatim
  def O (0,0,0) def I [1,0,0] def J [0,1,0] def K [0,0,1]
  def p0 (1,2) def p1 (1.5,0) def N 8
  def seg_rot rotate(360 / N, [J])
@end verbatim
@noindent
The points @code{p0} and @code{p1} are the end points of the line to
be swept.  The definition @code{seg_rot} is the sweep transformation.
With these, the cone itself is simple.
@sxindex sweep
@opindex cull
@sxindex line
@cindex swept line
@cindex line sweep
@verbatim
  sweep[cull=false] { N, [[seg_rot]] } line(p0)(p1)
@end verbatim

The axes are next and include an interesing trick that shows the
hidden parts as dotted lines.  The secret is draw the axes
twice---solid lines with the normal
@cindex hidden surface algorithm
hidden surface algorithm in effect, and then dotted with the
option
@opindex lay
@code{lay=over} so that no polygons can hide them.
@sxindex def
@sxindex line
@opindex arrows
@opindex linewidth
@opindex lay
@opindex linestyle
@sxindex special
@verbatim
  def ax (dx,0,0) % tips of the axes
  def ay (0,dy,0)
  def az (0,0,dz)
  line[arrows=<->,linewidth=.4pt](ax)(O)(ay)
  line[arrows=->,linewidth=.4pt](O)(az)
  % repeat dotted as an overlay to hint at the hidden lines
  line[lay=over,linestyle=dotted,linewidth=.4pt](ax)(O)(ay)
  line[lay=over,linestyle=dotted,linewidth=.4pt](O)(az)
  special|\footnotesize
          \uput[d]#1{$x$}\uput[u]#2{$y$}\uput[l]#3{$z$}|
    (ax)(ay)(az)
@end verbatim
@noindent
The labels are applied with @code{PSTricks} special objects 
@cindex special objects
as usual.

For the height dimension mark, the power of affine 
@cindex affine arithmetic
arithetic is very helpful.
@sxindex def
@sxindex unit
@sxindex scale
@sxindex line
@sxindex special
@verbatim
  def hdim_ref unit((p1) - (O)) then [[seg_rot]]^2
  def c0 (p0) then scale([J])
  def h00 (c0) + 1.1 * [hdim_ref]
  def h01 (c0) + 1.9 * [hdim_ref]
  def h02 (c0) + 1.8 * [hdim_ref]
  line(h00)(h01)
  def h10 (O) + 1.6 * [hdim_ref]
  def h11 (O) + 1.9 * [hdim_ref]
  def h12 (O) + 1.8 * [hdim_ref]
  line(h10)(h11)
  line[arrows=<->](h02)(h12)
  def hm2 ((h02) - (O) + (h12) - (O)) / 2 + (O)
  special|\footnotesize\rput*#1{$h$}|(hm2)
@end verbatim
The general idea employed here is to compute a unit ``reference
vector'' parallel to the @math{xz}-plane in the desired direction of
the dimension from the origin.  The transformation
@code{[[seg_rot]]^2} rotates two segments about the @math{y}-axis.
When applied to @code{(p1) - (O)}, the resulting vector points to the
right as shown.  In this manner, we can pick any vertex as the
location of the height dimension lines by varying the exponent of
@code{[[seg_rot]]}.  This is only one of many possible strategies.

The computation of @code{hm2} is a useful idiom for finding the
@cindex centroid
centroid of a set of points.

The two radius marks are done similarly, so we present the code
without comment.
@sxindex def
@sxindex line
@sxindex special
@sxindex scale
@opindex arrows
@verbatim
  % radius measurement marks
  def gap [0,.2,0]  % used to create small vertical gaps

  % first r1
  def up1 [0,3.1,0] % tick rises above dimension a little
  def r1 ((p1) then [[seg_rot]]^-2) + [up1]
  def r1c (r1) then scale([J])
  def r1t (r1) + [gap]
  def r1b ((r1t) then scale([1,0,1])) + [gap]
  line[arrows=<->](r1c)(r1)  % dimension line
  line(r1b)(r1t)             % tick
  def r1m ((r1) - (O) + (r1c) - (O)) / 2 + (O) % label position
  special |\footnotesize\rput*#1{$r_1$}|(r1m)  % label

  % same drill for r0, but must project down first
  def up0 [0,2.7,0]
  def r0 ((p0) then scale([1,0,1]) then [[seg_rot]]^-2) + [up0]
  def r0c (r0) then scale([J])
  def r0t (r0) + [gap]
  def r0b ((p0) then [[seg_rot]]^-2) + [gap]
  line[arrows=<->](r0c)(r0)
  line(r0b)(r0t)
  def r0m ((r0) - (O) + (r0c) - (O)) / 2 + (O)
  special |\footnotesize\rput*#1{$r_0$}|(r0m)
@end verbatim

The second drawing uses the same techniques.  Only the method for
drawing the elliptical arc is new.  Here is the code.
@sxindex def
@sxindex special
@opindex lay
@verbatim
  def mid ((p00)-(O)+(p10)-(O)+(p11)-(O)+(p01)-(O))/4+(O)
  special|\rput#1{\pscustom{
    \scale{1 1.3}
    \psarc[arrowlength=.5]{->}{.25}{-60}{240}}}|
    [lay=over](mid)
@end verbatim
@noindent
We could have swept a point to make the arc with @code{sketch}, but
using a @code{PSTricks} custom graphic was simpler.  Again we computed
the 
@cindex centroid
centroid of the quadrilateral by averaging points.  Note that scaling
in Postscript distorts the arrowhead, but in this case the distortion
actually looks better in the projection of the slanted face.  A
@code{sketch} arrowhead would not have been distorted.

The complete code for this example, which draws either figure
depending on the definition of the tag @code{<labeled>}, is included
in the @code{sketch} distribution in the file @file{truncatedcone.sk}.

@node A hierarchical model, Caveats, A technical drawing, Building a drawing
@comment  node-name,  next,  previous,  up
@section A hierarchical model
@cindex hierarchical model
While @code{sketch} was never meant to be a geometric modeling
language, it comes fairly close.  The following example puts all we
have seen to work in a very simple model of the human hand.  Start by
sweeping a line to make a truncated cone, which will be copied over
and over again to make the segments of fingers.
@sxindex def
@sxindex sweep
@sxindex rotate
@sxindex line
@verbatim
  def O (0,0,0) % origin
  def I [1,0,0] def J [0,1,0] def K [0,0,1] % canonical unit vectors
  def segment {
    def n_faces 8
    sweep { n_faces<>, rotate(360 / n_faces, [J]) } 
      line(proximal_rad, 0)(distal_rad, distal_len)
  }
@end verbatim
@noindent
In hand anatomy, @emph{distal} is ``at the tip'' and @emph{proximal}
is ``in the area of the palm.''  We have omitted all the scalar
constants.  You can find them in @file{hand.sk}, which is provided
in the @code{sketch} distribution. 

We also need a prototypical sphere to use for the joints themselves.
@sxindex def
@sxindex sweep
@sxindex rotate
@verbatim
  def joint_sphere {
    def n_joint_faces 8
    sweep [fillcolor=red] { n_joint_faces, rotate(360 / n_joint_faces, [J]) }
      sweep { n_joint_faces, rotate(180 / n_joint_faces) } 
        (0, -joint_rad)
  }
@end verbatim

We'll now design the index finger (number@w{ }1 in our notational
convention; finger@w{ }0 is the thumb).  The distal rotation for the
finger applies only to the tip, so we define the following.
@sxindex def
@sxindex put
@sxindex translate
@sxindex rotate
@sxindex scale
@verbatim
  def distal_1 {
    put { translate(joint_gap * joint_rad * [J]) 
          then rotate(distal_1_rot, [I]) 
          then translate((distal_len + joint_gap * joint_rad) * [J]) }
      {segment}
    put { rotate(distal_1_rot / 2, [I])
          then translate((distal_len + joint_gap * joint_rad) * [J]) } 
      {joint_sphere}
    put { scale( [J] + proximal_distal_ratio * ([I]+[K]) ) }
      {segment}
  }  
@end verbatim
@noindent
The identifiers here are for size and location constants. The
exception is @code{distal_rot_1}.  This rotation parameter models the
flexing of the finger tip.  The first @code{put} makes a copy of the
finger segment that is translated upward 
@cindex translation transform
@cindex transform, translation
just far enough to make room
for the spherical joint.  Then it applies the distal rotation.
@cindex rotation transform
@cindex transform, rotation
Finally it translates the whole assembly upward again to make room for
the middle phlanges (the next bone toward the palm).  The second
@code{put} positions the sphere.  There is a rotation to place the
grid on the sphere surface at an nice angle, then a translation to the
base of the distal phlanges, which is also center of its rotation.
Finally, the last @code{put} positions the middle segment itself.

The middle joint is the next one down, with rotation angle
@code{middle_rot_1}.  When this angle changes, we need all the objects
in @code{distal_1} to rotate as a unit.  
@cindex rotation transform
@cindex transform, rotation
This is the reasoning behind
the next definition.
@verbatim
  def finger_1 {
    put { translate(joint_gap * joint_rad * [J])
          then rotate(middle_1_rot, [I])
          then translate((middle_ratio * distal_len + 
                          joint_gap * joint_rad) * [J]) }
      {distal_1}
    put { scale(proximal_distal_ratio)
          then rotate(middle_1_rot / 2, [I])
          then translate((middle_ratio * distal_len + 
                          joint_gap * joint_rad) * [J]) } 
      {joint_sphere}
    put { scale( middle_ratio * [J] + 
                 proximal_distal_ratio^2 * ([I]+[K]) ) }
      {segment}
  }
@end verbatim
@noindent
This looks very similar to the previous definition, and it is.  The
important difference is that rather than positioning and rotating a
single segment, we position and rotate the entire ``assembly'' defined
as @code{distal_1}.  
@cindex rotation transform
@cindex transform, rotation
The rest is just arithmetic to compute sizes and
positions that look nice.  The last @code{put} places an appropriately
shaped segment that is the @emph{proximal phlanges}, the bone that
joins the palm of the hand.  This completes the finger itself.

All the other fingers are described identically to this one.  We
account for the fact that real fingers are different sizes in the next
step, which is to build the entire hand.

The @code{hand} definition that follows includes a section for each
finger.  We'll continue with finger@w{ }1 and omit all the others.
(Of note is that the thumb needs slightly special treatment---an extra
rotation to account for its opposing angle. This is clear in the full
source code.) Not surprisingly, the @code{hand} definition looks very
much like the previous two.  It should be no surprise that when the
rotation parameter @code{meta_1_rot} changes, the entire finger
rotates!
@cindex rotation transform
@cindex transform, rotation
There is an additional rotation that allows the fingers to spread
laterally.  We say these joints of the proximal phlanges have two
@emph{degrees of freedom}. The joints higher on the finger have only
one. Finally, each finger is scaled by a factor to lend it proportion.
@verbatim
  def hand {
    % finger 1 [all other fingers omitted]
    def scale_1 .85
    put { scale(scale_1) 
          then translate((joint_gap * joint_rad) * [J])
	  then rotate(meta_1_rot, [I])
          then rotate(-spread_rot, [K])
          then translate((proximal_1_loc) - (O)) } 
      {finger_1}
    put { scale(scale_1 * proximal_distal_ratio^2)
          then rotate(meta_1_rot / 2, [I])
          then rotate(-spread_rot, [K])
          then translate((proximal_1_loc) - (O)) } 
      {joint_sphere}

    % palm
    sweep { 1, rotate(6, (0,15,0), [I]) }
      put { rotate(-3, (0,15,0), [I]) } {
        polygon(proximal_1_loc)(proximal_2_loc)
               (proximal_3_loc)(proximal_4_loc)
               (h5)(h6)(h6a)(h9)(h10)
        polygon(h6a)(h7)(h8)(h9)
   }  }
@end verbatim
@noindent
The last section of the definition creates the polytope for the palm
of the hand by @code{sweep}ing 
@cindex swept polygon
@cindex polygon sweep
a 10-sided polygon through a very short
arc (9@w{ }degrees).  This provides a wedge-shaped profile when viewed
from the side. The thick end of the wedge is the wrist.  Because the
polygon is concave, it is split into into two convex shapes with nine
and four vertices.

We can now have fun positioning the hand by adjusting the various
rotation angles.  The complete source includes definitions with
alternatives that include the following views and more.

@center @image{ex210}@image{ex220}@image{ex230}@image{ex240}

@node Caveats,  , A hierarchical model, Building a drawing
@comment  node-name,  next,  previous,  up
@section Caveats
@code{Sketch} is a fairly powerful tool for drawing, but, just as with
@TeX{}, the power to create beautiful results comes along with the
power to make mistakes.  The following are some points where care is
necessary and where the current version of @code{sketch} is limited or
has known bugs.

@menu
* Limits on error detection::   What sketch doesn't do.
* Clipping::                    No clipping at present.
* Hidden surface removal::      Imperfections to fix.
@end menu

@node Limits on error detection, Clipping, Caveats, Caveats
@comment  node-name,  next,  previous,  up
@subsection Limits on @code{sketch} error detection

@code{Sketch} catches many kinds of errors, but not all. For example,
options that sketch does not recognize, even incorrect ones, are
quietly copied to @code{PSTricks} commands in the output.  It is also
unfortunately easy to produce @code{sketch} inputs that lead to no
picture at all (improper vertex ordering causes everything to be
culled), to pictures that are too big or too small for @code{PSTricks}
to draw (due to limits of @TeX{} math), and pictures that look nothing
like what was intended.  A picture with one of these problems can be
difficult to ``debug.''  We offer the following suggestions.
@itemize
@item 
Follow the suggested incremental development method described in
@ref{Overview}. 
@item
Always maintain one or two back-versions of a drawing so that it is
easy to fall back to a known-good increment.
@item
When using @code{perspective}, ensure all finally transformed objects
satisfy @math{z<0} and, in fact, do not come very close to the origin
at all.
@item
Temporarily use @code{cull=false} to see where vertex ordering
problems lie.
@item
Use temporary changes of color of one or more objects to ensure that
your understanding of the scene geometry is correct.
@item
If @code{PSTricks} complains about something, inspect the output
directly for clues.
@end itemize

@node Clipping, Hidden surface removal, Limits on error detection, Caveats
@comment  node-name,  next,  previous,  up
@subsection Clipping
The current version of @code{sketch} has no clipping 
@cindex clipping
operations.  The entire scene is always drawn.  This means that when a
perspective transform is employed, it is the user's responsibility to
make sure the entire scene remains in front of the viewer, the region
@math{z<0}.

@node Hidden surface removal,  , Clipping, Caveats
@comment  node-name,  next,  previous,  up
@subsection Hidden surface removal and polygon splitting
@code{Sketch} uses the @dfn{depth sort algorithm} 
@cindex depth sort
@cindex hidden surface algorithm
for hidden surface removal.  This is a very old technique due to
Newell.@footnote{Newell, M.E., R.G.@: Newell, and T.L.@: Sancha, A
solution to the hidden surface problem. @i{Proceedings of the ACM
annual conference - Volume 1}, page 443--450, ACM Press, 1972.}  It is
generally regarded as too slow for real time graphics, but it is
ideal for our purpose where speed is not very important.@footnote{We
have run @code{sketch} on the famous Stanford Bunny, which consists
of nearly @math{70,000} triangles.  Run time was about 6 seconds.
Most of this was spent writing the output file rather than in the
hidden surface algorithm.  @LaTeX{} took much longer to process the
resulting @code{PSTricks} code.  The obvious conclusion is that the
speed of the depth sort algorithm is not a worry.}

The depth sort algorithm merely sorts objects on a key of increasing
@math{z}-coordinate, equivalent to decreasing depth.  Objects are then
drawn in the sorted sequence so that those at the rear of the scene
are overwritten by those closer to the viewer. Since this is also
how oil painters practice their art, depth sort is sometimes called
``the painter's algorithm.''

In some cases it is impossible to strictly order polygons according to
depth.  Moreover, even if a correct depth ordering exists, the
computation needed to find it may be too complex and slow.  In these
cases, @code{sketch} splits 
@cindex splitting, line and surface
one or more polygons into pieces.  The
expectation is that the new, smaller polygons will be simpler to
order.  @code{Sketch} uses a @acronym{BSP,binary space partition} 
@cindex binary space partition
@cindex BSP, binary space partition
to handle the splitting operation.

@menu
* Statistics::                  Performance numbers on depth sort.
* Bugs and anomalies::          Imperfections in this implementation.
@end menu

@node Statistics, Bugs and anomalies, Hidden surface removal, Hidden surface removal
@comment  node-name,  next,  previous,  up
@subsubsection Statistics
For the curious, @code{sketch} writes one line of depth sort
statistics.  Here is an example for a large collection of triangles.
@verbatim
  remark, node=34824 probe=581.9 swap=5 split=2 (in=4 out=6) ols=24851/0
@end verbatim
@noindent
It means that @math{34,824} objects were depth sorted after culling.
For each, an average of @math{581.9} others had to be checked to
ensure that the initial, approximate ordering was correct.  Among all
these checks, only @math{5} resulted in swaps to reorder the initial
sort.  In two cases, a correct ordering could not be determined, so
binary space partitions
@cindex binary space partition
were constructed for splitting.  A total of @math{4}
objects (triangles in this case) were inserted in the partitions, and
@math{6} polygons were produced.  Finally, @math{24,851} ``last
resort'' polygon overlap checks were performed after simpler, faster
checks failed to yield conclusive results.  The final @math{/0} is for
line-polygon overlap checks.  For comparison, the statistics for the
last figure in @ref{Overview} follow.
@verbatim
  remark, node=27 probe=14.6 swap=36 split=15 (in=30 out=45) ols=0/69
@end verbatim
@noindent
Note that there was proportionally much more swapping and splitting
activity in this highly connected scene.

@node Bugs and anomalies,  , Statistics, Hidden surface removal
@comment  node-name,  next,  previous,  up
@subsubsection Bugs and anomalies
Polygon and line splitting can both cause anomalies in the output.  
@code{PSTricks} dash patterns, specified with @code{linestyle=dashed},
@opindex linestyle
can be disrupted by splitting.  This occurs when the depth sort
@cindex depth sort
gives up too early and splits a line where it is not really
necessary. 
A workaround is to use gray or finely dotted
lines instead.  If your drawing is small, you can also edit the
@code{sketch} output by hand to merge the pieces of the offending
line.

Another anomaly is tiny (or in degenerate cases not-so-tiny) notches
in the lines that border split polygons.  These derive from the way
each polygon is painted: first, all pixels within the boundary are
@dfn{filled} with color (perhaps white), then the same boundary is
@dfn{stroked} (a Postscript term) with a line.  The result is that
half the line lies inside the boundary and half outside, while the
Painter's algorithm assumes the polygon lies entirely within its
boundary.  The notches are due to one polygon fill operation
overwriting the already-drawn inside of the border of another
polygon.@footnote{I know how to fix this problem, but I don't like my
solution, and I'm interested in yours.}  One workaround is to make
border lines very thin.  In fact @code{linewidth=0pt} is guaranteed to
eliminate this problem, though this results in the thinnest line your
output device can draw, which is usually too thin.  You might get
lucky by merely reordering things in the input file, which is likely
to move the splits to different places.  The only sure-fire solution
is pretty terrible: custom fit @code{special} overlay lines (with
@code{\psline}) to cover the notches.

Polygon splitting also breaks @code{PSTricks} hatch patterns.  The
only known workaround is to substitute a solid fill for the hatch.

@node Command line, Installing sketch, Building a drawing, Top
@comment  node-name,  next,  previous,  up
@chapter Command line
@cindex command line, @code{sketch}
@strong{Synopsis:}
@example
sketch [-h][-V x.y][-v][-b][-d][t doctmp][-T[u|e][p[P|T][L|C]]][-o output.tex]
  [-D @var{tag} @dots{}] input1.sk [-U @var{tag} @dots{}] input2.sk @dots{}
@end example

@noindent
@strong{Description}
Processes the @code{sketch} input files in order to produce
@code{PSTricks} output code suitable for inclusion in a @TeX{} or
@LaTeX{} document.

@noindent
@strong{Options:}
@cindex options, command line
@cindex command line options
@table @code
@item -h
Print a short catalog of options.
@item -V
Set the @code{PSTricks} version assumed for output purposes to
@code{x.y}, for example 1.19.  Usually needed only if your
@code{PSTricks} is old compared to your @code{sketch}.  Use
@code{-v} to see what @code{sketch} assumes by default.
@item -v
Print version information to standard output, including the version
of @code{PSTricks} assumed for output (can be changed with @code{-V} above).
@item -b
Use a BSP 
@cindex binary space partition
@cindex BSP, binary space partition
(@xref{Hidden surface removal}.) for
@emph{all} hidden surface removal rather than the default, which is
the depth sort algorithm with BSPs used only for cycle resolution.
This may produce correct output in certain degenerate cases where the
depth sort cannot, but it also leads to many gratuitous splits, hence
more anomalies @ref{Bugs and anomalies} and big output files.
@item -d
Run @code{sketch}'s parser in debugging mode.  This is primarily for
development.
@item -t
Use contents of file @file{doctmp} as a document template 
@cindex document template
@cindex template, document
in which to enclose @code{PSTricks} output code.  The code is inserted
in place of the first instance of the escape string
@verb{|%%SKETCH_OUTPUT%%|}.  
@item -T
Causes @code{PSTricks} output to be enclosed in default US document
template text.  Option @option{-Tu} is a synonym.  Option @option{-Te}
causes the Euro standard document template to be used.  A @option{p}
appended to any of these options causes the respective default
@code{PSTricks} document template to be printed to standard output. An
appended @option{P} is a synonym.  An appended @option{T} causes the
the @code{TikZ/PGF} template to be printed.  An appended @option{L}
prints the @LaTeX{} version of the document template, a synonym for
the default.  A @option{C} prints the @code{ConTeXt} template.
@item -o
Use @file{output.tex} as the output file.  The default is standard output.
@item -D
Define a tag 
@cindex tag definition
@cindex definition, tag
for purposes of selecting definition alternatives.
@xref{Definitions}.  The definition applies for all input files that
follow unless the tag is undefined with @option{-U}.
@item input@math{i}.sk
Input files, read in the sequence they are given.
@item -U
Un-define a tag for purposes of selecting definition alternatives.
@end table

@node Installing sketch, Index of syntax, Command line, Top
@comment  node-name,  next,  previous,  up
@chapter Building and installing @code{sketch}
@code{Sketch} is so small that compiling by brute force is probably
best.  The following command ought to do the trick on any
systems where @code{gcc} is installed.  Make sure to first change
current directories to the place where you have unpacked the sources.
@example
gcc *.c -o sketch.exe -lm
@end example
@noindent
The @samp{.exe} at the end is necessary for Windows systems.  Drop it
if your system is some version of Unix.  Other C compilers ought
to work as just as well.  For example,
@example
cl *.c -o sketch.exe
@end example
@noindent
is the correct command for many versions of MS Visual C.  In the
latest versions, Microsoft has deprecated the @code{-o} option and, by
default, does not define the @code{__STDC__} macro.  This causes
problems with some versions of @code{flex}, @code{bison}, @code{lex},
and @code{yacc}, which are used to create the @code{sketch} scanner
and parser.  It's nearly always possible to find a set of options that
compiles with no errors or warnings, and this means @code{sketch} is
@emph{very} likely to work correctly.  For example, the Visual C++
2005 Express Edition compiler (available free of charge from the
Microsoft web site), @code{flex} version 2.5.4, and @code{bison}
version 2.1 build error-free with
@example
cl -DYY_NEVER_INTERACTIVE=1 -Za -Ox -Fesketch.exe *.c
@end example

For purists, there is also a @code{makefile} compatible with GNU
@code{make} and @code{gcc}. The command
@example
make
@end example
@noindent
will build the executable, including the rebuilding of the scanner and
parser with @code{flex} and @code{bison} if you have changed
@file{sketch.l} or @code{sketch.y} respectively.  

While not essential, building the manual is an excellent test of your
@code{sketch} installation.  To build this document in all its myriad
forms (assuming you have the necessary conversion programs on your
system), use
@example
make docs
@end example
@noindent
The possibilities are listed in this following table.
@multitable {@code{manual/index.html}}{texi2dvi,dvips}{@b{Pictures}}{wide column that needs to be as big as it needs to be,}
@headitem Format @tab Converter @tab Pictures @tab Description
@item manual.info 
  @tab makeinfo       @tab @file{.txt} @tab @acronym{GNU} Info.
@item manual.dvi
  @tab texi2dvi       @tab @file{.eps} @tab @TeX{} typeset output.
@item manual.ps   
  @tab texi2dvi,dvips @tab @file{.eps} @tab Postscript.
@item manual.pdf
  @tab texi2dvi       @tab @file{.pdf} @tab Adobe PDF.
@item manual.html 
  @tab makeinfo       @tab @file{.png} @tab A single web page.
@item manual/index.html
  @tab makeinfo       @tab @file{.png} @tab Linked web pages, one per node.
@end multitable
All the required tools are included in the standard @code{TeX Live}
distribution except that an additional open source program
@code{epstool} is needed to refine the Encapsulated Postscript
bounding boxes of @code{sketch}-generated figures in the manual.  This
tool is available in binary form for Windows, and it builds easily
with a simple @code{make} under linux or Cygwin installations that
have development tools installed or under @code{Mac OS X} with
@code{XCode} installed.

@node Index of syntax, Index, Installing sketch, Top
@comment  node-name,  next,  previous,  up
@unnumbered Index of syntax

@printindex sx

@node Index,  , Index of syntax, Top
@comment  node-name,  next,  previous,  up
@unnumbered Index of concepts

@printindex cp

@bye
