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