1 /* $Id: m_eval.h,v 1.2 2001/03/12 00:48:41 gareth Exp $ */
4 * Mesa 3-D graphics library
7 * Copyright (C) 1999-2001 Brian Paul All Rights Reserved.
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32 void _math_init_eval( void );
36 * Horner scheme for Bezier curves
38 * Bezier curves can be computed via a Horner scheme.
39 * Horner is numerically less stable than the de Casteljau
40 * algorithm, but it is faster. For curves of degree n
41 * the complexity of Horner is O(n) and de Casteljau is O(n^2).
42 * Since stability is not important for displaying curve
43 * points I decided to use the Horner scheme.
45 * A cubic Bezier curve with control points b0, b1, b2, b3 can be
48 * (([3] [3] ) [3] ) [3]
49 * c(t) = (([0]*s*b0 + [1]*t*b1)*s + [2]*t^2*b2)*s + [3]*t^2*b3
52 * where s=1-t and the binomial coefficients [i]. These can
53 * be computed iteratively using the identity:
56 * [i] = (n-i+1)/i * [i-1] and [0] = 1
61 _math_horner_bezier_curve(const GLfloat
*cp
, GLfloat
*out
, GLfloat t
,
62 GLuint dim
, GLuint order
);
66 * Tensor product Bezier surfaces
68 * Again the Horner scheme is used to compute a point on a
69 * TP Bezier surface. First a control polygon for a curve
70 * on the surface in one parameter direction is computed,
71 * then the point on the curve for the other parameter
72 * direction is evaluated.
74 * To store the curve control polygon additional storage
75 * for max(uorder,vorder) points is needed in the
80 _math_horner_bezier_surf(GLfloat
*cn
, GLfloat
*out
, GLfloat u
, GLfloat v
,
81 GLuint dim
, GLuint uorder
, GLuint vorder
);
85 * The direct de Casteljau algorithm is used when a point on the
86 * surface and the tangent directions spanning the tangent plane
87 * should be computed (this is needed to compute normals to the
88 * surface). In this case the de Casteljau algorithm approach is
89 * nicer because a point and the partial derivatives can be computed
90 * at the same time. To get the correct tangent length du and dv
91 * must be multiplied with the (u2-u1)/uorder-1 and (v2-v1)/vorder-1.
92 * Since only the directions are needed, this scaling step is omitted.
94 * De Casteljau needs additional storage for uorder*vorder
95 * values in the control net cn.
99 _math_de_casteljau_surf(GLfloat
*cn
, GLfloat
*out
, GLfloat
*du
, GLfloat
*dv
,
100 GLfloat u
, GLfloat v
, GLuint dim
,
101 GLuint uorder
, GLuint vorder
);