1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
|
/*
* Mesa 3-D graphics library
* Version: 3.5
*
* Copyright (C) 1999-2001 Brian Paul All Rights Reserved.
*
* Permission is hereby granted, free of charge, to any person obtaining a
* copy of this software and associated documentation files (the "Software"),
* to deal in the Software without restriction, including without limitation
* the rights to use, copy, modify, merge, publish, distribute, sublicense,
* and/or sell copies of the Software, and to permit persons to whom the
* Software is furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included
* in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
* OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
* THE AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN
* AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
*/
#ifndef _M_EVAL_H
#define _M_EVAL_H
#include "main/glheader.h"
void _math_init_eval( void );
/*
* Horner scheme for Bezier curves
*
* Bezier curves can be computed via a Horner scheme.
* Horner is numerically less stable than the de Casteljau
* algorithm, but it is faster. For curves of degree n
* the complexity of Horner is O(n) and de Casteljau is O(n^2).
* Since stability is not important for displaying curve
* points I decided to use the Horner scheme.
*
* A cubic Bezier curve with control points b0, b1, b2, b3 can be
* written as
*
* (([3] [3] ) [3] ) [3]
* c(t) = (([0]*s*b0 + [1]*t*b1)*s + [2]*t^2*b2)*s + [3]*t^2*b3
*
* [n]
* where s=1-t and the binomial coefficients [i]. These can
* be computed iteratively using the identity:
*
* [n] [n ] [n]
* [i] = (n-i+1)/i * [i-1] and [0] = 1
*/
void
_math_horner_bezier_curve(const GLfloat *cp, GLfloat *out, GLfloat t,
GLuint dim, GLuint order);
/*
* Tensor product Bezier surfaces
*
* Again the Horner scheme is used to compute a point on a
* TP Bezier surface. First a control polygon for a curve
* on the surface in one parameter direction is computed,
* then the point on the curve for the other parameter
* direction is evaluated.
*
* To store the curve control polygon additional storage
* for max(uorder,vorder) points is needed in the
* control net cn.
*/
void
_math_horner_bezier_surf(GLfloat *cn, GLfloat *out, GLfloat u, GLfloat v,
GLuint dim, GLuint uorder, GLuint vorder);
/*
* The direct de Casteljau algorithm is used when a point on the
* surface and the tangent directions spanning the tangent plane
* should be computed (this is needed to compute normals to the
* surface). In this case the de Casteljau algorithm approach is
* nicer because a point and the partial derivatives can be computed
* at the same time. To get the correct tangent length du and dv
* must be multiplied with the (u2-u1)/uorder-1 and (v2-v1)/vorder-1.
* Since only the directions are needed, this scaling step is omitted.
*
* De Casteljau needs additional storage for uorder*vorder
* values in the control net cn.
*/
void
_math_de_casteljau_surf(GLfloat *cn, GLfloat *out, GLfloat *du, GLfloat *dv,
GLfloat u, GLfloat v, GLuint dim,
GLuint uorder, GLuint vorder);
#endif
|
