src/mesa/math/m_eval.h

#ifndef _M_EVAL_H
#define _M_EVAL_H

#include "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