| |
| /* |
| * 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; |
| } |