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  1.  
  2. /*
  3.  * Mesa 3-D graphics library
  4.  * Version:  3.5
  5.  *
  6.  * Copyright (C) 1999-2001  Brian Paul   All Rights Reserved.
  7.  *
  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
  11.  * the rights to use, copy, modify, merge, publish, distribute, sublicense,
  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:
  14.  *
  15.  * The above copyright notice and this permission notice shall be included
  16.  * in all copies or substantial portions of the Software.
  17.  *
  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
  21.  * BRIAN PAUL BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN
  22.  * AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
  23.  * CONNECTION WITH THE SOFTWARE OR THE USE OR 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.