src/mesa/math/m_eval.c - platform/external/mesa3d - Git at Google


 /*
  * 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.
  */


 /*
  * eval.c was written by
  * Bernd Barsuhn (bdbarsuh@cip.informatik.uni-erlangen.de) and
  * Volker Weiss (vrweiss@cip.informatik.uni-erlangen.de).
  *
  * My original implementation of evaluators was simplistic and didn't
  * compute surface normal vectors properly.  Bernd and Volker applied
  * used more sophisticated methods to get better results.
  *
  * Thanks guys!
  */


 #include "main/glheader.h"
 #include "main/config.h"
 #include "m_eval.h"

 static GLfloat inv_tab[MAX_EVAL_ORDER];


 /*
  * 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)
 {
    GLfloat s, powert, bincoeff;
    GLuint i, k;

    if (order >= 2) {
       bincoeff = (GLfloat) (order - 1);
       s = 1.0F - t;

       for (k = 0; k < dim; k++)
 	 out[k] = s * cp[k] + bincoeff * t * cp[dim + k];

       for (i = 2, cp += 2 * dim, powert = t * t; i < order;
 	   i++, powert *= t, cp += dim) {
 	 bincoeff *= (GLfloat) (order - i);
 	 bincoeff *= inv_tab[i];

 	 for (k = 0; k < dim; k++)
 	    out[k] = s * out[k] + bincoeff * powert * cp[k];
       }
    }
    else {			/* order=1 -> constant curve */

       for (k = 0; k < dim; k++)
 	 out[k] = cp[k];
    }
 }

 /*
  * 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)
 {
    GLfloat *cp = cn + uorder * vorder * dim;
    GLuint i, uinc = vorder * dim;

    if (vorder > uorder) {
       if (uorder >= 2) {
 	 GLfloat s, poweru, bincoeff;
 	 GLuint j, k;

 	 /* Compute the control polygon for the surface-curve in u-direction */
 	 for (j = 0; j < vorder; j++) {
 	    GLfloat *ucp = &cn[j * dim];

 	    /* Each control point is the point for parameter u on a */
 	    /* curve defined by the control polygons in u-direction */
 	    bincoeff = (GLfloat) (uorder - 1);
 	    s = 1.0F - u;

 	    for (k = 0; k < dim; k++)
 	       cp[j * dim + k] = s * ucp[k] + bincoeff * u * ucp[uinc + k];

 	    for (i = 2, ucp += 2 * uinc, poweru = u * u; i < uorder;
 		 i++, poweru *= u, ucp += uinc) {
 	       bincoeff *= (GLfloat) (uorder - i);
 	       bincoeff *= inv_tab[i];

 	       for (k = 0; k < dim; k++)
 		  cp[j * dim + k] =
 		     s * cp[j * dim + k] + bincoeff * poweru * ucp[k];
 	    }
 	 }

 	 /* Evaluate curve point in v */
 	 _math_horner_bezier_curve(cp, out, v, dim, vorder);
       }
       else			/* uorder=1 -> cn defines a curve in v */
 	 _math_horner_bezier_curve(cn, out, v, dim, vorder);
    }
    else {			/* vorder <= uorder */

       if (vorder > 1) {
 	 GLuint i;

 	 /* Compute the control polygon for the surface-curve in u-direction */
 	 for (i = 0; i < uorder; i++, cn += uinc) {
 	    /* For constant i all cn[i][j] (j=0..vorder) are located */
 	    /* on consecutive memory locations, so we can use        */
 	    /* horner_bezier_curve to compute the control points     */

 	    _math_horner_bezier_curve(cn, &cp[i * dim], v, dim, vorder);
 	 }

 	 /* Evaluate curve point in u */
 	 _math_horner_bezier_curve(cp, out, u, dim, uorder);
       }
       else			/* vorder=1 -> cn defines a curve in u */
 	 _math_horner_bezier_curve(cn, out, u, dim, uorder);
    }
 }

 /*
  * 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)
 {
    GLfloat *dcn = cn + uorder * vorder * dim;
    GLfloat us = 1.0F - u, vs = 1.0F - v;
    GLuint h, i, j, k;
    GLuint minorder = uorder < vorder ? uorder : vorder;
    GLuint uinc = vorder * dim;
    GLuint dcuinc = vorder;

    /* Each component is evaluated separately to save buffer space  */
    /* This does not drasticaly decrease the performance of the     */
    /* algorithm. If additional storage for (uorder-1)*(vorder-1)   */
    /* points would be available, the components could be accessed  */
    /* in the innermost loop which could lead to less cache misses. */

 #define CN(I,J,K) cn[(I)*uinc+(J)*dim+(K)]
 #define DCN(I, J) dcn[(I)*dcuinc+(J)]
    if (minorder < 3) {
       if (uorder == vorder) {
 	 for (k = 0; k < dim; k++) {
 	    /* Derivative direction in u */
 	    du[k] = vs * (CN(1, 0, k) - CN(0, 0, k)) +
 	       v * (CN(1, 1, k) - CN(0, 1, k));

 	    /* Derivative direction in v */
 	    dv[k] = us * (CN(0, 1, k) - CN(0, 0, k)) +
 	       u * (CN(1, 1, k) - CN(1, 0, k));

 	    /* bilinear de Casteljau step */
 	    out[k] = us * (vs * CN(0, 0, k) + v * CN(0, 1, k)) +
 	       u * (vs * CN(1, 0, k) + v * CN(1, 1, k));
 	 }
       }
       else if (minorder == uorder) {
 	 for (k = 0; k < dim; k++) {
 	    /* bilinear de Casteljau step */
 	    DCN(1, 0) = CN(1, 0, k) - CN(0, 0, k);
 	    DCN(0, 0) = us * CN(0, 0, k) + u * CN(1, 0, k);

 	    for (j = 0; j < vorder - 1; j++) {
 	       /* for the derivative in u */
 	       DCN(1, j + 1) = CN(1, j + 1, k) - CN(0, j + 1, k);
 	       DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);

 	       /* for the `point' */
 	       DCN(0, j + 1) = us * CN(0, j + 1, k) + u * CN(1, j + 1, k);
 	       DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
 	    }

 	    /* remaining linear de Casteljau steps until the second last step */
 	    for (h = minorder; h < vorder - 1; h++)
 	       for (j = 0; j < vorder - h; j++) {
 		  /* for the derivative in u */
 		  DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);

 		  /* for the `point' */
 		  DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
 	       }

 	    /* derivative direction in v */
 	    dv[k] = DCN(0, 1) - DCN(0, 0);

 	    /* derivative direction in u */
 	    du[k] = vs * DCN(1, 0) + v * DCN(1, 1);

 	    /* last linear de Casteljau step */
 	    out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
 	 }
       }
       else {			/* minorder == vorder */

 	 for (k = 0; k < dim; k++) {
 	    /* bilinear de Casteljau step */
 	    DCN(0, 1) = CN(0, 1, k) - CN(0, 0, k);
 	    DCN(0, 0) = vs * CN(0, 0, k) + v * CN(0, 1, k);
 	    for (i = 0; i < uorder - 1; i++) {
 	       /* for the derivative in v */
 	       DCN(i + 1, 1) = CN(i + 1, 1, k) - CN(i + 1, 0, k);
 	       DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);

 	       /* for the `point' */
 	       DCN(i + 1, 0) = vs * CN(i + 1, 0, k) + v * CN(i + 1, 1, k);
 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
 	    }

 	    /* remaining linear de Casteljau steps until the second last step */
 	    for (h = minorder; h < uorder - 1; h++)
 	       for (i = 0; i < uorder - h; i++) {
 		  /* for the derivative in v */
 		  DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);

 		  /* for the `point' */
 		  DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
 	       }

 	    /* derivative direction in u */
 	    du[k] = DCN(1, 0) - DCN(0, 0);

 	    /* derivative direction in v */
 	    dv[k] = us * DCN(0, 1) + u * DCN(1, 1);

 	    /* last linear de Casteljau step */
 	    out[k] = us * DCN(0, 0) + u * DCN(1, 0);
 	 }
       }
    }
    else if (uorder == vorder) {
       for (k = 0; k < dim; k++) {
 	 /* first bilinear de Casteljau step */
 	 for (i = 0; i < uorder - 1; i++) {
 	    DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
 	    for (j = 0; j < vorder - 1; j++) {
 	       DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
 	       DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
 	    }
 	 }

 	 /* remaining bilinear de Casteljau steps until the second last step */
 	 for (h = 2; h < minorder - 1; h++)
 	    for (i = 0; i < uorder - h; i++) {
 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
 	       for (j = 0; j < vorder - h; j++) {
 		  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
 		  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
 	       }
 	    }

 	 /* derivative direction in u */
 	 du[k] = vs * (DCN(1, 0) - DCN(0, 0)) + v * (DCN(1, 1) - DCN(0, 1));

 	 /* derivative direction in v */
 	 dv[k] = us * (DCN(0, 1) - DCN(0, 0)) + u * (DCN(1, 1) - DCN(1, 0));

 	 /* last bilinear de Casteljau step */
 	 out[k] = us * (vs * DCN(0, 0) + v * DCN(0, 1)) +
 	    u * (vs * DCN(1, 0) + v * DCN(1, 1));
       }
    }
    else if (minorder == uorder) {
       for (k = 0; k < dim; k++) {
 	 /* first bilinear de Casteljau step */
 	 for (i = 0; i < uorder - 1; i++) {
 	    DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
 	    for (j = 0; j < vorder - 1; j++) {
 	       DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
 	       DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
 	    }
 	 }

 	 /* remaining bilinear de Casteljau steps until the second last step */
 	 for (h = 2; h < minorder - 1; h++)
 	    for (i = 0; i < uorder - h; i++) {
 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
 	       for (j = 0; j < vorder - h; j++) {
 		  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
 		  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
 	       }
 	    }

 	 /* last bilinear de Casteljau step */
 	 DCN(2, 0) = DCN(1, 0) - DCN(0, 0);
 	 DCN(0, 0) = us * DCN(0, 0) + u * DCN(1, 0);
 	 for (j = 0; j < vorder - 1; j++) {
 	    /* for the derivative in u */
 	    DCN(2, j + 1) = DCN(1, j + 1) - DCN(0, j + 1);
 	    DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);

 	    /* for the `point' */
 	    DCN(0, j + 1) = us * DCN(0, j + 1) + u * DCN(1, j + 1);
 	    DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
 	 }

 	 /* remaining linear de Casteljau steps until the second last step */
 	 for (h = minorder; h < vorder - 1; h++)
 	    for (j = 0; j < vorder - h; j++) {
 	       /* for the derivative in u */
 	       DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);

 	       /* for the `point' */
 	       DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
 	    }

 	 /* derivative direction in v */
 	 dv[k] = DCN(0, 1) - DCN(0, 0);

 	 /* derivative direction in u */
 	 du[k] = vs * DCN(2, 0) + v * DCN(2, 1);

 	 /* last linear de Casteljau step */
 	 out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
       }
    }
    else {			/* minorder == vorder */

       for (k = 0; k < dim; k++) {
 	 /* first bilinear de Casteljau step */
 	 for (i = 0; i < uorder - 1; i++) {
 	    DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
 	    for (j = 0; j < vorder - 1; j++) {
 	       DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
 	       DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
 	    }
 	 }

 	 /* remaining bilinear de Casteljau steps until the second last step */
 	 for (h = 2; h < minorder - 1; h++)
 	    for (i = 0; i < uorder - h; i++) {
 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
 	       for (j = 0; j < vorder - h; j++) {
 		  DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
 		  DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
 	       }
 	    }

 	 /* last bilinear de Casteljau step */
 	 DCN(0, 2) = DCN(0, 1) - DCN(0, 0);
 	 DCN(0, 0) = vs * DCN(0, 0) + v * DCN(0, 1);
 	 for (i = 0; i < uorder - 1; i++) {
 	    /* for the derivative in v */
 	    DCN(i + 1, 2) = DCN(i + 1, 1) - DCN(i + 1, 0);
 	    DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);

 	    /* for the `point' */
 	    DCN(i + 1, 0) = vs * DCN(i + 1, 0) + v * DCN(i + 1, 1);
 	    DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
 	 }

 	 /* remaining linear de Casteljau steps until the second last step */
 	 for (h = minorder; h < uorder - 1; h++)
 	    for (i = 0; i < uorder - h; i++) {
 	       /* for the derivative in v */
 	       DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);

 	       /* for the `point' */
 	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
 	    }

 	 /* derivative direction in u */
 	 du[k] = DCN(1, 0) - DCN(0, 0);

 	 /* derivative direction in v */
 	 dv[k] = us * DCN(0, 2) + u * DCN(1, 2);

 	 /* last linear de Casteljau step */
 	 out[k] = us * DCN(0, 0) + u * DCN(1, 0);
       }
    }
 #undef DCN
 #undef CN
 }


 /*
  * Do one-time initialization for evaluators.
  */
 void
 _math_init_eval(void)
 {
    GLuint i;

    /* KW: precompute 1/x for useful x.
     */
    for (i = 1; i < MAX_EVAL_ORDER; i++)
       inv_tab[i] = 1.0F / i;
 }

	/*
	* 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.
	*/


	/*
	* eval.c was written by
	* Bernd Barsuhn (bdbarsuh@cip.informatik.uni-erlangen.de) and
	* Volker Weiss (vrweiss@cip.informatik.uni-erlangen.de).
	*
	* My original implementation of evaluators was simplistic and didn't
	* compute surface normal vectors properly. Bernd and Volker applied
	* used more sophisticated methods to get better results.
	*
	* Thanks guys!
	*/


	#include "main/glheader.h"
	#include "main/config.h"
	#include "m_eval.h"

	static GLfloat inv_tab[MAX_EVAL_ORDER];



	/*
	* 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]sb0 + [1]tb1)s + [2]t^2b2)s + [3]t^2b3
	*
	* [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)
	{
	GLfloat s, powert, bincoeff;
	GLuint i, k;

	if (order >= 2) {
	bincoeff = (GLfloat) (order - 1);
	s = 1.0F - t;

	for (k = 0; k < dim; k++)
	out[k] = s * cp[k] + bincoeff * t * cp[dim + k];

	for (i = 2, cp += 2 * dim, powert = t * t; i < order;
	i++, powert *= t, cp += dim) {
	bincoeff *= (GLfloat) (order - i);
	bincoeff *= inv_tab[i];

	for (k = 0; k < dim; k++)
	out[k] = s * out[k] + bincoeff * powert * cp[k];
	}
	}
	else { /* order=1 -> constant curve */

	for (k = 0; k < dim; k++)
	out[k] = cp[k];
	}
	}

	/*
	* 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)
	{
	GLfloat cp = cn + uorder vorder * dim;
	GLuint i, uinc = vorder * dim;

	if (vorder > uorder) {
	if (uorder >= 2) {
	GLfloat s, poweru, bincoeff;
	GLuint j, k;

	/* Compute the control polygon for the surface-curve in u-direction */
	for (j = 0; j < vorder; j++) {
	GLfloat ucp = &cn[j dim];

	/* Each control point is the point for parameter u on a */
	/* curve defined by the control polygons in u-direction */
	bincoeff = (GLfloat) (uorder - 1);
	s = 1.0F - u;

	for (k = 0; k < dim; k++)
	cp[j * dim + k] = s * ucp[k] + bincoeff * u * ucp[uinc + k];

	for (i = 2, ucp += 2 * uinc, poweru = u * u; i < uorder;
	i++, poweru *= u, ucp += uinc) {
	bincoeff *= (GLfloat) (uorder - i);
	bincoeff *= inv_tab[i];

	for (k = 0; k < dim; k++)
	cp[j * dim + k] =
	s * cp[j * dim + k] + bincoeff * poweru * ucp[k];
	}
	}

	/* Evaluate curve point in v */
	_math_horner_bezier_curve(cp, out, v, dim, vorder);
	}
	else /* uorder=1 -> cn defines a curve in v */
	_math_horner_bezier_curve(cn, out, v, dim, vorder);
	}
	else { /* vorder <= uorder */

	if (vorder > 1) {
	GLuint i;

	/* Compute the control polygon for the surface-curve in u-direction */
	for (i = 0; i < uorder; i++, cn += uinc) {
	/* For constant i all cn[i][j] (j=0..vorder) are located */
	/* on consecutive memory locations, so we can use */
	/* horner_bezier_curve to compute the control points */

	_math_horner_bezier_curve(cn, &cp[i * dim], v, dim, vorder);
	}

	/* Evaluate curve point in u */
	_math_horner_bezier_curve(cp, out, u, dim, uorder);
	}
	else /* vorder=1 -> cn defines a curve in u */
	_math_horner_bezier_curve(cn, out, u, dim, uorder);
	}
	}

	/*
	* 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)
	{
	GLfloat dcn = cn + uorder vorder * dim;
	GLfloat us = 1.0F - u, vs = 1.0F - v;
	GLuint h, i, j, k;
	GLuint minorder = uorder < vorder ? uorder : vorder;
	GLuint uinc = vorder * dim;
	GLuint dcuinc = vorder;

	/* Each component is evaluated separately to save buffer space */
	/* This does not drasticaly decrease the performance of the */
	/* algorithm. If additional storage for (uorder-1)(vorder-1) /
	/* points would be available, the components could be accessed */
	/* in the innermost loop which could lead to less cache misses. */

	#define CN(I,J,K) cn[(I)uinc+(J)dim+(K)]
	#define DCN(I, J) dcn[(I)*dcuinc+(J)]
	if (minorder < 3) {
	if (uorder == vorder) {
	for (k = 0; k < dim; k++) {
	/* Derivative direction in u */
	du[k] = vs * (CN(1, 0, k) - CN(0, 0, k)) +
	v * (CN(1, 1, k) - CN(0, 1, k));

	/* Derivative direction in v */
	dv[k] = us * (CN(0, 1, k) - CN(0, 0, k)) +
	u * (CN(1, 1, k) - CN(1, 0, k));

	/* bilinear de Casteljau step */
	out[k] = us * (vs * CN(0, 0, k) + v * CN(0, 1, k)) +
	u * (vs * CN(1, 0, k) + v * CN(1, 1, k));
	}
	}
	else if (minorder == uorder) {
	for (k = 0; k < dim; k++) {
	/* bilinear de Casteljau step */
	DCN(1, 0) = CN(1, 0, k) - CN(0, 0, k);
	DCN(0, 0) = us * CN(0, 0, k) + u * CN(1, 0, k);

	for (j = 0; j < vorder - 1; j++) {
	/* for the derivative in u */
	DCN(1, j + 1) = CN(1, j + 1, k) - CN(0, j + 1, k);
	DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);

	/* for the `point' */
	DCN(0, j + 1) = us * CN(0, j + 1, k) + u * CN(1, j + 1, k);
	DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
	}

	/* remaining linear de Casteljau steps until the second last step */
	for (h = minorder; h < vorder - 1; h++)
	for (j = 0; j < vorder - h; j++) {
	/* for the derivative in u */
	DCN(1, j) = vs * DCN(1, j) + v * DCN(1, j + 1);

	/* for the `point' */
	DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
	}

	/* derivative direction in v */
	dv[k] = DCN(0, 1) - DCN(0, 0);

	/* derivative direction in u */
	du[k] = vs * DCN(1, 0) + v * DCN(1, 1);

	/* last linear de Casteljau step */
	out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
	}
	}
	else { /* minorder == vorder */

	for (k = 0; k < dim; k++) {
	/* bilinear de Casteljau step */
	DCN(0, 1) = CN(0, 1, k) - CN(0, 0, k);
	DCN(0, 0) = vs * CN(0, 0, k) + v * CN(0, 1, k);
	for (i = 0; i < uorder - 1; i++) {
	/* for the derivative in v */
	DCN(i + 1, 1) = CN(i + 1, 1, k) - CN(i + 1, 0, k);
	DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);

	/* for the `point' */
	DCN(i + 1, 0) = vs * CN(i + 1, 0, k) + v * CN(i + 1, 1, k);
	DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
	}

	/* remaining linear de Casteljau steps until the second last step */
	for (h = minorder; h < uorder - 1; h++)
	for (i = 0; i < uorder - h; i++) {
	/* for the derivative in v */
	DCN(i, 1) = us * DCN(i, 1) + u * DCN(i + 1, 1);

	/* for the `point' */
	DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
	}

	/* derivative direction in u */
	du[k] = DCN(1, 0) - DCN(0, 0);

	/* derivative direction in v */
	dv[k] = us * DCN(0, 1) + u * DCN(1, 1);

	/* last linear de Casteljau step */
	out[k] = us * DCN(0, 0) + u * DCN(1, 0);
	}
	}
	}
	else if (uorder == vorder) {
	for (k = 0; k < dim; k++) {
	/* first bilinear de Casteljau step */
	for (i = 0; i < uorder - 1; i++) {
	DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
	for (j = 0; j < vorder - 1; j++) {
	DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
	DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
	}
	}

	/* remaining bilinear de Casteljau steps until the second last step */
	for (h = 2; h < minorder - 1; h++)
	for (i = 0; i < uorder - h; i++) {
	DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
	for (j = 0; j < vorder - h; j++) {
	DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
	DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
	}
	}

	/* derivative direction in u */
	du[k] = vs * (DCN(1, 0) - DCN(0, 0)) + v * (DCN(1, 1) - DCN(0, 1));

	/* derivative direction in v */
	dv[k] = us * (DCN(0, 1) - DCN(0, 0)) + u * (DCN(1, 1) - DCN(1, 0));

	/* last bilinear de Casteljau step */
	out[k] = us * (vs * DCN(0, 0) + v * DCN(0, 1)) +
	u * (vs * DCN(1, 0) + v * DCN(1, 1));
	}
	}
	else if (minorder == uorder) {
	for (k = 0; k < dim; k++) {
	/* first bilinear de Casteljau step */
	for (i = 0; i < uorder - 1; i++) {
	DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
	for (j = 0; j < vorder - 1; j++) {
	DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
	DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
	}
	}

	/* remaining bilinear de Casteljau steps until the second last step */
	for (h = 2; h < minorder - 1; h++)
	for (i = 0; i < uorder - h; i++) {
	DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
	for (j = 0; j < vorder - h; j++) {
	DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
	DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
	}
	}

	/* last bilinear de Casteljau step */
	DCN(2, 0) = DCN(1, 0) - DCN(0, 0);
	DCN(0, 0) = us * DCN(0, 0) + u * DCN(1, 0);
	for (j = 0; j < vorder - 1; j++) {
	/* for the derivative in u */
	DCN(2, j + 1) = DCN(1, j + 1) - DCN(0, j + 1);
	DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);

	/* for the `point' */
	DCN(0, j + 1) = us * DCN(0, j + 1) + u * DCN(1, j + 1);
	DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
	}

	/* remaining linear de Casteljau steps until the second last step */
	for (h = minorder; h < vorder - 1; h++)
	for (j = 0; j < vorder - h; j++) {
	/* for the derivative in u */
	DCN(2, j) = vs * DCN(2, j) + v * DCN(2, j + 1);

	/* for the `point' */
	DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);
	}

	/* derivative direction in v */
	dv[k] = DCN(0, 1) - DCN(0, 0);

	/* derivative direction in u */
	du[k] = vs * DCN(2, 0) + v * DCN(2, 1);

	/* last linear de Casteljau step */
	out[k] = vs * DCN(0, 0) + v * DCN(0, 1);
	}
	}
	else { /* minorder == vorder */

	for (k = 0; k < dim; k++) {
	/* first bilinear de Casteljau step */
	for (i = 0; i < uorder - 1; i++) {
	DCN(i, 0) = us * CN(i, 0, k) + u * CN(i + 1, 0, k);
	for (j = 0; j < vorder - 1; j++) {
	DCN(i, j + 1) = us * CN(i, j + 1, k) + u * CN(i + 1, j + 1, k);
	DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
	}
	}

	/* remaining bilinear de Casteljau steps until the second last step */
	for (h = 2; h < minorder - 1; h++)
	for (i = 0; i < uorder - h; i++) {
	DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
	for (j = 0; j < vorder - h; j++) {
	DCN(i, j + 1) = us * DCN(i, j + 1) + u * DCN(i + 1, j + 1);
	DCN(i, j) = vs * DCN(i, j) + v * DCN(i, j + 1);
	}
	}

	/* last bilinear de Casteljau step */
	DCN(0, 2) = DCN(0, 1) - DCN(0, 0);
	DCN(0, 0) = vs * DCN(0, 0) + v * DCN(0, 1);
	for (i = 0; i < uorder - 1; i++) {
	/* for the derivative in v */
	DCN(i + 1, 2) = DCN(i + 1, 1) - DCN(i + 1, 0);
	DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);

	/* for the `point' */
	DCN(i + 1, 0) = vs * DCN(i + 1, 0) + v * DCN(i + 1, 1);
	DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
	}

	/* remaining linear de Casteljau steps until the second last step */
	for (h = minorder; h < uorder - 1; h++)
	for (i = 0; i < uorder - h; i++) {
	/* for the derivative in v */
	DCN(i, 2) = us * DCN(i, 2) + u * DCN(i + 1, 2);

	/* for the `point' */
	DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);
	}

	/* derivative direction in u */
	du[k] = DCN(1, 0) - DCN(0, 0);

	/* derivative direction in v */
	dv[k] = us * DCN(0, 2) + u * DCN(1, 2);

	/* last linear de Casteljau step */
	out[k] = us * DCN(0, 0) + u * DCN(1, 0);
	}
	}
	#undef DCN
	#undef CN
	}


	/*
	* Do one-time initialization for evaluators.
	*/
	void
	_math_init_eval(void)
	{
	GLuint i;

	/* KW: precompute 1/x for useful x.
	*/
	for (i = 1; i < MAX_EVAL_ORDER; i++)
	inv_tab[i] = 1.0F / i;
	}