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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 "glheader.h"
+#include "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;
+}
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