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<H1> <EM>surf</EM> version 1.0.1</H1>

<H2>Stephan Endrass
         
<A HREF="mailto:endrass@mathematik.uni-mainz.de">&lt;endrass@mathematik.uni-mainz.de&gt;</A></H2> May 28, 2000
<HR>
<EM>The aim was to have a tool to visualize some real algebraic geometry:
plane algebraic curves given as zero
locus of a polynomial in two variables,
algebraic surfaces given as zero locus of
a polynomial in three variables,
hyperplane sections of surfaces: algebraic space curves
given as zero locus of two polynomials in three variables:
a polynomial of arbitrary degree (the surface) and
a linear polynomial (the hyperplane),
and lines on surfaces given by two points on a surface.
The algorithms should be stable enough not to be confused by
curve/surface singularities in codimension greater than one and
the degree of the surface or curve.
This has been achieved quite a bit. We have drawn curves of degree up to 30
and surfaces of degree up to 20 successfully. However, there are examples
of curves/surfaces of lower degree where surf fails to produce perfect
images. This happens especially if the equation of the curve/surface is not
reduced. Best results are achieved using reduced equations. On the other
hand, surf displays the Fermat-curves accurately for degree up to 98.</EM>
<HR>
<P>
<H2><A NAME="toc1">1.</A> <A HREF="surf-alggeo.html#s1">Overview</A></H2>

<UL>
<LI><A NAME="toc1.1">1.1</A> <A HREF="surf-alggeo.html#ss1.1">Acknowledgements</A>
<LI><A NAME="toc1.2">1.2</A> <A HREF="surf-alggeo.html#ss1.2">Copyright</A>
<LI><A NAME="toc1.3">1.3</A> <A HREF="surf-alggeo.html#ss1.3">How to get <EM>surf</EM></A>
<LI><A NAME="toc1.4">1.4</A> <A HREF="surf-alggeo.html#ss1.4">System requirements</A>
<LI><A NAME="toc1.5">1.5</A> <A HREF="surf-alggeo.html#ss1.5">Starting <EM>surf</EM></A>
<LI><A NAME="toc1.6">1.6</A> <A HREF="surf-alggeo.html#ss1.6">Scripts versus graphical user interface</A>
<LI><A NAME="toc1.7">1.7</A> <A HREF="surf-alggeo.html#ss1.7">Scripts</A>
<LI><A NAME="toc1.8">1.8</A> <A HREF="surf-alggeo.html#ss1.8">Output</A>
<LI><A NAME="toc1.9">1.9</A> <A HREF="surf-alggeo.html#ss1.9">Sample scripts</A>
<LI><A NAME="toc1.10">1.10</A> <A HREF="surf-alggeo.html#ss1.10"><EM>surf</EM> and make</A>
<LI><A NAME="toc1.11">1.11</A> <A HREF="surf-alggeo.html#ss1.11">Oddities, bugs and bug reports</A>
</UL>
<P>
<H2><A NAME="toc2">2.</A> <A HREF="surf-alggeo.html#s2">Introduction to <EM>surf</EM>'s command language</A></H2>

<UL>
<LI><A NAME="toc2.1">2.1</A> <A HREF="surf-alggeo.html#ss2.1">Data types</A>
<LI><A NAME="toc2.2">2.2</A> <A HREF="surf-alggeo.html#ss2.2">Operators</A>
<LI><A NAME="toc2.3">2.3</A> <A HREF="surf-alggeo.html#ss2.3">Mathematical functions</A>
<LI><A NAME="toc2.4">2.4</A> <A HREF="surf-alggeo.html#ss2.4">String functions</A>
<LI><A NAME="toc2.5">2.5</A> <A HREF="surf-alggeo.html#ss2.5">Polynomial functions</A>
<LI><A NAME="toc2.6">2.6</A> <A HREF="surf-alggeo.html#ss2.6">First examples</A>
<LI><A NAME="toc2.7">2.7</A> <A HREF="surf-alggeo.html#ss2.7">Conditional statements</A>
</UL>
<P>
<H2><A NAME="toc3">3.</A> <A HREF="surf-alggeo.html#s3">Features</A></H2>

<UL>
<LI><A NAME="toc3.1">3.1</A> <A HREF="surf-alggeo.html#ss3.1">Plane curves</A>
<LI><A NAME="toc3.2">3.2</A> <A HREF="surf-alggeo.html#ss3.2">Surfaces</A>
<LI><A NAME="toc3.3">3.3</A> <A HREF="surf-alggeo.html#ss3.3">Hyperplane sections</A>
<LI><A NAME="toc3.4">3.4</A> <A HREF="surf-alggeo.html#ss3.4">Multiple curves/surfaces</A>
<LI><A NAME="toc3.5">3.5</A> <A HREF="surf-alggeo.html#ss3.5">Graphs and isolines</A>
<LI><A NAME="toc3.6">3.6</A> <A HREF="surf-alggeo.html#ss3.6">Interactive positioning</A>
<LI><A NAME="toc3.7">3.7</A> <A HREF="surf-alggeo.html#ss3.7">Preview</A>
<LI><A NAME="toc3.8">3.8</A> <A HREF="surf-alggeo.html#ss3.8">Anti aliasing surfaces</A>
<LI><A NAME="toc3.9">3.9</A> <A HREF="surf-alggeo.html#ss3.9">Animations</A>
<LI><A NAME="toc3.10">3.10</A> <A HREF="surf-alggeo.html#ss3.10">Stereo pictures</A>
<LI><A NAME="toc3.11">3.11</A> <A HREF="surf-alggeo.html#ss3.11">Black &amp; white images</A>
<LI><A NAME="toc3.12">3.12</A> <A HREF="surf-alggeo.html#ss3.12">Algorithms</A>
<LI><A NAME="toc3.13">3.13</A> <A HREF="surf-alggeo.html#ss3.13">Output</A>
</UL>
<P>
<H2><A NAME="toc4">4.</A> <A HREF="surf-alggeo.html#s4">List of all reserved words</A></H2>

<UL>
<LI><A NAME="toc4.1">4.1</A> <A HREF="surf-alggeo.html#ss4.1">Reserved words corresponding to the main window</A>
<LI><A NAME="toc4.2">4.2</A> <A HREF="surf-alggeo.html#ss4.2">Reserved words corresponding to the position window</A>
<LI><A NAME="toc4.3">4.3</A> <A HREF="surf-alggeo.html#ss4.3">Reserved words corresponding to the display window</A>
<LI><A NAME="toc4.4">4.4</A> <A HREF="surf-alggeo.html#ss4.4">Reserved words corresponding to the light window</A>
<LI><A NAME="toc4.5">4.5</A> <A HREF="surf-alggeo.html#ss4.5">Reserved words corresponding to the clip window</A>
<LI><A NAME="toc4.6">4.6</A> <A HREF="surf-alggeo.html#ss4.6">Reserved words corresponding to the dither window</A>
<LI><A NAME="toc4.7">4.7</A> <A HREF="surf-alggeo.html#ss4.7">Reserved words corresponding to the save color image window</A>
<LI><A NAME="toc4.8">4.8</A> <A HREF="surf-alggeo.html#ss4.8">Reserved words corresponding to the save dithered image window</A>
<LI><A NAME="toc4.9">4.9</A> <A HREF="surf-alggeo.html#ss4.9">Reserved words corresponding to the numeric window</A>
<LI><A NAME="toc4.10">4.10</A> <A HREF="surf-alggeo.html#ss4.10">Reserved words corresponding to the curve window</A>
</UL>

<HR>
<H2><A NAME="s1">1.</A> <A HREF="#toc1">Overview</A></H2>



<H2><A NAME="ss1.1">1.1</A> <A HREF="#toc1.1">Acknowledgements</A>
</H2>


<P>I thank Prof. W. Barth (University Erlangen)
for (en)forcing me to start this project. 
Hans H&uuml;lf, R&uuml;diger &Ouml;rtel and Kai Schneider have
spent lots of time on coding parts of <EM>surf</EM>.
Some of the code has been copied from other places:
<UL>
<LI> Writing SUN rasterfiles
and XWD files has been copied from
Michael L. Mauldin's Fuzzy PixMap (fbm) library version 1.2.</LI>
<LI> Writing a TIFF  file has been copied from John Cristy's
Image Magick version 3.0.</LI>
<LI> The octree color reduction algorithm is
copied from Ian Ashdown's article <EM>Octree Color Quantization</EM>
in the C/C++ Users Journal Vol. 13, Number 3, pp. 31-43.</LI>
</UL>

We thank all these people who made their code free so that
we could use it.</P>

<H2><A NAME="ss1.2">1.2</A> <A HREF="#toc1.2">Copyright</A>
</H2>

<P><EM>surf</EM> 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 2 of the License, or
(at your option) any later version.</P>
<P>This program 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.</P>

<H2><A NAME="ss1.3">1.3</A> <A HREF="#toc1.3">How to get <EM>surf</EM></A>
</H2>

<P><EM>surf</EM> is available via http/ftp at the surf home page
<A HREF="http://surf.sourceforge.net">http://surf.sourceforge.net</A>.</P>

<H2><A NAME="ss1.4">1.4</A> <A HREF="#toc1.4">System requirements</A>
</H2>


<P>To compile <EM>surf</EM>, the following software is needed:
<UL>
<LI><B>GNU gcc/g++</B> version 2.7.x or higher (any ISO C++ compiler should work..),</LI>
<LI><B>make</B>,</LI>
<LI>GNU <B>flex</B> version 2.5 or higher (minor versions should
work also, but lex does <EM>not</EM> suffice),</LI>
<LI>Berkeley <B>yacc</B> (GNU bison should work also),</LI>
<LI><B>GTK+</B> version 1.2.0 or later (only required if you want to
compile <B>surf</B> with GUI support),</LI>
<LI><B>POSIX threads</B> (If you have Linux make sure you use glibc2),</LI>
<LI><B>GNU MP (gmp)</B> version 2 or later,</LI>
</UL>

<B>Warning:</B> Be prepared, the memory consumption of <EM>surf</EM> is about
<UL>
<LI>  15 MB for a 1000x1000 pixel image,</LI>
<LI>  60 MB for a 2000x2000 pixel image and</LI>
<LI> 130 MB for a 3000x3000 pixel image (gobble!!).</LI>
</UL>
</P>

<H2><A NAME="ss1.5">1.5</A> <A HREF="#toc1.5">Starting <EM>surf</EM></A>
</H2>

<P><EM>surf</EM> is started by typing <CODE>surf</CODE> on the command line.
Optional arguments are <CODE>--no-gui</CODE> (or <CODE>-n</CODE>) for starting
<EM>surf</EM> without graphical user interface, <CODE>--exec</CODE> (or
<CODE>-x</CODE>) to immediately execute the first passed script file and -
when using surf <EM>with</EM> GUI - <CODE>--progress-dialog</CODE>
which tells surf to use a progress dialog
instead of a status bar, <CODE>--auto-resize</CODE> which forces the image
windows to get automatically resized to the size of the image, and the
usual GTK+ options. <CODE>--help</CODE> prints out the usage information:
<PRE>
    surf -n | --no-gui FILE...
    surf [GTK-OPTIONS] [-x | --exec] [--progress-dialog]
         [--auto-resize] [FILE]...
    surf --help
</PRE>
</P>

<H2><A NAME="ss1.6">1.6</A> <A HREF="#toc1.6">Scripts versus graphical user interface</A>
</H2>

<P><EM>surf</EM> is designed to visualize algebraic curves and surfaces.
This can be
done either by <EM>writing scripts</EM> in surf's command language
and executing them interactively or from another program
(for example make), or by <EM>using
surf's graphical user interface</EM>.
By using scripts one can draw series of pictures where each picture
consists of several surfaces/curves at a high resolution.</P>


<H2><A NAME="ss1.7">1.7</A> <A HREF="#toc1.7">Scripts</A>
</H2>

<P>Scripts in surf's command language are stored in files with the
suffix <CODE>.pic</CODE>. These files consist of descriptions of curves and/or
surfaces and some commands. They can be invoked in two ways:
<UL>
<LI>by loading a script and pressing the button <EM>execute script</EM> or</LI>
<LI>by starting <EM>surf</EM> with the name of a script file from the
command line. If you compiled/started <EM>surf</EM> without GUI
support, it automatically starts interpreting the script. Otherwise
you have to press the <EM>execute script</EM> button.</LI>
</UL>
</P>

<H2><A NAME="ss1.8">1.8</A> <A HREF="#toc1.8">Output</A>
</H2>

<P><EM>surf</EM> calculates both color and black &amp; white images.
Color images can currently be
stored in the following formats:
<UL>
<LI><B>XWD</B>,</LI>
<LI><B>SUN rasterfile</B>,</LI>
<LI><B>PPM</B> and</LI>
<LI><B>JPEG</B></LI>
</UL>

Additionally one can choose a convenient colormap among
<UL>
<LI>Netscape 216 color cube (8 bit),</LI>
<LI>optimized by an octree algorithm (8 bit) and</LI>
<LI>true color (24 bit).</LI>
</UL>

Black &amp; white images can be stored in the following formats:
<UL>
<LI><B>Postscript</B>,</LI>
<LI><B>Encapsulated Postscript</B>,</LI>
<LI><B>TIFF</B>,</LI>
<LI><B>XBM</B>,</LI>
<LI><B>PGM</B> and</LI>
<LI><B>PBM</B>.</LI>
</UL>
</P>

<H2><A NAME="ss1.9">1.9</A> <A HREF="#toc1.9">Sample scripts</A>
</H2>


<P>You will find some sample scripts together with <EM>surf</EM>'s
distribution. They are stored in the <CODE>examples</CODE> directory.</P>


<H2><A NAME="ss1.10">1.10</A> <A HREF="#toc1.10"><EM>surf</EM> and make</A>
</H2>


<P><EM>surf</EM> can be invoked from make. This comes in quite handy when
visualising a series of curves/surfaces. Suppose there are script
files s1.pic, s2.pic, ... , sn.pic which create during execution
images s1.xwd, s2.xwd, ... , sn.xwd. If for example gif is the
desired image file format, an appropriate makefile might look like:
<HR>
<PRE>
    #!/bin/bash
    #
    SURF    = surf
    RM      = /bin/rm -f
    CONVERT = convert
    #
    OBJS    = s1.gif s2.gif .... sn.gif
    #
    .SUFFIXES: .pic .gif
    #
    .pic.gif:
        ${SURF}    -n $&lt;
        ${CONVERT} $*.xwd $*.gif
        ${RM}      $*.xwd
    #
    dummy:
        @echo ' '
        @echo 'usage:'
        @echo ' '
        @echo '    print this message:'
        @echo '        make'
        @echo ' '
        @echo '    build images:'
        @echo '        make all'
        @echo ' '
        @echo '    remove images:'
        @echo '        make clean'
        @echo ' '
    #
    all: ${OBJS}
    #
    clean:
        ${RM} *.gif
    #
    # end of makefile
</PRE>
<HR>

Here convert is the Image Magick image format converter.</P>

<H2><A NAME="ss1.11">1.11</A> <A HREF="#toc1.11">Oddities, bugs and bug reports</A>
</H2>


<P>In case you find any bug, please use the excellent 
<A HREF="http://sourceforge.net/bugs/group_id=3275">Bug Tracking System</A> on <EM>surf</EM>'s project page at Sourceforge.</P>

<H2><A NAME="s2">2.</A> <A HREF="#toc2">Introduction to <EM>surf</EM>'s command language</A></H2>



<H2><A NAME="ss2.1">2.1</A> <A HREF="#toc2.1">Data types</A>
</H2>


<P>The language used in <EM>surf</EM>'s scripts is quite simple.
It has got a (very restricted) C-like syntax and provides the four data types
<UL>
<LI><CODE>int</CODE>         (integer),</LI>
<LI><CODE>double</CODE>      (double precision float value),</LI>
<LI><CODE>string</CODE>      (any &#34;&#34;-quoted string) and</LI>
<LI><CODE>poly</CODE>        (any polynomial in x, y and z).</LI>
</UL>

So a valid declaration/initialisation looks like:
<UL>
<LI><CODE>int a=3;</CODE> or <CODE>int a; a=3;</CODE></LI>
<LI><CODE>double b=3.3;</CODE> or <CODE>double b; b=3.3;</CODE></LI>
<LI><CODE>string c=&#34;test.xwd&#34;;</CODE>
or <CODE>string c; c=&#34;test.xwd&#34;;</CODE></LI>
<LI><CODE>poly d=(x-3)^3-y^2+z;</CODE>
or <CODE>poly d; d=(x-3)^3-y^2+z;</CODE></LI>
</UL>

There is no comma separator like in C. Declaring a name twice results in
an error. The scope of the name begins at the point of its declaration and
lasts until the end of the file. There is no method of undeclaring a name.</P>

<H2><A NAME="ss2.2">2.2</A> <A HREF="#toc2.2">Operators</A>
</H2>


<P>The following arithmetic operators are implemented:
<PRE>
    operator | meaning          | valid data types 
    -----------------------------------------------------------------------
    +        | binary plus      | {int,double,poly}+{int,double,poly}
    +        | concatenation    | {string}+{string}
    +        | unary plus       | +{int,double,poly}
    -        | binary minus     | {int,double,poly}-{int,double,poly}
    -        | unary minus      | -{int,double,poly}
    *        | multiplication   | {int,double,poly}*{int,double,poly}
    /        | division         | {int,double,poly}/{int,double}
    %        | remainder        | {int}%{int}
    ^        | power            | {int,double}^{int,double}
             |                  | {poly}^{int}
    ( )      | brackets         | ({int,double,poly})
    =        | equals           | {poly}={int,double,poly}
             |                  | {double}={int,double}
             |                  | {int}={int}
             |                  | {string}={string}
    ==       | equal            | {int,double}=={int,double}
    !=       | not equal        | {int,double}!={int,double}
    &lt;        | smaller than     | {int,double}&lt;{int,double}
    &lt;=       | smaller or equal | {int,double}&lt;={int,double}
    >        | greater than     | {int,double}>{int,double}
    >=       | greater or equal | {int,double}>={int,double}
</PRE>

The precedence of operators copied from C.</P>


<H2><A NAME="ss2.3">2.3</A> <A HREF="#toc2.3">Mathematical functions</A>
</H2>


<P>There are some built-in math functions:
<PRE>
    function | meaning       | valid arguments            | returns
    ---------------------------------------------------------------
    sqrt     | square root   | sqrt({int,double})         | double
    pow      | power         | pow({int},{int,double})    | double
             |               | pow({double},{int,double}) | double
    sin      | sinus         | sin({int,double})          | double
    cos      | cosinus       | cos({int,double})          | double
    arcsin   | arcus sinus   | arcsin({int,double})       | double
    arccos   | arcus cosinus | arccos({int,double})       | double
    tan      | tangens       | tan({int,double})          | double
    arctan   | arcus tangens | arctan({int,double})       | double
</PRE>

They take int and double as argument.</P>

<H2><A NAME="ss2.4">2.4</A> <A HREF="#toc2.4">String functions</A>
</H2>


<P>There are also two functions returning strings:
<PRE>
    function | meaning       | valid arguments      | returns
    ------------------------------------------------------------------------
    itostr   | int to string | itostr({int})        | string
    itostrn  | int to string | itostrn({int},{int}) | string of spec. length
</PRE>

itostr converts its argument to a string without blanks. For example
<CODE>itostr( 31 )</CODE> returns <CODE>&#34;31&#34;</CODE>.
itostrn allows to specify the length of the string.
For example:
<UL>
<LI><CODE>itostrn( 3,88 )</CODE>  returns  <CODE>&#34;088&#34;</CODE></LI>
<LI><CODE>itostrn( 4,88 )</CODE>  returns  <CODE>&#34;0088&#34;</CODE></LI>
</UL>
</P>

<H2><A NAME="ss2.5">2.5</A> <A HREF="#toc2.5">Polynomial functions</A>
</H2>


<P>Some functions work on polynomials:
<PRE>
    function | meaning       | valid arguments           | returns
    --------------------------------------------------------------
    deg      | degree        | deg({poly})            | int
    len      | length        | len({poly})            | int
    diff     | derivative    | diff({poly},{x,y,z})   | poly
    rotate   | rotation      | rotate({poly},{double} |
             |               |   {xAxis,yAxis,zAxis}) | poly
    hesse    | hesse surface | hesse({poly})          | poly
</PRE>

This enables you to work out arbitrary polynomials. </P>


<H2><A NAME="ss2.6">2.6</A> <A HREF="#toc2.6">First examples</A>
</H2>


<P>Values can be passed to <EM>surf</EM> by setting global variables.
The most important two global variables are <CODE>curve</CODE> and
<CODE>surface</CODE>, which should be set to the
polynomial whose zero set should be visualized. So the
shortest effective script contains only three lines, for example:
<UL>
<LI>1st example: draw the newton knot
<HR>
<PRE>
        clear_screen;
        curve=y^2-x^2*(x+1);
        draw_curve;
</PRE>
<HR>
</LI>
<LI>2nd example: draw a sphere
<HR>
<PRE>
        clear_screen;
        surface=x^2+y^2+z^2-80;
        draw_surface;
</PRE>
<HR>
</LI>
</UL>

Both examples can be invoked by pressing the button <EM>execute script</EM>.
The command <CODE>draw_curve</CODE> is somehow equivalent to pressing the
button <EM>draw curve</EM>. The command <CODE>draw_surface</CODE> is somehow
equivalent to pressing the button <EM>draw surface</EM>.</P>


<H2><A NAME="ss2.7">2.7</A> <A HREF="#toc2.7">Conditional statements</A>
</H2>


<P><B>CAUTION</B>: There are no <CODE>for</CODE> and no <CODE>while</CODE> statements.
There is only the crude
<PRE>
    if( INTEGER-EXPRESSION ) goto LABEL;
</PRE>

which you might remember from your early BASIC sessions.
Here <CODE>INTEGER-EXPRESSION</CODE> can be arbitrary complicated as long as it
results in an integer. <CODE>LABEL</CODE> is something like <CODE>NAME:</CODE>
which has occurred
before. Consider the example
<HR>
<PRE>
    int i=0;
    loop:
        surface=x^2+y^2+z^2-(i+1.0)/2.0;
        clear_screen;
        draw_surface;
        filename="sphere"+itostrn( 2,i )+".ras";
        save_color_image;
        i=i+1;
    if( i&lt;50 ) goto loop;
</PRE>
<HR>

which obviously draws fifty spheres of increasing radius 
and saves them into the <B>SUN rasterfiles</B>:
<BLOCKQUOTE>
sphere00.ras ... sphere49.ras
</BLOCKQUOTE>

There exist some more commands explained briefly afterwards.
C++ comments are welcome.
<B>Warning:</B> Check if your loop terminates!</P>



<H2><A NAME="s3">3.</A> <A HREF="#toc3">Features</A></H2>


<P>In this section most features of <EM>surf</EM> are explained.
Many of these features
can be invoked from the graphical user interface. All features can be
invoked through <EM>surf</EM>'s command language. Command language features
are only explained if not accessible through the GUI. For a complete
reference to the command language, have a look at the next section.</P>

<H2><A NAME="ss3.1">3.1</A> <A HREF="#toc3.1">Plane curves</A>
</H2>


<P>To draw a plane curve, enter the equation into <EM>surf</EM>'s text
window preceded by <CODE>curve=</CODE> and followed by a semicolon.
Then press the button <EM>draw curve</EM>.
Some seconds later the curve will show up in the window titled
<EM>color image</EM>.
By default the curve is drawn inside the rectangle
<BLOCKQUOTE>
-10.0 &lt;= x,y &lt;= 10.0
</BLOCKQUOTE>

and is clipped at a circle with radius 10.0.
The x-axis is horizontal pointing to the right, the y-axis is vertical and
points upwards. By default the image size is 200 x 200 pixels.
The image size can be altered by setting 
<EM>width</EM> and <EM>height</EM> in the main window.</P>
<P>The view can be altered in the <EM>position window</EM>:
A different origin can be specified by setting
<EM>origin x</EM> and <EM>origin x</EM>.
A rotation with center at (0,0) can be specified by setting
<EM>rotation about z-axis</EM>. The curve may be scaled by setting
<EM>scale factor x</EM> and <EM>scale factor y</EM>.
The appearance of the curve can be altered in the <EM>curve window</EM>.</P>
<P>The clipping area can be specified in the <EM>clip window</EM>.
For a curve the only reasonable values are
<EM>sphere</EM> and <EM>none</EM>.</P>
<P>An arbitrary color can be given to the curve by setting
<EM>curve red</EM>,<EM>curve green</EM> and <EM>curve blue</EM>
to appropriate values in the <EM>curve window</EM>.
The curve width can be set by changing <EM>curve width</EM>.
A high value of <EM>curve gamma</EM> sharpens the curve, whereas a low
value blurs the curve.</P>

<H2><A NAME="ss3.2">3.2</A> <A HREF="#toc3.2">Surfaces</A>
</H2>


<P>To draw a surface, enter its equation into <EM>surf</EM>'s text window
preceded by <CODE>surface=</CODE> and followed by a semicolon. Then press the button
<EM>draw surface</EM>.
Some more seconds later the surface will appear. By default, the surface
is calculated inside the cube
<BLOCKQUOTE>
-10.0 &lt;= x,y,z &lt;= 10.0
</BLOCKQUOTE>

and clipped at a sphere of radius 10.0.
The x-axis is horizontal pointing to the right, the y-axis is vertical and
points upwards. The z-axis points to you. The spectator
is located at (0,0,25) by default.</P>
<P>Changing the view can be done by altering the settings in the
<EM>position window</EM>. A different origin may be specified by setting
<EM>origin x</EM>, <EM>origin y</EM> and <EM>origin z</EM>.
To rotate the surface one can set <EM>rotation about x-axis</EM>,
<EM>rotation about y-axis</EM> and <EM>rotation about z-axis</EM> 
to appropriate values.
Rotation is performed on the following order: y-axis, x-axis, z-axis.
To scale the surface set <EM>scale factor x</EM>, <EM>scale factor y</EM>
and <EM>scale factor z</EM> to desired
values. It is also possible to switch from central perspective to parallel
perspective.</P>
<P>Illumination and color can be altered in the <EM>light window</EM>.
The direction of the normal vector given by the gradient
of the surface equation defines one side of the surface which
is regarded as outside. You can specify a color for this
side by setting <EM>surface red</EM>, <EM>surface green</EM>
and <EM>surface blue</EM>. The other side of the surface
(inside) can be given a different color by specifying
<EM>inside red</EM>, <EM>inside green</EM> and <EM>inside blue</EM>.</P>
<P>Currently only the Phong illumination model is implemented.
Therefore the intensity of the surface in one point consists of
four components which are calculated separately:
<UL>
<LI> ambient light,</LI>
<LI> diffuse light,</LI>
<LI> reflected light and</LI>
<LI> transmitted light.</LI>
</UL>

Ambient light is a constant which represents the light a
point on the surface receives from the whole environment
(the sky, the floor, the lawn ...)
but not from the light sources. Diffuse light is the
light the point receives from the light sources and
which is reflected equally in every possible direction.
The amount of diffuse light is independent of the
spectator position, it is proportional to
the cosine of the angle between the normal vector and
the vector from the point to the light source.
Reflected light is the light from the light sources which is
reflected specular from the surface point.
Its amount is proportional to a power of the cosine of
the angle between the vector from the point to the
spectator and the specular reflection vector from the
light source. If a high power of the cosine is taken,
the surface will appear shiny, whereas a low power
of the cosine lets the surface look rough.
Therfore this power is labelled <EM>smoothness</EM>.
Transmitted light comes in if a surface is transparent.
A constant called <EM>transparence</EM> specifies the
percentage of light which passes through the surface.
Algebraic surfaces are infinitesimally thin. However our
eye is not used to such objects, so we pretend that
our surfaces have a constant <EM>thickness</EM>. Specifying
a positive thickness for a transparent surface results
in a loss of transparency in the places where the
surface normal does not point to the spectator.</P>
<P>These four light components are added with weights
<EM>ambient</EM>, <EM>diffuse</EM>, <EM>reflected</EM>
and <EM>transmitted</EM>.</P>
<P>The number of light sources is limited to nine. For every light
source, the position, the color and the intensity
can be specified.</P>
<P>The <EM>clip window</EM> allows to specify a different clipping area. Here the
center and radius of the clipping area may be specified. Additionally
a front and a back clipping plane may be specified.</P>

<H2><A NAME="ss3.3">3.3</A> <A HREF="#toc3.3">Hyperplane sections</A>
</H2>


<P>To draw one or more hyperplane sections of an algebraic surface,
just specify the hyperplane by setting the global variable
<CODE>plane</CODE> to its equation.
The section is drawn when the command <CODE>cut_with_plane</CODE> is interpreted.
For example:
<HR>
<PRE>
    rot_x=0.3;                        // a nice rotation
    rot_y=0.2;
    surface=x^2*y^2+y^2*z^2+z^2*x^2-16*x*y*z;
    clear_screen;                     // draw the steiner roman surface
    draw_surface;
    curve_red=0;
    curve_green=255;
    curve_blue=0;
    curve_width=5;
    curve_gamma=1.2;
    plane=x+y+z;                      // draw a green hyperplane section
    cut_with_plane;
    plane=x+y+z+4.0;                  // draw another one
    cut_with_plane;
</PRE>
<HR>

The color of the hyperplane section can be set by specifying
<CODE>curve_red</CODE>, <CODE>curve_green</CODE> and <CODE>curve_blue</CODE>.
The width of the section is altered by setting <CODE>curve_width</CODE>
to any suitable value. A high value of <CODE>curve_gamma</CODE>
(eg. 10.0) makes the curve
look very pixelized, whereas a small value (eg. 1.0) makes the section
look blurred.</P>

<H2><A NAME="ss3.4">3.4</A> <A HREF="#toc3.4">Multiple curves/surfaces</A>
</H2>


<P>Multiple curves can be drawn in script files just by <EM>NOT</EM> clearing
the screen. This works fine for plane curves.
Just consider the following example:
<HR>
<PRE>
    do_background=yes;
    clear_screen;
    curve=y^2-x^2*(x-1);
    draw_curve;             // draw a cubic
    do_background=no;
    curve=x;
    draw_curve;             // draw y-axis
    curve=y;
    draw_curve;             // draw y-axis
</PRE>
<HR>

Not that every curve will be drawn just over all curves that have been
draw so far.</P>
<P>Multiple surfaces can be drawn by specifying up to 9 surfaces
in the variables <CODE>surface</CODE>, <CODE>surface2</CODE> ...
<CODE>surface9</CODE>. Additionally it is possible to draw on every surface
any number of hyperplane sections.
<HR>
<PRE>
    rot_x=0.69;                      // a nice rotation
    rot_y=0.35;
    illumination=ambient_light +     // specify illumination 
                 diffuse_light +     // model
                 reflected_light +
                 transmitted_light;
    transparence=35;                 // set transparence for surface no 1
    transparence2=35;                // set transparence for surface no 2
    surface=x^2+y^2+z^2-30;          // first surface: a sphere
    surface2_red=255;                // second surface: a red steiner surface
    surface2_green=0;
    surface2_blue=0;
    surface2=x^2*y^2+x^2*z^2+y^2*z^2-16*x*y*z;
    clear_screen;
    draw_surface;                    // draw the surface
    curve_width=5;
    curve_red=0;
    curve_green=255;
    curve_blue=0;
    plane=x+y+z-6.0;                 // draw a green hyperplane section
    surf_nr=1;                       // on the sphere
    cut_with_plane;
    curve_red=0;
    curve_green=255;
    curve_blue=255;
    plane=x+y+z+4.0;                 // draw a turquoise hyperplane section
    surf_nr=2;                       // on the steiner surface
    cut_with_plane;
</PRE>
<HR>
</P>


<H2><A NAME="ss3.5">3.5</A> <A HREF="#toc3.5">Graphs and isolines</A>
</H2>


<P>Given a polynomial function <CODE>f(x,y)</CODE> and a set of levels
<CODE>z1</CODE>, ... ,<CODE>zn</CODE>,  <B>surf</B> can visualize the graph
<CODE>z=f(x,y)</CODE> and all isoline for the levels
<CODE>z1</CODE>, ... ,<CODE>zn</CODE> as follows:
<HR>
<PRE>
    rot_x=-0.8;
    clear_screen;
    poly f=x^2+y^2; // graph of (x,y)->x^2+y^2
    surface=z-f;
    draw_surface;      // draw the graph
    curve_width=3;     // width of isoline
    plane=z-1;
    cut_with_plane;    // draw isoline f(x,y)=1
    plane=z-2;
    cut_with_plane;    // draw isoline f(x,y)=2  
    plane=z-3;
    cut_with_plane;    // draw isoline f(x,y)=3
    plane=z-4;
    cut_with_plane;    // draw isoline f(x,y)=4
    plane=z-5;
    cut_with_plane;    // draw isoline f(x,y)=5
    plane=z-6;
    cut_with_plane;    // draw isoline f(x,y)=6
    plane=z-7;
    cut_with_plane;    // draw isoline f(x,y)=7
    plane=z-8;
    cut_with_plane;    // draw isoline f(x,y)=8
    plane=z-9;
    cut_with_plane;    // draw isoline f(x,y)=9
</PRE>
<HR>

If however your function  <CODE>f</CODE>  is not polynomial, try to expand
calculate its Taylor series. Since the new root algorithms work fine
with polynomials of degree up to 30, you might approximate <CODE>f</CODE>
by its Taylor series. If your function is piecewise defined, better
use another program.</P>

<H2><A NAME="ss3.6">3.6</A> <A HREF="#toc3.6">Interactive positioning</A>
</H2>

<P>The <EM>position</EM> window provides an interface to adjust the
curve/surface position. You can set the 9 buttons into the
three modes <EM>translate</EM>, <EM>rotate</EM> and 
<EM>scale</EM>.</P>

<H2><A NAME="ss3.7">3.7</A> <A HREF="#toc3.7">Preview</A>
</H2>


<P>If you try to draw a surface and give the equation to <EM>surf</EM>,
the resulting image normally does not look nice at all.
You have to find the right scaling, rotation and so on.
Often you want to see immediately what happens if you change
some value. But it simply takes <EM>surf</EM> too long
to calculate one image. Here comes the preview in.
Setting the preview buttons in the <EM>main window</EM> to
<EM>3x3</EM> has the effect that only every 9th pixel
is calculated, setting it to <EM>9x9</EM> only every
81st pixel is calculated. But one can still get an impression
of what the image looks like, AND computation is speeded up
by the factor 9 resp. 81.</P>
<P>Up to two preview buttons can be pressed at one time.
If for example <EM>9x9</EM> and <EM>1x1</EM> are pressed,
then the image will be calculated in three steps.
First, every 81st pixel, after that every 9th pixel
and finally every pixel will be calculated.</P>

<H2><A NAME="ss3.8">3.8</A> <A HREF="#toc3.8">Anti aliasing surfaces</A>
</H2>


<P>Especially in animations aliasing is very disturbing. Therefore if
in the <EM>display window</EM>,
<EM>antialiasing level</EM> is set to a value n &gt; 1,
then in a second pass all
pixels differing by a value of at least <EM>antialiasing threshold</EM>
from one of their neighbours are refined. Exactly n^2+1 intensity
values are calculated. In most cases an antialiasing level of
4 will remove aliasing.</P>

<H2><A NAME="ss3.9">3.9</A> <A HREF="#toc3.9">Animations</A>
</H2>


<P>On a nifty machine <EM>surf</EM> is fast enough to provide a real time
animation of an algebraic curve of degree &lt; 5. For example
<HR>
<PRE>
    // --------------------------
    // animation of a cubic curve
    // --------------------------
    clear_screen;
    double a=-10.0;
    loop:
        curve=y^2-(x^2-1)*(x-a);
        clear_pixmap;
        draw_curve;
        a=a+0.1;
    if( a &lt;= 10.0 ) goto loop;
</PRE>
<HR>

calculates some 200 curves. In a 200x200 window, <EM>surf</EM> shows me about
five frames per second on a sparc 20.
However, real time animations of algebraic surfaces are still
beyond computation power (or do you call a 200-processor-machine your own?).
But you can calculate a series of images with <EM>surf</EM> and convert this
series of images to the movie format of your choice.
<HR>
<PRE>
    // --------------------------
    // the 4-nodal cubic rotating
    // --------------------------
    width=200;
    height=200;                  // set image size
    double sf=0.3;
    scale_x=sf;
    scale_y=sf;
    scale_z=sf;                  // set scaling
    double Pi=2*arccos(0);
    double w2=sqrt(2);           // define some constants
    poly p=1-z-w2*x;
    poly q=1-z+w2*x;
    poly r=1+z+w2*y;
    poly s=1+z-w2*y;             // define tetrahedral coordinates
    poly cubic=4*(p^3+q^3+r^3+s^3)-(p+q+r+s)^3; // the cubic
    int i=0;
    loop:
        surface=rotate(cubic,2*Pi/100*i,zAxis); // rotate the cubic
        clear_screen;
        draw_surface;                           // draw the cubic
        filename="cubic"+itostrn(3,i)+".ras";
        save_color_image;                       // save the image
        i=i+1;
    if( i &lt; 100 ) goto loop;                    // repeat 100 times
</PRE>
<HR>

Here some 100 SUN rasterfiles are created.  Afterwards you could use
some tool to convert these single images to a movie.</P>


<H2><A NAME="ss3.10">3.10</A> <A HREF="#toc3.10">Stereo pictures</A>
</H2>

<P>Have you ever watched one of those films  with that red and green
glasses? <EM>surf</EM> tries to accomplish exactly this effect
when you set <EM>eye distance</EM> in the
<EM>display window</EM> to a value greater than zero.
The following situation is simulated: The spectator is located
at (0,0,<EM>spectator z</EM>) and the distance between his
eyes is <EM>eye distance</EM>. The surface will appear at
the z-coordinate <EM>distance from screen</EM>.
Furthermore it is possible to adjust to specific
red-green or red-blue glasses by setting
<EM>left eye red value</EM>, <EM>right eye green value</EM> end
<EM>right eye blue value</EM>. In particular it is assumed that
the right eye wears the red glass.</P>

<H2><A NAME="ss3.11">3.11</A> <A HREF="#toc3.11">Black &amp; white images</A>
</H2>


<P>If a color image of a surface/curve has been calculated, this image can
be mapped to a black and white image by pressing the button
<EM>dither surface</EM> or <EM>dither curve</EM>.
The second one is just designed for dithering curves.
The appearance of the black and white image can be altered/adjusted in several
ways in the <EM>dither window</EM>.
Since the mapping itself is done by dithering, the dithering algorithm
can be specified. Currently available are seven algorithms coming in three
groups:</P>

<H3>Dithering with blue noise</H3>

<P>
<UL>
<LI>Floyd-Steinberg filter</LI>
<LI>Jarvis, Judis and Ninke filter</LI>
<LI>Stucki filter</LI>
</UL>

All three filters are based on the same idea of error
distribution. Floyd Steinberg is the simplest one, whereas Stucki
differs from Jarvis only by its weights.
They tend to produce disturbing patterns if they process large
areas of intensity near 0.5. Therefore one can let them proceed
in a serpentine fashion, which reduces the patterns. Nearly all patterns
disappear if the weights are disturbed randomly.
The algorithms are best for use with low resolution printers,
typically 300 dpi. Some (most?) 600 dpi laser printers do not like these
algorithms, since they do not like isolated pixels.</P>

<H3>Dithering with ordered dither</H3>

<P>
<UL>
<LI>Clustered dot ordered dither</LI>
<LI>Dispersed dot ordered dither</LI>
</UL>

The clustered dot ordered dither is a fast method and produces
satisfying results in combination with high resolution printers
(600 dpi and more). The second algorithm is for use with low
resolution printers. Both perform  no error distribution.
Depending on the printer resolution and the number of emulated
gray levels, one can choose the pattern size:
<UL>
<LI>4 x 4 pixels: 16 gray levels,</LI>
<LI>8 x 8 pixels: 64 gray levels or</LI>
<LI>16 x 16 pixels: 256 gray levels.</LI>
</UL>
</P>

<H3>Hybrid methods</H3>

<P>
<UL>
<LI>Knuth's dot diffusion</LI>
<LI>Knuth's smooth dot diffusion</LI>
</UL>

Both algorithms combine clustered dot ordered dither and error
distribution. Depending on the printer resolution on can choose
the number of barons in a 8x8 matrix to be
<UL>
<LI>1 for resolutions of 1200 dpi or above or</LI>
<LI>2 for resolutions of 600 dpi or above.</LI>
</UL>

The barons are the bad guys in a matrix which get all the
error left over from the good guys.</P>
<H3>The black &amp; white problem</H3>


<P>The surfaces on black and white images often don't look very impressive;
often it is hard to detect the edges of a surface. An algorithm called
enhancing the edges avoids this drawback. This algorithm takes a value
<EM>alpha</EM> in [0,1] as input.
Best results are achieved with alpha around 0.9.</P>
<P>The intensity of the background on the black and white image can be
specified by altering the value <EM>background</EM> to any value
in [0.1]. Here 0 is black whereas 1 means white.</P>
<P>The <EM>tone scale adjustment</EM> maps intensity values
between 0 and 0.1 to 0, values between 0.1 and 0.9 linear to [0,1]
and values between 0.9 and 1 to 1.
This is used to enhance the contrast of an image. An additional gamma
correction can be also performed to correct the linearity of an output
device.</P>
<P>By specifying <EM>pixel size</EM> one can correct the printer pixel size:
A value of 50 means that the radius of a pixel is exactly
half the distance between two neighbouring pixels. A value of 100
says that the radius of a pixel is exactly the distance between two
neighbouring pixels.</P>

<H2><A NAME="ss3.12">3.12</A> <A HREF="#toc3.12">Algorithms</A>
</H2>


<P>The heart of <EM>surf</EM> is an algorithm which determines all
roots of a polynomial in one variable. Currently you can choose between
seven methods in the <EM>numeric window</EM>.
The first six methods use a chain of derivatives
to determine intervals where the polynomial has exactly one root.
They differ by the iteration method which is used to find the roots
in these intervals. Some of the iteration methods were just implemented
out of academic interest. However, they all work. The last method uses
Rockwoods all roots algorithm: the polynomial is converted into a
bezier function and the roots of the bezier function are approximated by
the roots of the control polygon.</P>
<P>For curves/surfaces of degree less than ten, all methods work.
When the degree gets higher, best results are achieved by
the bisection, the Newton and the bezier all roots method. At last, for
a degree higher than 30 only the bisection methods seems to work (up
to degree 50). If a curve has multiple components, the bisection
and the Newton method tend to produce the best results.</P>
<P>Moreover it is possible to specify a numerical precision
<EM>epsilon</EM> which is used in all root finders.
Additionally the maximal number of <EM>iterations</EM> of the
iteration methods can be specified.</P>

<H2><A NAME="ss3.13">3.13</A> <A HREF="#toc3.13">Output</A>
</H2>


<P><EM>surf</EM> can store color images in one of several
file formats. In the <EM>save color image window</EM>
you can choose between
<UL>
<LI><B>XWD</B> (X Window Dump),</LI>
<LI><B>SUN rasterfile</B>,</LI>
<LI><B>PPM</B> (Portable PixMap) and</LI>
<LI><B>JPEG</B>.</LI>
</UL>

Additionally the color space can be chosen among
<UL>
<LI>Netscape 216 color cube (8 bit),</LI>
<LI>optimized by an octree algorithm (8 bit) and</LI>
<LI>True color (24 bit).</LI>
</UL>

The first colormap is just the 6x6x6 colormap Netscape uses.
The second one results from an octree algorithm which chooses
the most used 216 colors among all colors of the image.
Storing an image in True color results in better quality,
but bigger file size.</P>
<P><EM>surf</EM> can store black and white images
in different file formats. We have implemented
<UL>
<LI><B>Postscript</B>,</LI>
<LI><B>EPS</B> (Encapsulated Postscript),</LI>
<LI><B>TIFF</B>,</LI>
<LI><B>XBM</B>,</LI>
<LI><B>PGM</B> and</LI>
<LI><B>PBM</B>.</LI>
</UL>

For postscript and encapsulated postscript also
the resolution may be specified among
<UL>
<LI>75 dpi,</LI>
<LI>100 dpi,</LI>
<LI>150 dpi,</LI>
<LI>300 dpi,</LI>
<LI>600 dpi and</LI>
<LI>1200 dpi.</LI>
</UL>

These settings may be chosen in the <EM>save dithered image window</EM>.
When using postscript, the image will (regardless its size)
appear centred on the side (which is assumed to be a4).</P>

<H2><A NAME="s4">4.</A> <A HREF="#toc4">List of all reserved words</A></H2>


<P>A reserved word in <EM>surf</EM>'s language is either a command or a global
variable. A command is invoked mostly without parameters. Global variables
are either constant or may be altered. The commands correspond to
pushbuttons of <EM>surf</EM>'s GUI, global variables correspond to other panel
items.</P>


<H2><A NAME="ss4.1">4.1</A> <A HREF="#toc4.1">Reserved words corresponding to the main window</A>
</H2>

<P>
<PRE>
    res. word           | type    | description
    --------------------------------------------------------------------------
    clear_screen        | command | erase the image
    clear_pixmap        | command | erase the image in memory (useful for
                        |         | real-time-animations of algebraic curves)
    draw_curve          | command | draw the curve defined by the global
                        |         | polynomial curve
    draw_surface        | command | draw the surfaces defined by the global
                        |         | polynomials surface, surface2, ...
    cut_with_plane      | command | draw the hyperplane section defined
                        |         | defined by the linear polynomial plane
    dither_surface      | command | convert color image to a dithered
                        |         | black and white image
    dither_curve        | command | convert color image to a dithered
                        |         | black and white image (for curves only)
    save_color_image    | command | save color image in file defined by the
                        |         | global string filename
    save_dithered_image | command | save dithered black and white image in
                        |         | file defined by the global string filename
    set_size            | command | not needed any more (still there for
                        |         | compatibility issues)
</PRE>
</P>
<P>
<PRE>
    res. word | type    | range         | default | description
    -------------------------------------------------------------------------
    curve     | poly    |           any |       0 | polynomial of curve
    surface   | poly    |           any |       0 | polynomial of surface
    surface2  | poly    |           any |       0 | polynomial of surface2
    ...       | ...     |           ... |     ... | ...
    surface9  | poly    |           any |       0 | polynomial of surface9
    plane     | poly    |        linear |       0 | equation of hyperplane
    width     | int     | {64,...,3000} |     200 | width of surface image
    height    | int     | {64,...,3000} |     200 | height of surface image
    filename  | string  |           any |      "" | filename used in 
              |         |               |         | save_color_image,
              |         |               |         | save_dithered_image
    surf_nr   | int     |     {1,...,9} |       1 | surface which is used
              |         |               |         | for cut_with_plane
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    width=400;                // Set image width
    height=300;               //  and height
    surface=x^2+y^2+z^2-81;   // Set global variable surface to a sphere
    draw_surface;             // Draw the sphere onto the screen
    plane=x+y+z;              // Choose a hyperplane
    cut_with_plane;           // Draw the hyperplane section
    filename="sphere.ras";
    save_color_image;         // Save the color image in file sphere.xwd
    dither_surface;           // Perform dithering on the color image
    filename="sphere.ps";
    save_dithered_image;      // Save the dithered image in sphere.ps
</PRE>
<HR>
</P>

<H2><A NAME="ss4.2">4.2</A> <A HREF="#toc4.2">Reserved words corresponding to the position window</A>
</H2>


<P>
<PRE>
    res. word    | type   | range        | def. | description
    -------------------------------------------------------------------------
    origin_x     | double | ]-9999,9999[ |    0 |  \
    origin_y     | double | ]-9999,9999[ |    0 |   > position of origin 
    origin_z     | double | ]-9999,9999[ |    0 |  / 
    spec_z       | double |     ]0,9999[ |  100 | spectator dist. from origin
    rot_x        | double | ]-9999,9999[ |    0 | rotation angle of surface
                 |        |              |      | about the x-axis
    rot_y        | double | ]-9999,9999[ |    0 | rotation of surface
                 |        |              |      | about the y-axis
    rot_z        | double | ]-9999,9999[ |    0 | rotation of surface
                 |        |              |      | about the z-axis
    scale_x      | double | ]-9999,9999[ |    1 | ratio surface is scaled in
                 |        |              |      | direction of the x-axis
    scale_y      | double | ]-9999,9999[ |    1 | ratio surface is scaled in
                 |        |              |      | direction of the y-axis
    scale_z      | double | ]-9999,9999[ |    1 | ratio surface is scaled in
                 |        |              |      | direction of the z-axis
    perspective  |    int |        {0,1} |    0 | perspective to use
    parallel     |    int |            0 |    0 | constant
    central      |    int |            1 |    1 | constant
    first        |    int |      {0,1,2} |    0 | first performed  \
    second       |    int |      {0,1,2} |    1 | second performed  > action
    third        |    int |      {0,1,2} |    2 | third performed  /
    translate    |    int |            0 |    0 | constant
    rotate       |    int |            1 |    1 | constant
    scale        |    int |            2 |    2 | constant
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    double Pi=2*arccos(0);
    origin_x = -3;
    origin_y = -4;                   // Set origin to point (-3,-4,2)
    origin_z =  2;
    spec_z = 25;                     // Spectator is now at (-3,-4,27)
    rot_x = Pi/2;                    // Rotate 90 degrees about x-axis
    rot_y = Pi/4;                    // Rotate 45 degrees about y-axis
    rot_z = Pi;                      // Rotate 180 degrees about z-axis
    scale_x = 1.0;                   // Don't scale in x-direction
    scale_y = 1.5;                   // Shrink surface in y-direction
    scale_z = 1/2;                   // Oversize surface in z-direction
    first   = rotate;                // rotate first
    second  = scale;                 // then scale
    third   = translate;             // then translate
</PRE>
<HR>
</P>

<H2><A NAME="ss4.3">4.3</A> <A HREF="#toc4.3">Reserved words corresponding to the display window</A>
</H2>

<P>
<PRE>
    res. word              | type   | range       | def.  | description
    -------------------------------------------------------------------------
    dither_colors          | int    |    {yes,no} |   yes | color dithering
    dither_steps           | double | [5,...,255] |  20.0 | steps of dithering
    normalize              | int    |    {yes,no} |    no | normalize image
    normalize_factor       | double |   ]0,...,5] |   1.0 | multiply with
    antialiasing           | int    |    {1,..,8} |     1 | level of
                           |        |             |       | antialiasing
    antialiasing_threshold | double |       ]0,1[ |   0.1 | threshold
    antialiasing_radius    | double | [0.5,...,2] |   2.0 | radius 
    depth_cueing           | int    |    {yes,no} |    no | use depth cueing
    depth_value            | double |  [-1000,10[ | -14.0 | depth of mist
    stereo_eye             | double |  [-100,100] |   0.0 | eye distance
    stereo_z               | double |    [-30,30] |   5.0 | dist. from screen
    stereo_red             | double |       [0,1] |   1.0 | left eye red
    stereo_green           | double |       [0,1] |   0.7 | right eye green
    stereo_blue            | double |       [0,1] |   0.0 | right eye blue
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    dither_colors          = yes;  // perform color dithering
    dither_steps           = 60.0; // use soft dithering
    normalize              = yes;  
    normalize_factor       = 1.5;  // light up image
    antialiasing           = 4;    // do 4 fold antialiasing
    antialiasing_threshold = 0.05; // with a low threshold
    antialiasing_radius    = 1.5;  // and a small radius
    depth_cueing           = yes;  // perform depth cueing
    depth_value            = -11.0;// from -11 on everythin is dark
    stereo_eye             = 5.0;  // make a red-blue image
    stereo_z               = 2.0;  // object 2 units before screen
    stereo_red             = 1.0;  
    stereo_green           = 0.0;
    stereo_blue            = 1.0;
</PRE>
<HR>
</P>

<H2><A NAME="ss4.4">4.4</A> <A HREF="#toc4.4">Reserved words corresponding to the light window</A>
</H2>


<P>
<PRE>
    res. word         | cat.   | range        | def. | description
    -------------------------------------------------------------------------
    illumination      | int    |     {0,..15} |    7 | illumination model
    ambient_light     | int    |            1 |    1 | constant
    diffuse_light     | int    |            2 |    2 | constant
    reflected_light   | int    |            4 |    4 | constant
    transmitted_light | int    |            8 |    8 | constant
    surface_red       | int    |  {0,...,255} |  123 | \  outside
    surface_green     | int    |  {0,...,255} |  104 |  > color of surface
    surface_blue      | int    |  {0,...,255} |  238 | /  (medium slate blue)
    inside_red        | int    |  {0,...,255} |  230 | \  inside
    inside_green      | int    |  {0,...,255} |  180 |  > color of surface
    inside_blue       | int    |  {0,...,255} |   30 | /  (golden)
    surface2_red      | int    |  {0,...,255} |  123 | \  outside
    surface2_green    | int    |  {0,...,255} |  104 |  > color of surface2
    surface2_blue     | int    |  {0,...,255} |  238 | /  (medium slate blue)
    inside2_red       | int    |  {0,...,255} |  230 | \  inside
    inside2_green     | int    |  {0,...,255} |  180 |  > color of surface2
    inside2_blue      | int    |  {0,...,255} |   30 | /  (golden)
    ...               | ...    | ...          |  ... | ...
    ...               | ...    | ...          |  ... | ...
    ...               | ...    | ...          |  ... | ...
    surface9_red      | int    |  {0,...,255} |  123 | \  outside
    surface9_green    | int    |  {0,...,255} |  104 |  > color of surface9
    surface9_blue     | int    |  {0,...,255} |  238 | /  (medium slate blue)
    inside9_red       | int    |  {0,...,255} |  230 | \  inside
    inside9_green     | int    |  {0,...,255} |  180 |  > color of surface9
    inside9_blue      | int    |  {0,...,255} |   30 | /  (golden)
    ambient           | int    |  {0,...,100} |   35 | amount of ambient light
    diffuse           | int    |  {0,...,100} |   60 | diffuse reflected light
    reflected         | int    |  {0,...,100} |   60 | specular reflected light
    transmitted       | int    |  {0,...,100} |   60 | spec. transmitted light
    smoothness        | int    |  {0,...,100} |   13 | roughness of surface
    transparency      | int    |  {0,...,100} |   80 | transparency of surface
    ambient2          | int    |  {0,...,100} |   35 | amount of ambient light
    diffuse2          | int    |  {0,...,100} |   60 | diffuse reflected light
    reflected2        | int    |  {0,...,100} |   60 | specular reflected light
    transmitted2      | int    |  {0,...,100} |   60 | spec. transmitted light
    smoothness2       | int    |  {0,...,100} |   13 | roughness of surface2
    transparency2     | int    |  {0,...,100} |   80 | transparency of surface2
    ...               | ...    | ...          |  ... | ...
    ...               | ...    | ...          |  ... | ...
    ...               | ...    | ...          |  ... | ...
    ambient9          | int    |  {0,...,100} |   35 | amount of ambient light
    diffuse9          | int    |  {0,...,100} |   60 | diffuse reflected light
    reflected9        | int    |  {0,...,100} |   60 | specular reflected light
    transmitted9      | int    |  {0,...,100} |   60 | spec. transmitted light
    smoothness9       | int    |  {0,...,100} |   13 | roughness of surface9
    transparency9     | int    |  {0,...,100} |   80 | transparency of surface9
    light1_x          | double | [-9999,9999] | -100 | \
    light1_y          | double | [-9999,9999] |  100 |  \ position and volume
    light1_z          | double | [-9999,9999] |  100 |  / of the first light
    light1_vol        | int    | {0,...,100}  |   50 | /  source
    light1_red        | int    |   {0,..,255} |  255 | \
    light1_green      | int    |   {0,..,255} |  255 |  > color of first
    light1_blue       | int    |   {0,..,255} |  255 | /  light source
    light2_x          | double | [-9999,9999] |    0 | \
    light2_y          | double | [-9999,9999] |  100 |  \ position and volume
    light2_z          | double | [-9999,9999] |  100 |  / of the second light
    light2_vol        | int    | {0,...,100}  |    0 | /  source
    light2_red        | int    |   {0,..,255} |  255 | \
    light2_green      | int    |   {0,..,255} |  255 |  > color of second
    light2_blue       | int    |   {0,..,255} |  255 | /  light source
    ...               | ...    | ...          |  ... | ...
    ...               | ...    | ...          |  ... | ...
    ...               | ...    | ...          |  ... | ...
    light9_x          | double | [-9999,9999] |  100 | \
    light9_y          | double | [-9999,9999] | -100 |  \ position and volume
    light9_z          | double | [-9999,9999] |  100 |  / of the ninteh light
    light9_vol        | int    | {0,...,100}  |    0 | /  source
    light9_red        | int    |   {0,..,255} |  255 | \
    light9_green      | int    |   {0,..,255} |  255 |  > color of nineth
    light9_blue       | int    |   {0,..,255} |  255 | /  light source
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    illumination = ambient_light
                 + diffuse_light
                 + reflected_light 
                 + transmitted_light; // Select illumination
    surface_red   = 205;
    surface_green = 92;
    surface_blue  = 92;               // Select indian red for surface outside
    inside_red    = surface_red;
    inside_green  = surface_green;
    inside_blue   = surface_blue;     // Select indian red for surface inside
    ambient      = 10;                // 40% ambient light
    diffuse      = 60;                // 60% diffuse light
    reflected    = 60;                // 60% reflected light
    transmitted  = 70;                // 60% reflected light
    smoothness   = 50;                // make surface shiny
    transparence = 90;                // very transparent
    thickness    = 20;                // but also very thick
    light2_x = 100;
    light2_y = 0;
    light2_z = 200;
    light2_volume = 100;              // turn on light no. 2 red at (100,0,200)
    light2_red   = 255;
    light2_green = 0;
    light2_blue  = 0;
</PRE>
<HR>
</P>

<H2><A NAME="ss4.5">4.5</A> <A HREF="#toc4.5">Reserved words corresponding to the clip window</A>
</H2>


<P>
<PRE>
    reserved word  | cat.   | range        | def. | description
    -------------------------------------------------------------------------
    clip           | int    |    {0,...,5} |    0 | clipping area
    ball           | int    |            0 |    0 | constant
    cylinder_xaxis | int    |            1 |    1 | constant
    cylinder_yaxis | int    |            2 |    2 | constant
    cylinder_zaxis | int    |            3 |    3 | constant
    cube           | int    |            4 |    4 | constant
    none           | int    |            5 |    5 | constant
    clip_front     | double | [-9999,9999] |   10 | \ additional clip region
    clip_back      | double | [-9999,9999] |  -10 | /
    radius         | double |     ]0,9999] |   10 | radius of clip region
    center_x       | double | [-9999,9999] |    0 | \
    center_x       | double | [-9999,9999] |    0 |  > center of clip region
    center_x       | double | [-9999,9999] |    0 | /
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    clip = cube;
    radius = 7;
    center_x = -3;           // Set clipping area to cube with center at
    center_y = 2;            // (-3,2,1) and edge length 14
    center_z = 1;
    clip_front = 4;          // Clip off points with z > 4
    clip_back = -10;         // Clip off points with z > -10
</PRE>
<HR>
</P>


<H2><A NAME="ss4.6">4.6</A> <A HREF="#toc4.6">Reserved words corresponding to the dither window</A>
</H2>


<P>
<PRE>
    reserved word                | cat.   | range     | def. | description
    -------------------------------------------------------------------------
    dithering_method             | int    | {0,...,6} |    1 | dithering method
    floyd_steinberg_filter       | int    |         0 |    0 | constant
    jarvis_judis_ninke_filter    | int    |         1 |    1 | constant
    stucki_filter                | int    |         2 |    2 | constant
    clustered_dot_ordered_dither | int    |         3 |    3 | constant
    dispersed_dot_ordered_dither | int    |         4 |    4 | constant
    dot_diffusion                | int    |         5 |    5 | constant
    smooth_dot_diffusion         | int    |         6 |    6 | constant
</PRE>
</P>
<P>
<PRE>
    reserved word     | cat.   | range     | def. | description
    -------------------------------------------------------------------------
    serpentine_raster | int    |  {yes,no} |  yes | use of serpentine raster
    random_weights    | int    |  {yes,no} |  yes | use of random weights
    weight            | double |     [0,1] |  0.5 | amount of random weights
    barons            | int    |     {0,1} |    1 | number of barons
    one_baron         | int    |         0 |    0 | constant
    two_baron         | int    |         1 |    1 | constant
    pattern_size      | int    |   {0,1,2} |    1 | size of dithering tile
    pattern_4x4       | int    |         0 |    0 | constant
    pattern_8x8       | int    |         1 |    1 | constant
    pattern_16x16     | int    |         2 |    2 | constant
    enhance_edges     | int    |  {yes,no} |  yes | enhance edges of b w image
    alpha             | double |     [0,1] |  0.9 | filter coefficient used in
                      |        |           |      | for enhancing the edges
    background        | double |     [0,1] |  1.0 | background intensity of
                      |        |           |      | b w image
tone_scale_adjustment | int    |  {yes,no} |  yes | perform tone scale adjust.
    gamma             | double |    ]0,oo[ |  1.3 | gamma-correction
    pixel_size        | int    |  ]50,100] |   73 | correction for printers
                      |        |           |      | that produce too fat pixels
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    dithering_method = stucki_filter;    // select stucki filter
    serpentine_raster = yes;             // turn on serpentine raster
    random_weights = yes;                // turn on random weights
    weight = 0.5;                        // select 50% weights
    enhance_edges = yes;                 // turn on enhancing edges
    alpha = 0.8;                         // edges less visible than default
    background = 0.5;                    // gray background for b w image
    tone_scale_adjustment = yes;         // perform tone scale adjustment
    gamma = 1.5;                         // more gamma-correction than default

    dithering_method = dispersed_dot;    // select dispersed dot ordered dither
    pattern_size = pattern_16x16;        // select a 16x16-tile

    dithering_method = dot_diffusion;    // select dot-diffusion
    barons = two_barons;                 // select a 2-barons tile
</PRE>
<HR>
</P>

<H2><A NAME="ss4.7">4.7</A> <A HREF="#toc4.7">Reserved words corresponding to the save color image window</A>
</H2>

<P>
<PRE>
    reserved word       | type | range   | def. | description
    -------------------------------------------------------------------------
    color_file_format   | int  |   {0,1} |    1 | file format
    xwd                 | int  |       0 |    0 | constant
    sun                 | int  |       1 |    1 | constant
    color_file_colormap | int  | {0,1,2} |    0 | colormap type
    netscape            | int  |       0 |    0 | constant
    optimized           | int  |       1 |    1 | constant
    truecolor           | int  |       2 |    2 | constant
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    color_file_format   = xwd;
    color_file_colormap = truecolor; // format is 24 bit XWD
</PRE>
<HR>
</P>

<H2><A NAME="ss4.8">4.8</A> <A HREF="#toc4.8">Reserved words corresponding to the save dithered image window</A>
</H2>

<P>
<PRE>
    reserved word        | type | range     | def. | description
    -------------------------------------------------------------------------
    resolution           | int  | {0,...,5} |    3 | (printer) resolution
    res_75dpi            | int  |         0 |    0 | constant
    res_100dpi           | int  |         1 |    1 | constant
    res_150dpi           | int  |         2 |    2 | constant
    res_300dpi           | int  |         3 |    3 | constant
    res_600dpi           | int  |         4 |    4 | constant
    res_1200dpi          | int  |         5 |    5 | constant
    dithered_file_format | int  | {0,...,4} |    2 | file format
    postscript           | int  |         0 |    0 | constant
    encapsulated         | int  |         1 |    1 | constant
    xbm                  | int  |         2 |    2 | constant
    tiff                 | int  |         3 |    3 | constant
    bm2font              | int  |         4 |    4 | constant
</PRE>
</P>

<H3>Examples</H3>

<P>
<HR>
<PRE>
    resolution           = res_300dpi;     // select 300 dpi
    dithered_file_format = bm2font;        // TeX pk
</PRE>
<HR>
</P>

<H2><A NAME="ss4.9">4.9</A> <A HREF="#toc4.9">Reserved words corresponding to the numeric window</A>
</H2>


<P>
<PRE>
    reserved word               | cat.   | range     | def. | description
    ---------------------------------------------------------------------------
    root_finder                 | int    | {0,...,6} |    6 | used root finder
    d_chain_bisection           | int    |         0 |    0 | constant
    d_chain_regula_falsi        | int    |         1 |    1 | constant
    d_chain_pegasus             | int    |         2 |    2 | constant
    d_chain_illinois            | int    |         3 |    3 | constant
    d_chain_anderson_bjoerck    | int    |         4 |    4 | constant
    d_chain_newton              | int    |         5 |    5 | constant
    bezier_all_roots            | int    |         6 |    6 | constant
    epsilon                     | double |     ]0,1[ | 1e-4 | precision of
                                |        |           |      | root finder
    iterations                  | int    |  [1,2000] |  200 | max. number of
                                |        |           |      | iterations
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    root_finder = d_chain_bisection;      // Slow, but safe
    epsilon = 1.0e-7;                     // Work very precise
    iterations = 80;                      // max. 80 iterations on each root
</PRE>
<HR>
</P>


<H2><A NAME="ss4.10">4.10</A> <A HREF="#toc4.10">Reserved words corresponding to the curve window</A>
</H2>


<P>
<PRE>
    reserved word | type   | range       | def. | description
    ---------------------------------------------------------------------------
    curve_red     | int    | {0,...,255} |  255 | \
    curve_green   | int    | {0,...,255} |  255 |  > curve color
    curve_blue    | int    | {0,...,255} |  255 | /
    curve_width   | double | {1,2,...}   |    1 | width of curve
    curve_gamma   | double | ]0,oo[      |  4.0 | 
</PRE>
</P>
<H3>Examples</H3>

<P>
<HR>
<PRE>
    curve_red  =0;
    curve_green=255;
    curve_blue =0;                       // make the curve look green
    curve_width=6.0;                     // thick curve
    curve_gamma=2.0;                     // intensity increases slower
</PRE>
<HR>
</P>

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