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971 lines
50 KiB
XML
971 lines
50 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<!DOCTYPE book PUBLIC "-//OASIS//DTD DocBook MathML Module V1.1b1//EN"
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"http://www.oasis-open.org/docbook/xml/mathml/1.1CR1/dbmathml.dtd">
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<refentry id="glMap2">
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<refmeta>
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<refmetainfo>
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<copyright>
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<year>1991-2006</year>
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<holder>Silicon Graphics, Inc.</holder>
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</copyright>
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</refmetainfo>
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<refentrytitle>glMap2</refentrytitle>
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<manvolnum>3G</manvolnum>
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</refmeta>
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<refnamediv>
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<refname>glMap2</refname>
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<refpurpose>define a two-dimensional evaluator</refpurpose>
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</refnamediv>
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<refsynopsisdiv><title>C Specification</title>
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<funcsynopsis>
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<funcprototype>
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<funcdef>void <function>glMap2f</function></funcdef>
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<paramdef>GLenum <parameter>target</parameter></paramdef>
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<paramdef>GLfloat <parameter>u1</parameter></paramdef>
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<paramdef>GLfloat <parameter>u2</parameter></paramdef>
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<paramdef>GLint <parameter>ustride</parameter></paramdef>
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<paramdef>GLint <parameter>uorder</parameter></paramdef>
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<paramdef>GLfloat <parameter>v1</parameter></paramdef>
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<paramdef>GLfloat <parameter>v2</parameter></paramdef>
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<paramdef>GLint <parameter>vstride</parameter></paramdef>
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<paramdef>GLint <parameter>vorder</parameter></paramdef>
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<paramdef>const GLfloat * <parameter>points</parameter></paramdef>
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</funcprototype>
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</funcsynopsis>
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<funcsynopsis>
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<funcprototype>
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<funcdef>void <function>glMap2d</function></funcdef>
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<paramdef>GLenum <parameter>target</parameter></paramdef>
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<paramdef>GLdouble <parameter>u1</parameter></paramdef>
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<paramdef>GLdouble <parameter>u2</parameter></paramdef>
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<paramdef>GLint <parameter>ustride</parameter></paramdef>
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<paramdef>GLint <parameter>uorder</parameter></paramdef>
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<paramdef>GLdouble <parameter>v1</parameter></paramdef>
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<paramdef>GLdouble <parameter>v2</parameter></paramdef>
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<paramdef>GLint <parameter>vstride</parameter></paramdef>
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<paramdef>GLint <parameter>vorder</parameter></paramdef>
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<paramdef>const GLdouble * <parameter>points</parameter></paramdef>
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</funcprototype>
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</funcsynopsis>
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</refsynopsisdiv>
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<!-- eqn: ignoring delim $$ -->
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<refsect1 id="parameters"><title>Parameters</title>
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<variablelist>
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<varlistentry>
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<term><parameter>target</parameter></term>
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<listitem>
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<para>
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Specifies the kind of values that are generated by the evaluator.
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Symbolic constants
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<constant>GL_MAP2_VERTEX_3</constant>,
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<constant>GL_MAP2_VERTEX_4</constant>,
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<constant>GL_MAP2_INDEX</constant>,
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<constant>GL_MAP2_COLOR_4</constant>,
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<constant>GL_MAP2_NORMAL</constant>,
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<constant>GL_MAP2_TEXTURE_COORD_1</constant>,
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<constant>GL_MAP2_TEXTURE_COORD_2</constant>,
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<constant>GL_MAP2_TEXTURE_COORD_3</constant>, and
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<constant>GL_MAP2_TEXTURE_COORD_4</constant> are accepted.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><parameter>u1</parameter></term>
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<term><parameter>u2</parameter></term>
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<listitem>
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<para>
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Specify a linear mapping of
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<inlineequation><mml:math><mml:mi mathvariant="italic">u</mml:mi></mml:math></inlineequation>,
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as presented to <citerefentry><refentrytitle>glEvalCoord2</refentrytitle></citerefentry>,
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to
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<inlineequation><mml:math>
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<!-- eqn: u hat:-->
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<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
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</mml:math></inlineequation>,
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one of the two variables that are evaluated by the equations specified
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by this command. Initially, <parameter>u1</parameter> is 0 and <parameter>u2</parameter> is 1.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><parameter>ustride</parameter></term>
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<listitem>
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<para>
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Specifies the number of floats or doubles between
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the beginning of control point
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<inlineequation><mml:math>
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<!-- eqn: R sub ij:-->
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<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
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<mml:mi mathvariant="italic">ij</mml:mi>
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</mml:msub>
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</mml:math></inlineequation>
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and the beginning of control point
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<inlineequation><mml:math>
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<!-- eqn: R sub { (i+1) j }:-->
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<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
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<mml:mfenced open="" close="">
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<mml:mrow>
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<mml:mfenced open="(" close=")">
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<mml:mrow>
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<mml:mi mathvariant="italic">i</mml:mi>
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<mml:mo>+</mml:mo>
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<mml:mn>1</mml:mn>
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</mml:mrow>
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</mml:mfenced>
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<mml:mo>⁢</mml:mo>
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<mml:mi mathvariant="italic">j</mml:mi>
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</mml:mrow>
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</mml:mfenced>
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</mml:msub>
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</mml:math></inlineequation>,
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where
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<inlineequation><mml:math><mml:mi mathvariant="italic">i</mml:mi></mml:math></inlineequation>
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and
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<inlineequation><mml:math><mml:mi mathvariant="italic">j</mml:mi></mml:math></inlineequation>
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are the
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<inlineequation><mml:math><mml:mi mathvariant="italic">u</mml:mi></mml:math></inlineequation>
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and
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<inlineequation><mml:math><mml:mi mathvariant="italic">v</mml:mi></mml:math></inlineequation>
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control point indices, respectively.
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This allows control points to be embedded in arbitrary data structures.
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The only constraint is that the values for a particular control point
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must occupy contiguous memory locations. The initial value of <parameter>ustride</parameter> is 0.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><parameter>uorder</parameter></term>
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<listitem>
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<para>
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Specifies the dimension of the control point array in the
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<inlineequation><mml:math><mml:mi mathvariant="italic">u</mml:mi></mml:math></inlineequation>
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axis.
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Must be positive. The initial value is 1.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><parameter>v1</parameter></term>
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<term><parameter>v2</parameter></term>
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<listitem>
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<para>
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Specify a linear mapping of
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<inlineequation><mml:math><mml:mi mathvariant="italic">v</mml:mi></mml:math></inlineequation>,
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as presented to <citerefentry><refentrytitle>glEvalCoord2</refentrytitle></citerefentry>,
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to
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<inlineequation><mml:math>
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<!-- eqn: v hat:-->
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<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
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</mml:math></inlineequation>,
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one of the two variables that are evaluated by the equations specified
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by this command. Initially, <parameter>v1</parameter> is 0 and <parameter>v2</parameter> is 1.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><parameter>vstride</parameter></term>
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<listitem>
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<para>
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Specifies the number of floats or doubles between
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the beginning of control point
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<inlineequation><mml:math>
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<!-- eqn: R sub ij:-->
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<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
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<mml:mi mathvariant="italic">ij</mml:mi>
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</mml:msub>
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</mml:math></inlineequation>
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and the beginning of control point
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<inlineequation><mml:math>
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<!-- eqn: R sub { i (j+1) }:-->
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<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
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<mml:mfenced open="" close="">
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<mml:mrow>
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<mml:mi mathvariant="italic">i</mml:mi>
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<mml:mo>⁡</mml:mo>
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<mml:mfenced open="(" close=")">
|
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<mml:mrow>
|
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<mml:mi mathvariant="italic">j</mml:mi>
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<mml:mo>+</mml:mo>
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<mml:mn>1</mml:mn>
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</mml:mrow>
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</mml:mfenced>
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</mml:mrow>
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</mml:mfenced>
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</mml:msub>
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</mml:math></inlineequation>,
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where
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<inlineequation><mml:math><mml:mi mathvariant="italic">i</mml:mi></mml:math></inlineequation>
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and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">j</mml:mi></mml:math></inlineequation>
|
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are the
|
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<inlineequation><mml:math><mml:mi mathvariant="italic">u</mml:mi></mml:math></inlineequation>
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and
|
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<inlineequation><mml:math><mml:mi mathvariant="italic">v</mml:mi></mml:math></inlineequation>
|
|
control point indices, respectively.
|
|
This allows control points to be embedded in arbitrary data structures.
|
|
The only constraint is that the values for a particular control point
|
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must occupy contiguous memory locations. The initial value of <parameter>vstride</parameter> is 0.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><parameter>vorder</parameter></term>
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<listitem>
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<para>
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Specifies the dimension of the control point array in the
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<inlineequation><mml:math><mml:mi mathvariant="italic">v</mml:mi></mml:math></inlineequation>
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axis.
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Must be positive. The initial value is 1.
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</para>
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</listitem>
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</varlistentry>
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<varlistentry>
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<term><parameter>points</parameter></term>
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<listitem>
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<para>
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Specifies a pointer to the array of control points.
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</para>
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</listitem>
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</varlistentry>
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</variablelist>
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</refsect1>
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<refsect1 id="description"><title>Description</title>
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<para>
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Evaluators provide a way to use polynomial or rational polynomial mapping
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to produce vertices,
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normals,
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texture coordinates,
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and colors.
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The values produced by an evaluator are sent on to further stages
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of GL processing just as if they had been presented using
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<citerefentry><refentrytitle>glVertex</refentrytitle></citerefentry>,
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<citerefentry><refentrytitle>glNormal</refentrytitle></citerefentry>,
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<citerefentry><refentrytitle>glTexCoord</refentrytitle></citerefentry>, and
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<citerefentry><refentrytitle>glColor</refentrytitle></citerefentry> commands,
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except that the generated values do not update the current normal,
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texture coordinates,
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or color.
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</para>
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<para>
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All polynomial or rational polynomial splines of any degree
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(up to the maximum degree supported by the GL implementation)
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can be described using evaluators.
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These include almost all surfaces used in computer graphics,
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including B-spline surfaces,
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NURBS surfaces,
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Bezier surfaces, and so on.
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</para>
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<para>
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Evaluators define surfaces based on bivariate Bernstein polynomials.
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Define
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<inlineequation><mml:math>
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<!-- eqn: p(u hat, v hat):-->
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<mml:mrow>
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<mml:mi mathvariant="italic">p</mml:mi>
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<mml:mo>⁡</mml:mo>
|
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<mml:mfenced open="(" close=")">
|
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<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
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<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
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</mml:mfenced>
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</mml:mrow>
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</mml:math></inlineequation>
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as
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</para>
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<para>
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<informalequation><mml:math>
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<!-- eqn: p(u hat, v hat) = sum from {i = 0} to {n} { sum from {j = 0} to {m} { {B sub i} sup n ( u hat ) {B sub j} sup m ( v hat ) R sub ij } }:-->
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<mml:mrow>
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<mml:mrow>
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<mml:mi mathvariant="italic">p</mml:mi>
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<mml:mo>⁡</mml:mo>
|
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<mml:mfenced open="(" close=")">
|
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<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
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<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
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</mml:mfenced>
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</mml:mrow>
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<mml:mo>=</mml:mo>
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<mml:mrow>
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<mml:munderover>
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<mml:mo>Σ</mml:mo>
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<mml:mrow>
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<mml:mi mathvariant="italic">i</mml:mi>
|
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<mml:mo>=</mml:mo>
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<mml:mn>0</mml:mn>
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</mml:mrow>
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<mml:mi mathvariant="italic">n</mml:mi>
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</mml:munderover>
|
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<mml:mrow>
|
|
<mml:munderover>
|
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<mml:mo>Σ</mml:mo>
|
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<mml:mrow>
|
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<mml:mi mathvariant="italic">j</mml:mi>
|
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<mml:mo>=</mml:mo>
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<mml:mn>0</mml:mn>
|
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</mml:mrow>
|
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<mml:mi mathvariant="italic">m</mml:mi>
|
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</mml:munderover>
|
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<mml:mrow>
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<mml:mrow>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:msub><mml:mi mathvariant="italic">B</mml:mi>
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<mml:mi mathvariant="italic">i</mml:mi>
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</mml:msub>
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</mml:mfenced>
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<mml:mi mathvariant="italic">n</mml:mi>
|
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</mml:msup>
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<mml:mo>⁡</mml:mo>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:mrow>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:msub><mml:mi mathvariant="italic">B</mml:mi>
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<mml:mi mathvariant="italic">j</mml:mi>
|
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</mml:msub>
|
|
</mml:mfenced>
|
|
<mml:mi mathvariant="italic">m</mml:mi>
|
|
</mml:msup>
|
|
<mml:mo>⁡</mml:mo>
|
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<mml:mfenced open="(" close=")">
|
|
<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
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</mml:mfenced>
|
|
</mml:mrow>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
|
|
<mml:mi mathvariant="italic">ij</mml:mi>
|
|
</mml:msub>
|
|
</mml:mrow>
|
|
</mml:mrow>
|
|
</mml:mrow>
|
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</mml:mrow>
|
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</mml:math></informalequation>
|
|
</para>
|
|
<para>
|
|
</para>
|
|
<para>
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where
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: R sub ij:-->
|
|
<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
|
|
<mml:mi mathvariant="italic">ij</mml:mi>
|
|
</mml:msub>
|
|
</mml:math></inlineequation>
|
|
is a control point,
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: {B sub i} sup n ( u hat ):-->
|
|
<mml:mrow>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:msub><mml:mi mathvariant="italic">B</mml:mi>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
</mml:msub>
|
|
</mml:mfenced>
|
|
<mml:mi mathvariant="italic">n</mml:mi>
|
|
</mml:msup>
|
|
<mml:mo>⁡</mml:mo>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>
|
|
is the
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">i</mml:mi></mml:math></inlineequation>th
|
|
Bernstein polynomial of degree
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">n</mml:mi></mml:math></inlineequation>
|
|
(<parameter>uorder</parameter> =
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: n + 1:-->
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">n</mml:mi>
|
|
<mml:mo>+</mml:mo>
|
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<mml:mn>1</mml:mn>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>)
|
|
</para>
|
|
<para>
|
|
<informalequation><mml:math>
|
|
<!-- eqn: {B sub i} sup n ( u hat ) = left ( {cpile { n above i }} right) {u hat} sup i { ( 1 - u hat ) sup { n - i } }:-->
|
|
<mml:mrow>
|
|
<mml:mrow>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:msub><mml:mi mathvariant="italic">B</mml:mi>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
</mml:msub>
|
|
</mml:mfenced>
|
|
<mml:mi mathvariant="italic">n</mml:mi>
|
|
</mml:msup>
|
|
<mml:mo>⁡</mml:mo>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
<mml:mo>=</mml:mo>
|
|
<mml:mrow>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mfenced open="" close="">
|
|
<mml:mtable>
|
|
<mml:mtr><mml:mtd>
|
|
<mml:mi mathvariant="italic">n</mml:mi>
|
|
</mml:mtd></mml:mtr>
|
|
<mml:mtr><mml:mtd>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
</mml:mtd></mml:mtr>
|
|
</mml:mtable>
|
|
</mml:mfenced>
|
|
</mml:mfenced>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mfenced>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
</mml:msup>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:mfenced open="" close="">
|
|
<mml:msup><mml:mfenced open="(" close=")">
|
|
<mml:mrow>
|
|
<mml:mn>1</mml:mn>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">n</mml:mi>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:msup>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
</mml:mrow>
|
|
</mml:math></informalequation>
|
|
</para>
|
|
<para>
|
|
and
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: {B sub j} sup m ( v hat ):-->
|
|
<mml:mrow>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:msub><mml:mi mathvariant="italic">B</mml:mi>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:msub>
|
|
</mml:mfenced>
|
|
<mml:mi mathvariant="italic">m</mml:mi>
|
|
</mml:msup>
|
|
<mml:mo>⁡</mml:mo>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>
|
|
is the
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">j</mml:mi></mml:math></inlineequation>th
|
|
Bernstein polynomial of degree
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">m</mml:mi></mml:math></inlineequation>
|
|
(<parameter>vorder</parameter> =
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: m + 1:-->
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">m</mml:mi>
|
|
<mml:mo>+</mml:mo>
|
|
<mml:mn>1</mml:mn>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>)
|
|
</para>
|
|
<para>
|
|
<informalequation><mml:math>
|
|
<!-- eqn: {B sub j} sup m ( v hat ) = left ( {cpile { m above j }} right ) {v hat} sup j { ( 1 - v hat ) sup { m - j } }:-->
|
|
<mml:mrow>
|
|
<mml:mrow>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:msub><mml:mi mathvariant="italic">B</mml:mi>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:msub>
|
|
</mml:mfenced>
|
|
<mml:mi mathvariant="italic">m</mml:mi>
|
|
</mml:msup>
|
|
<mml:mo>⁡</mml:mo>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
<mml:mo>=</mml:mo>
|
|
<mml:mrow>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mfenced open="" close="">
|
|
<mml:mtable>
|
|
<mml:mtr><mml:mtd>
|
|
<mml:mi mathvariant="italic">m</mml:mi>
|
|
</mml:mtd></mml:mtr>
|
|
<mml:mtr><mml:mtd>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:mtd></mml:mtr>
|
|
</mml:mtable>
|
|
</mml:mfenced>
|
|
</mml:mfenced>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:msup><mml:mfenced open="" close="">
|
|
<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mfenced>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:msup>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:mfenced open="" close="">
|
|
<mml:msup><mml:mfenced open="(" close=")">
|
|
<mml:mrow>
|
|
<mml:mn>1</mml:mn>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">m</mml:mi>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:msup>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
</mml:mrow>
|
|
</mml:math></informalequation>
|
|
</para>
|
|
<para>
|
|
Recall that
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: 0 sup 0 == 1:-->
|
|
<mml:mrow>
|
|
<mml:msup><mml:mn>0</mml:mn>
|
|
<mml:mn>0</mml:mn>
|
|
</mml:msup>
|
|
<mml:mo>==</mml:mo>
|
|
<mml:mn>1</mml:mn>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>
|
|
and
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: left ( {cpile { n above 0 }} right ) == 1:-->
|
|
<mml:mrow>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mfenced open="" close="">
|
|
<mml:mtable>
|
|
<mml:mtr><mml:mtd>
|
|
<mml:mi mathvariant="italic">n</mml:mi>
|
|
</mml:mtd></mml:mtr>
|
|
<mml:mtr><mml:mtd>
|
|
<mml:mn>0</mml:mn>
|
|
</mml:mtd></mml:mtr>
|
|
</mml:mtable>
|
|
</mml:mfenced>
|
|
</mml:mfenced>
|
|
<mml:mo>==</mml:mo>
|
|
<mml:mn>1</mml:mn>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>
|
|
</para>
|
|
<para>
|
|
<function>glMap2</function> is used to define the basis and to specify what kind of values
|
|
are produced.
|
|
Once defined,
|
|
a map can be enabled and disabled by calling <citerefentry><refentrytitle>glEnable</refentrytitle></citerefentry> and <citerefentry><refentrytitle>glDisable</refentrytitle></citerefentry>
|
|
with the map name, one of the nine predefined values for <parameter>target</parameter>,
|
|
described below.
|
|
When <citerefentry><refentrytitle>glEvalCoord2</refentrytitle></citerefentry> presents values
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">u</mml:mi></mml:math></inlineequation>
|
|
and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">v</mml:mi></mml:math></inlineequation>,
|
|
the bivariate Bernstein polynomials are evaluated using
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: u hat:-->
|
|
<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:math></inlineequation>
|
|
and
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: v hat:-->
|
|
<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
</mml:math></inlineequation>,
|
|
where
|
|
</para>
|
|
<para>
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: u hat = {u - u1} over {u2 - u1}:-->
|
|
<mml:mrow>
|
|
<mml:mover><mml:mi mathvariant="italic">u</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
<mml:mo>=</mml:mo>
|
|
<mml:mfrac>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">u</mml:mi>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mi mathvariant="italic">u1</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">u2</mml:mi>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mi mathvariant="italic">u1</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:mfrac>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>
|
|
</para>
|
|
<para>
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: v hat = {v - v1} over {v2 - v1}:-->
|
|
<mml:mrow>
|
|
<mml:mover><mml:mi mathvariant="italic">v</mml:mi><mml:mo>^</mml:mo></mml:mover>
|
|
<mml:mo>=</mml:mo>
|
|
<mml:mfrac>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">v</mml:mi>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mi mathvariant="italic">v1</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">v2</mml:mi>
|
|
<mml:mo>-</mml:mo>
|
|
<mml:mi mathvariant="italic">v1</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:mfrac>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>
|
|
</para>
|
|
<para>
|
|
<parameter>target</parameter> is a symbolic constant that indicates what kind of control points
|
|
are provided in <parameter>points</parameter>,
|
|
and what output is generated when the map is evaluated.
|
|
It can assume one of nine predefined values:
|
|
</para>
|
|
<variablelist>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_VERTEX_3</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is three floating-point values representing
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">x</mml:mi></mml:math></inlineequation>,
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">y</mml:mi></mml:math></inlineequation>,
|
|
and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">z</mml:mi></mml:math></inlineequation>.
|
|
Internal <citerefentry><refentrytitle>glVertex3</refentrytitle></citerefentry> commands are generated when the map is evaluated.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_VERTEX_4</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is four floating-point values representing
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">x</mml:mi></mml:math></inlineequation>,
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">y</mml:mi></mml:math></inlineequation>,
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">z</mml:mi></mml:math></inlineequation>,
|
|
and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">w</mml:mi></mml:math></inlineequation>.
|
|
Internal <citerefentry><refentrytitle>glVertex4</refentrytitle></citerefentry> commands are generated when the map is evaluated.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_INDEX</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is a single floating-point value representing a color index.
|
|
Internal <citerefentry><refentrytitle>glIndex</refentrytitle></citerefentry> commands are generated when the map is evaluated
|
|
but the current index is not updated with the value of these
|
|
<citerefentry><refentrytitle>glIndex</refentrytitle></citerefentry> commands.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_COLOR_4</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is four floating-point values representing
|
|
red, green, blue, and alpha.
|
|
Internal <citerefentry><refentrytitle>glColor4</refentrytitle></citerefentry> commands are generated when the map is
|
|
evaluated but the current color is not updated with the value of
|
|
these <citerefentry><refentrytitle>glColor4</refentrytitle></citerefentry> commands.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_NORMAL</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is three floating-point values representing
|
|
the
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">x</mml:mi></mml:math></inlineequation>,
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">y</mml:mi></mml:math></inlineequation>,
|
|
and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">z</mml:mi></mml:math></inlineequation>
|
|
components of a normal vector.
|
|
Internal <citerefentry><refentrytitle>glNormal</refentrytitle></citerefentry> commands are generated when the map is
|
|
evaluated but the current normal is not updated with the value of
|
|
these <citerefentry><refentrytitle>glNormal</refentrytitle></citerefentry> commands.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_TEXTURE_COORD_1</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is a single floating-point value representing
|
|
the
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">s</mml:mi></mml:math></inlineequation>
|
|
texture coordinate.
|
|
Internal
|
|
<citerefentry><refentrytitle>glTexCoord1</refentrytitle></citerefentry> commands are generated when the map is evaluated but
|
|
the current texture coordinates are not updated with the value
|
|
of these <citerefentry><refentrytitle>glTexCoord</refentrytitle></citerefentry> commands.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_TEXTURE_COORD_2</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is two floating-point values representing
|
|
the
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">s</mml:mi></mml:math></inlineequation>
|
|
and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">t</mml:mi></mml:math></inlineequation>
|
|
texture coordinates.
|
|
Internal
|
|
<citerefentry><refentrytitle>glTexCoord2</refentrytitle></citerefentry> commands are generated when the map is evaluated but
|
|
the current texture coordinates are not updated with the value
|
|
of these <citerefentry><refentrytitle>glTexCoord</refentrytitle></citerefentry> commands.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_TEXTURE_COORD_3</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is three floating-point values representing
|
|
the
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">s</mml:mi></mml:math></inlineequation>,
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">t</mml:mi></mml:math></inlineequation>,
|
|
and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">r</mml:mi></mml:math></inlineequation>
|
|
texture coordinates.
|
|
Internal <citerefentry><refentrytitle>glTexCoord3</refentrytitle></citerefentry> commands are generated when the map is
|
|
evaluated but the current texture coordinates are not updated with the value
|
|
of these <citerefentry><refentrytitle>glTexCoord</refentrytitle></citerefentry> commands.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
<varlistentry>
|
|
<term><constant>GL_MAP2_TEXTURE_COORD_4</constant></term>
|
|
<listitem>
|
|
<para>
|
|
Each control point is four floating-point values representing
|
|
the
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">s</mml:mi></mml:math></inlineequation>,
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">t</mml:mi></mml:math></inlineequation>,
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">r</mml:mi></mml:math></inlineequation>,
|
|
and
|
|
<inlineequation><mml:math><mml:mi mathvariant="italic">q</mml:mi></mml:math></inlineequation>
|
|
texture coordinates.
|
|
Internal
|
|
<citerefentry><refentrytitle>glTexCoord4</refentrytitle></citerefentry> commands are generated when the map is evaluated but the current texture coordinates are not updated with the value
|
|
of these <citerefentry><refentrytitle>glTexCoord</refentrytitle></citerefentry> commands.
|
|
</para>
|
|
</listitem>
|
|
</varlistentry>
|
|
</variablelist>
|
|
<para>
|
|
<parameter>ustride</parameter>,
|
|
<parameter>uorder</parameter>,
|
|
<parameter>vstride</parameter>,
|
|
<parameter>vorder</parameter>, and
|
|
<parameter>points</parameter> define the array addressing for accessing the control points.
|
|
<parameter>points</parameter> is the location of the first control point,
|
|
which occupies one, two, three, or four contiguous memory locations,
|
|
depending on which map is being defined.
|
|
There are
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: uorder times vorder:-->
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">uorder</mml:mi>
|
|
<mml:mo>×</mml:mo>
|
|
<mml:mi mathvariant="italic">vorder</mml:mi>
|
|
</mml:mrow>
|
|
</mml:math></inlineequation>
|
|
control points in the array.
|
|
<parameter>ustride</parameter> specifies how many float or double locations are skipped to advance
|
|
the internal memory pointer from control point
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: R sub {i j}:-->
|
|
<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:msub>
|
|
</mml:math></inlineequation>
|
|
to control point
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: R sub {(i+1) j}:-->
|
|
<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
<mml:mo>+</mml:mo>
|
|
<mml:mn>1</mml:mn>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:msub>
|
|
</mml:math></inlineequation>.
|
|
<parameter>vstride</parameter> specifies how many float or double locations are skipped to advance
|
|
the internal memory pointer from control point
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: R sub {i j}:-->
|
|
<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
<mml:mo>⁢</mml:mo>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:msub>
|
|
</mml:math></inlineequation>
|
|
to control point
|
|
<inlineequation><mml:math>
|
|
<!-- eqn: R sub {i (j+1) }:-->
|
|
<mml:msub><mml:mi mathvariant="italic">R</mml:mi>
|
|
<mml:mfenced open="" close="">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">i</mml:mi>
|
|
<mml:mo>⁡</mml:mo>
|
|
<mml:mfenced open="(" close=")">
|
|
<mml:mrow>
|
|
<mml:mi mathvariant="italic">j</mml:mi>
|
|
<mml:mo>+</mml:mo>
|
|
<mml:mn>1</mml:mn>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:mrow>
|
|
</mml:mfenced>
|
|
</mml:msub>
|
|
</mml:math></inlineequation>.
|
|
</para>
|
|
</refsect1>
|
|
<refsect1 id="notes"><title>Notes</title>
|
|
<para>
|
|
As is the case with all GL commands that accept pointers to data,
|
|
it is as if the contents of <parameter>points</parameter> were copied by <function>glMap2</function> before <function>glMap2</function>
|
|
returns.
|
|
Changes to the contents of <parameter>points</parameter> have no effect after <function>glMap2</function> is called.
|
|
</para>
|
|
<para>
|
|
Initially, <constant>GL_AUTO_NORMAL</constant> is enabled. If <constant>GL_AUTO_NORMAL</constant> is enabled,
|
|
normal vectors are generated when either
|
|
<constant>GL_MAP2_VERTEX_3</constant> or <constant>GL_MAP2_VERTEX_4</constant> is used to generate
|
|
vertices.
|
|
</para>
|
|
</refsect1>
|
|
<refsect1 id="errors"><title>Errors</title>
|
|
<para>
|
|
<constant>GL_INVALID_ENUM</constant> is generated if <parameter>target</parameter> is not an accepted value.
|
|
</para>
|
|
<para>
|
|
<constant>GL_INVALID_VALUE</constant> is generated if <parameter>u1</parameter> is equal to <parameter>u2</parameter>,
|
|
or if <parameter>v1</parameter> is equal to <parameter>v2</parameter>.
|
|
</para>
|
|
<para>
|
|
<constant>GL_INVALID_VALUE</constant> is generated if either <parameter>ustride</parameter> or <parameter>vstride</parameter>
|
|
is less than the number of values in a control point.
|
|
</para>
|
|
<para>
|
|
<constant>GL_INVALID_VALUE</constant> is generated if either <parameter>uorder</parameter> or <parameter>vorder</parameter>
|
|
is less than 1 or greater than the return value of <constant>GL_MAX_EVAL_ORDER</constant>.
|
|
</para>
|
|
<para>
|
|
<constant>GL_INVALID_OPERATION</constant> is generated if <function>glMap2</function>
|
|
is executed between the execution of
|
|
<citerefentry><refentrytitle>glBegin</refentrytitle></citerefentry>
|
|
and the corresponding execution of <citerefentry><refentrytitle>glEnd</refentrytitle></citerefentry>.
|
|
</para>
|
|
<para>
|
|
<constant>GL_INVALID_OPERATION</constant> is generated if <function>glMap2</function> is called and the value
|
|
of <constant>GL_ACTIVE_TEXTURE</constant> is not <constant>GL_TEXTURE0</constant>.
|
|
</para>
|
|
</refsect1>
|
|
<refsect1 id="associatedgets"><title>Associated Gets</title>
|
|
<para>
|
|
<citerefentry><refentrytitle>glGetMap</refentrytitle></citerefentry>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glGet</refentrytitle></citerefentry> with argument <constant>GL_MAX_EVAL_ORDER</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_VERTEX_3</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_VERTEX_4</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_INDEX</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_COLOR_4</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_NORMAL</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_TEXTURE_COORD_1</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_TEXTURE_COORD_2</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_TEXTURE_COORD_3</constant>
|
|
</para>
|
|
<para>
|
|
<citerefentry><refentrytitle>glIsEnabled</refentrytitle></citerefentry> with argument <constant>GL_MAP2_TEXTURE_COORD_4</constant>
|
|
</para>
|
|
</refsect1>
|
|
<refsect1 id="seealso"><title>See Also</title>
|
|
<para>
|
|
<citerefentry><refentrytitle>glBegin</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glColor</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glEnable</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glEvalCoord</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glEvalMesh</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glEvalPoint</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glMap1</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glMapGrid</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glNormal</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glTexCoord</refentrytitle></citerefentry>,
|
|
<citerefentry><refentrytitle>glVertex</refentrytitle></citerefentry>
|
|
</para>
|
|
</refsect1>
|
|
<refsect1 id="Copyright"><title>Copyright</title>
|
|
<para>
|
|
Copyright <trademark class="copyright"></trademark> 1991-2006
|
|
Silicon Graphics, Inc. This document is licensed under the SGI
|
|
Free Software B License. For details, see
|
|
<ulink url="http://oss.sgi.com/projects/FreeB/">http://oss.sgi.com/projects/FreeB/</ulink>.
|
|
</para>
|
|
</refsect1>
|
|
</refentry>
|