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1 
2 /*
3  * Mesa 3-D graphics library
4  *
5  * Copyright (C) 1999-2001  Brian Paul   All Rights Reserved.
6  *
7  * Permission is hereby granted, free of charge, to any person obtaining a
8  * copy of this software and associated documentation files (the "Software"),
9  * to deal in the Software without restriction, including without limitation
10  * the rights to use, copy, modify, merge, publish, distribute, sublicense,
11  * and/or sell copies of the Software, and to permit persons to whom the
12  * Software is furnished to do so, subject to the following conditions:
13  *
14  * The above copyright notice and this permission notice shall be included
15  * in all copies or substantial portions of the Software.
16  *
17  * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
18  * OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
19  * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL
20  * THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR
21  * OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
22  * ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
23  * OTHER DEALINGS IN THE SOFTWARE.
24  */
25 
26 #ifndef _M_EVAL_H
27 #define _M_EVAL_H
28 
29 #include "main/glheader.h"
30 
31 void _math_init_eval( void );
32 
33 
34 /*
35  * Horner scheme for Bezier curves
36  *
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.
43  *
44  * A cubic Bezier curve with control points b0, b1, b2, b3 can be
45  * written as
46  *
47  *        (([3]        [3]     )     [3]       )     [3]
48  * c(t) = (([0]*s*b0 + [1]*t*b1)*s + [2]*t^2*b2)*s + [3]*t^2*b3
49  *
50  *                                           [n]
51  * where s=1-t and the binomial coefficients [i]. These can
52  * be computed iteratively using the identity:
53  *
54  * [n]               [n  ]             [n]
55  * [i] = (n-i+1)/i * [i-1]     and     [0] = 1
56  */
57 
58 
59 void
60 _math_horner_bezier_curve(const GLfloat *cp, GLfloat *out, GLfloat t,
61 			  GLuint dim, GLuint order);
62 
63 
64 /*
65  * Tensor product Bezier surfaces
66  *
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.
72  *
73  * To store the curve control polygon additional storage
74  * for max(uorder,vorder) points is needed in the
75  * control net cn.
76  */
77 
78 void
79 _math_horner_bezier_surf(GLfloat *cn, GLfloat *out, GLfloat u, GLfloat v,
80 			 GLuint dim, GLuint uorder, GLuint vorder);
81 
82 
83 /*
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.
92  *
93  * De Casteljau needs additional storage for uorder*vorder
94  * values in the control net cn.
95  */
96 
97 void
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);
101 
102 
103 #endif
104