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 * Mesa 3-D graphics library

 * Version:  3.5

 *

 * 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

 * BRIAN PAUL 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/config.h"

#include "m_eval.h"

 * 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

 */

_math_horner_bezier_curve(const GLfloat * cp, GLfloat * out, GLfloat t,

			  GLuint dim, GLuint order)

{

   GLfloat s, powert, bincoeff;

   GLuint i, k;

      bincoeff = (GLfloat) (order - 1);

      s = 1.0F - t;

	 out[k] = s * cp[k] + bincoeff * t * cp[dim + k];

	   i++, powert *= t, cp += dim) {

	 bincoeff *= (GLfloat) (order - i);

	 bincoeff *= inv_tab[i];

	    out[k] = s * out[k] + bincoeff * powert * cp[k];

      }

   }

   else {			/* order=1 -> constant curve */

	 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.

 */

_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 (uorder >= 2) {

	 GLfloat s, poweru, bincoeff;

	 GLuint j, k;

	 for (j = 0; j < vorder; j++) {

	    GLfloat *ucp = &cn[j * dim];

	    /* curve defined by the control polygons in u-direction */

	    bincoeff = (GLfloat) (uorder - 1);

	    s = 1.0F - u;

	       cp[j * dim + k] = s * ucp[k] + bincoeff * u * ucp[uinc + k];

		 i++, poweru *= u, ucp += uinc) {

	       bincoeff *= (GLfloat) (uorder - i);

	       bincoeff *= inv_tab[i];

		  cp[j * dim + k] =

		     s * cp[j * dim + k] + bincoeff * poweru * ucp[k];

	    }

	 }

	 _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 */

	 GLuint i;

	 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(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.

 */

_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;

   /* 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 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));

	    dv[k] = us * (CN(0, 1, k) - CN(0, 0, k)) +

	       u * (CN(1, 1, k) - CN(1, 0, k));

	    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 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);

	       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);

	    }

	    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);

		  DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);

	       }

	    dv[k] = DCN(0, 1) - DCN(0, 0);

	    du[k] = vs * DCN(1, 0) + v * DCN(1, 1);

	    out[k] = vs * DCN(0, 0) + v * DCN(0, 1);

	 }

      }

      else {			/* minorder == vorder */

	    /* 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);

	       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);

	    }

	    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);

		  DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);

	       }

	    du[k] = DCN(1, 0) - DCN(0, 0);

	    dv[k] = us * DCN(0, 1) + u * DCN(1, 1);

	    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);

	    }

	 }

	 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);

	       }

	    }

	 du[k] = vs * (DCN(1, 0) - DCN(0, 0)) + v * (DCN(1, 1) - DCN(0, 1));

	 dv[k] = us * (DCN(0, 1) - DCN(0, 0)) + u * (DCN(1, 1) - DCN(1, 0));

	 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);

	    }

	 }

	 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);

	       }

	    }

	 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);

	    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);

	 }

	 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);

	       DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1);

	    }

	 dv[k] = DCN(0, 1) - DCN(0, 0);

	 du[k] = vs * DCN(2, 0) + v * DCN(2, 1);

	 out[k] = vs * DCN(0, 0) + v * DCN(0, 1);

      }

   }

   else {			/* minorder == vorder */

	 /* 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);

	    }

	 }

	 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);

	       }

	    }

	 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);

	    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);

	 }

	 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);

	       DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0);

	    }

	 du[k] = DCN(1, 0) - DCN(0, 0);

	 dv[k] = us * DCN(0, 2) + u * DCN(1, 2);

	 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;

    */

   for (i = 1; i < MAX_EVAL_ORDER; i++)

      inv_tab[i] = 1.0F / i;

}