3 * Mesa 3-D graphics library
6 * Copyright (C) 1999-2001 Brian Paul All Rights Reserved.
8 * Permission is hereby granted, free of charge, to any person obtaining a
9 * copy of this software and associated documentation files (the "Software"),
10 * to deal in the Software without restriction, including without limitation
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12 * and/or sell copies of the Software, and to permit persons to whom the
13 * Software is furnished to do so, subject to the following conditions:
15 * The above copyright notice and this permission notice shall be included
16 * in all copies or substantial portions of the Software.
18 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
19 * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
20 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
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29 #include "main/glheader.h"
31 void _math_init_eval( void );
35 * Horner scheme for Bezier curves
37 * Bezier curves can be computed via a Horner scheme.
38 * Horner is numerically less stable than the de Casteljau
39 * algorithm, but it is faster. For curves of degree n
40 * the complexity of Horner is O(n) and de Casteljau is O(n^2).
41 * Since stability is not important for displaying curve
42 * points I decided to use the Horner scheme.
44 * A cubic Bezier curve with control points b0, b1, b2, b3 can be
47 * (([3] [3] ) [3] ) [3]
48 * c(t) = (([0]*s*b0 + [1]*t*b1)*s + [2]*t^2*b2)*s + [3]*t^2*b3
51 * where s=1-t and the binomial coefficients [i]. These can
52 * be computed iteratively using the identity:
55 * [i] = (n-i+1)/i * [i-1] and [0] = 1
60 _math_horner_bezier_curve(const GLfloat
*cp
, GLfloat
*out
, GLfloat t
,
61 GLuint dim
, GLuint order
);
65 * Tensor product Bezier surfaces
67 * Again the Horner scheme is used to compute a point on a
68 * TP Bezier surface. First a control polygon for a curve
69 * on the surface in one parameter direction is computed,
70 * then the point on the curve for the other parameter
71 * direction is evaluated.
73 * To store the curve control polygon additional storage
74 * for max(uorder,vorder) points is needed in the
79 _math_horner_bezier_surf(GLfloat
*cn
, GLfloat
*out
, GLfloat u
, GLfloat v
,
80 GLuint dim
, GLuint uorder
, GLuint vorder
);
84 * The direct de Casteljau algorithm is used when a point on the
85 * surface and the tangent directions spanning the tangent plane
86 * should be computed (this is needed to compute normals to the
87 * surface). In this case the de Casteljau algorithm approach is
88 * nicer because a point and the partial derivatives can be computed
89 * at the same time. To get the correct tangent length du and dv
90 * must be multiplied with the (u2-u1)/uorder-1 and (v2-v1)/vorder-1.
91 * Since only the directions are needed, this scaling step is omitted.
93 * De Casteljau needs additional storage for uorder*vorder
94 * values in the control net cn.
98 _math_de_casteljau_surf(GLfloat
*cn
, GLfloat
*out
, GLfloat
*du
, GLfloat
*dv
,
99 GLfloat u
, GLfloat v
, GLuint dim
,
100 GLuint uorder
, GLuint vorder
);