/* * Mesa 3-D graphics library * * 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 OR COPYRIGHT HOLDERS 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