m_eval.c

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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/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;
}