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#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
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