0,0 → 1,461 |
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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. |
*/ |
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/* |
* 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! |
*/ |
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#include "main/glheader.h" |
#include "main/config.h" |
#include "m_eval.h" |
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static GLfloat inv_tab[MAX_EVAL_ORDER]; |
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/* |
* 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 |
*/ |
|
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void |
_math_horner_bezier_curve(const GLfloat * cp, GLfloat * out, GLfloat t, |
GLuint dim, GLuint order) |
{ |
GLfloat s, powert, bincoeff; |
GLuint i, k; |
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if (order >= 2) { |
bincoeff = (GLfloat) (order - 1); |
s = 1.0F - t; |
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for (k = 0; k < dim; k++) |
out[k] = s * cp[k] + bincoeff * t * cp[dim + k]; |
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for (i = 2, cp += 2 * dim, powert = t * t; i < order; |
i++, powert *= t, cp += dim) { |
bincoeff *= (GLfloat) (order - i); |
bincoeff *= inv_tab[i]; |
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for (k = 0; k < dim; k++) |
out[k] = s * out[k] + bincoeff * powert * cp[k]; |
} |
} |
else { /* order=1 -> constant curve */ |
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for (k = 0; k < dim; k++) |
out[k] = cp[k]; |
} |
} |
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/* |
* 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. |
*/ |
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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; |
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if (vorder > uorder) { |
if (uorder >= 2) { |
GLfloat s, poweru, bincoeff; |
GLuint j, k; |
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/* Compute the control polygon for the surface-curve in u-direction */ |
for (j = 0; j < vorder; j++) { |
GLfloat *ucp = &cn[j * dim]; |
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/* 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; |
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for (k = 0; k < dim; k++) |
cp[j * dim + k] = s * ucp[k] + bincoeff * u * ucp[uinc + k]; |
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for (i = 2, ucp += 2 * uinc, poweru = u * u; i < uorder; |
i++, poweru *= u, ucp += uinc) { |
bincoeff *= (GLfloat) (uorder - i); |
bincoeff *= inv_tab[i]; |
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for (k = 0; k < dim; k++) |
cp[j * dim + k] = |
s * cp[j * dim + k] + bincoeff * poweru * ucp[k]; |
} |
} |
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/* 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 */ |
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if (vorder > 1) { |
GLuint i; |
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/* 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 */ |
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_math_horner_bezier_curve(cn, &cp[i * dim], v, dim, vorder); |
} |
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/* 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); |
} |
} |
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/* |
* 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. |
*/ |
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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; |
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/* 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. */ |
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#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)); |
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/* Derivative direction in v */ |
dv[k] = us * (CN(0, 1, k) - CN(0, 0, k)) + |
u * (CN(1, 1, k) - CN(1, 0, k)); |
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/* 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); |
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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); |
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/* 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); |
} |
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/* 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); |
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/* for the `point' */ |
DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1); |
} |
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/* derivative direction in v */ |
dv[k] = DCN(0, 1) - DCN(0, 0); |
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/* derivative direction in u */ |
du[k] = vs * DCN(1, 0) + v * DCN(1, 1); |
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/* last linear de Casteljau step */ |
out[k] = vs * DCN(0, 0) + v * DCN(0, 1); |
} |
} |
else { /* minorder == vorder */ |
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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); |
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/* 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); |
} |
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/* 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); |
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/* for the `point' */ |
DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0); |
} |
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/* derivative direction in u */ |
du[k] = DCN(1, 0) - DCN(0, 0); |
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/* derivative direction in v */ |
dv[k] = us * DCN(0, 1) + u * DCN(1, 1); |
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/* 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); |
} |
} |
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/* 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); |
} |
} |
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/* derivative direction in u */ |
du[k] = vs * (DCN(1, 0) - DCN(0, 0)) + v * (DCN(1, 1) - DCN(0, 1)); |
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/* derivative direction in v */ |
dv[k] = us * (DCN(0, 1) - DCN(0, 0)) + u * (DCN(1, 1) - DCN(1, 0)); |
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/* 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); |
} |
} |
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/* 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); |
} |
} |
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/* 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); |
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/* 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); |
} |
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/* 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); |
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/* for the `point' */ |
DCN(0, j) = vs * DCN(0, j) + v * DCN(0, j + 1); |
} |
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/* derivative direction in v */ |
dv[k] = DCN(0, 1) - DCN(0, 0); |
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/* derivative direction in u */ |
du[k] = vs * DCN(2, 0) + v * DCN(2, 1); |
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/* last linear de Casteljau step */ |
out[k] = vs * DCN(0, 0) + v * DCN(0, 1); |
} |
} |
else { /* minorder == vorder */ |
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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); |
} |
} |
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/* 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); |
} |
} |
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/* 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); |
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/* 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); |
} |
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/* 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); |
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/* for the `point' */ |
DCN(i, 0) = us * DCN(i, 0) + u * DCN(i + 1, 0); |
} |
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/* derivative direction in u */ |
du[k] = DCN(1, 0) - DCN(0, 0); |
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/* derivative direction in v */ |
dv[k] = us * DCN(0, 2) + u * DCN(1, 2); |
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/* last linear de Casteljau step */ |
out[k] = us * DCN(0, 0) + u * DCN(1, 0); |
} |
} |
#undef DCN |
#undef CN |
} |
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/* |
* Do one-time initialization for evaluators. |
*/ |
void |
_math_init_eval(void) |
{ |
GLuint i; |
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/* KW: precompute 1/x for useful x. |
*/ |
for (i = 1; i < MAX_EVAL_ORDER; i++) |
inv_tab[i] = 1.0F / i; |
} |