/* $Id: m_eval.c,v 1.1 2000/12/26 05:09:31 keithw Exp $ */
/*
* Mesa 3-D graphics library
* Version: 3.5
*
* Copyright (C) 1999-2000 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;
GLuint i, k, bincoeff;
if(order >= 2)
{
bincoeff = order-1;
s = 1.0-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 *= 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;
GLuint j, k, bincoeff;
/* 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 = uorder-1;
s = 1.0-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 *= 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 = 1.0-u, vs = 1.0-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] = 1.0 / i;
}