::::::::::::::
aitken.c
::::::::::::::
float aitken(degree,coeff,t)
/*
     uses  Aitken's algorithm to compute one coordinate
     value of a  Lagrange curve. Has to be called 
     for each coordinate  (x,y, and/or z) of data set.
     Input:   degree: degree of curve.
              coeff:  array with coordinates to be interpolated.
              t:      parameter value where to interpolate.
      Output: coordinate value.

      Note: we  assume a uniform knot sequence!
*/
     float coeff[];
     float t;
     int degree; 
     {
     int r,i;
     float t1;
     float coeffa[30];
   
     for (i=0; i<=degree; i++) coeffa[i]=coeff[i]; /* save input */
     

     for (r=1; r<= degree; r++)
     for (i=0; i<= degree - r; i++)
     {
	  t1=(degree*t-i)/((float) r);
          coeffa[i]= (1.0-t1)* coeffa[i]  +   t1 * coeffa[i+1] ;
     }     

     return (coeffa[0]);
     }

::::::::::::::
area.c
::::::::::::::
float area(p1,p2,p3)
/* find area of 2D triangle p1,p2,p3 */

	float p1[2], p2[2], p3[2];
{
	return( (p2[0]-p1[0])*(p3[1]-p1[1])-(p2[1]-p1[1])*(p3[0]-p1[0]))/2.0;
}
::::::::::::::
bessel_ends.c
::::::::::::::
void bessel_ends(data,knot,l)
/*	Computes B-spline points data[1] and data[l+]
	according to Bessel end condition.

input:	data:	sequence of data coordinates data[0] to data[l+2].
		Note that data[1] and data[l+1] are expected to
		be empty, as they will be filled by this routine.
	knot:	knot sequence
	l:	number of intervals
 
output:	data:	completed, as above.
*/
	float data[], knot[];
	int l;
{
	float alpha, beta; int i;
	
	if (l==1)
	{/*  This is not really Bessel, but then what do you do
	     when you have only one interval? -- make it linear!
	 */
		data[1]= (2.0*data[0] + data[3])/3.0;
		data[2]= (2.0*data[3] + data[0])/3.0;
	}

	else if (l==2)
	{
		/* beginning:    */
		alpha= (knot[2]-knot[1])/(knot[2]-knot[0]);
		beta = 1.0 - alpha;

		data[1]=(data[2]-alpha*alpha*data[0]-beta*beta*data[4])
		/(2.0*alpha*beta);
		data[1]=2.0*(alpha*data[0]+beta*data[1])/3.0 + data[0]/3.0;

		/* end:  */
		alpha= (knot[2]-knot[1])/(knot[2]-knot[0]);
		beta = 1.0 - alpha;

		data[3]=(data[2]-alpha*alpha*data[0]-beta*beta*data[4])
		/(2.0*alpha*beta);

		data[3]=2.0*(alpha*data[3]+beta*data[4])/3.0+data[4]/3.0;	
	}
	
	else

	{


		/* beginning:    */
		alpha= (knot[2]-knot[1])/(knot[2]-knot[0]);
		beta = 1.0 - alpha;

		data[1]=(data[2]-alpha*alpha*data[0]-beta*beta*data[3])
		/(2.0*alpha*beta);
		data[1]=2.0*(alpha*data[0]+beta*data[1])/3.0 + data[0]/3.0;


		/* end:  */
		alpha= (knot[l]-knot[l-1])/(knot[l]-knot[l-2]);
		beta = 1.0 - alpha;

		data[l+1]=(data[l]-alpha*alpha*data[l-1]-beta*beta*data[l+2])
		/(2.0*alpha*beta);

		data[l+1]=2.0*(alpha*data[l+1]+beta*data[l+2])/3.0+data[l+2]/3.0;

	}

}
::::::::::::::
bez_to_points.
::::::::::::::
void bez_to_points(degree,npoints,coeff,points)
/*	Converts Bezier curve into point sequence. Works on
	one coordinate only.
 
	Input:   degree:  degree of curve.
	         npoints: # of coordinates to be generated. (counting
		          from 0!)
	         coeff:   coordinates of control polygon.
	Output:  points:  coordinates of points on curve.

	Remark: For a 2D curve, this routine needs to be called twice,
		once for the x-coordinates and once for y.
*/

	int degree, npoints;
	float coeff[],points[];

{	
	float t,delt;
	int i;
	float hornbez();
	int k;

	delt=1.0/(float)npoints;
	t=0.0;
	for(i=0; i<=npoints; i++) 
	{
		points[i]=hornbez(degree,coeff,t);
		t = t+delt;
	}
}
::::::::::::::
bez_to_power.
::::::::::::::


void bezier_to_power(degree,bez,coeff)
/*	Converts Bezier form to power (monomial) form. Works on
	one coordinate only.
 
	Input:   degree:  degree of curve.
	         bez:	  coefficients of Bezier form
	Output:  coeff:   coefficients of power form.

	Remark: For a 2D curve, this routine needs to be called twice,
		once for the x-coordinates and once for y.
*/

	int degree;
	float coeff[], bez[];

{	
	float  i_factorial, n_r;
	int i;

	differences(degree,bez,coeff); /* compute forward differences    */
				       /* and store them in coeff.       */

					/* Note that i_factorial is int. */
					/* For high degrees: danger!     */
	coeff[0]=bez[0];
	i_factorial=1;	
	n_r=1;
	for (i=1; i<=degree; i++)
	{
		i_factorial=i_factorial*i;
		n_r= n_r*(degree-i+1);

		coeff[i]=n_r*coeff[i]/i_factorial;
	}


}




::::::::::::::
bspl.dat
::::::::::::::
43
3  1 15 


1.2   0
1.2  0.7
1.2  1.4
1.2  2
2.1  2
3    2.3
3    3
3    8
3    14
3    18
3    19.2 
2.5  20 
0.8  20 
0.8  20.7
0.8  21.4
0.8  22
2.2  22
4.8  22
6.5  22
6.5  21.3
6.5  20.5
6.5  19.8
8.1  22
9    23
10.7 23
11.9 23
13   21.5
13   19
13   16
12   15
11   15
10   15
9.3  16.5
9    19
7.7  16.9
7    15
7    10
7    4
7.8  2.2
10   2
10   1.4
10   0.7
10   0
7    0
4    0
1.2  0


0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9
10 10 10
11 11 11
12 12 12
13 13 13
14 14 14
15 15 15
16 16 16
17 17 17

This file generates the letter 'r', as piecewise cubic B-spline curve
with triple knots everywhere. It is read by the main progam deboormain.c.

::::::::::::::
bspl_kappas.c
::::::::::::::
#include <stdio.h>
#include <math.h>
extern FILE *outfile;

void bspl_kappas(bspl_x,bspl_y,bspl_w,knot,l,dense)
/*	writes curvatures of cubic rational  B-spline curve into 
	a file.
input:
	bspl_x,bspl_y:  2D rat. B-spline polygon
	bspl_w:	the  B-spline weights
	knot:   the knot sequence
	dense:	how many curvature values to compute per interval
	l:      no. of intervals
output:
	written into file outfile
*/

	float bspl_x[],bspl_y[],bspl_w[],knot[];
	int  dense,l;
{
	float bez_x[300],bez_y[300], bez_w[300];
	float bleftx[4],blefty[4],brightx[4],brighty[4];
	float wleft[4], wright[4],coeffx[4],coeffy[4],weight[4];
	float dist,t,delt,h,u,diff;
	int i,j,i3;
	float curvature_0(),curvature_1(),abs();

	/* first, convert B-spline to Bezier: */

	ratbspline_to_bezier(bspl_x,bspl_y,bspl_w,knot,l,
	                     bez_x,bez_y,bez_w);

	/* Now plot kappas for each interval. */

	fprintf(outfile,"%d  %d\n",dense*l,dense);
	for(i=0; i<l; i++)
	{	i3=i*3;
		u= knot[i];  /*starting value for i-th segment*/
		diff=knot[i+1]-knot[i];
		for(j=0;j<=3;j++)
		{       /* create i-th cubic Bezier curve: */ 
			coeffx[j]=bez_x[i3+j]; 
			coeffy[j]=bez_y[i3+j];
			weight[j]=bez_w[i3+j];
		}


		delt=1.0/(float)dense;

		for(t=0.0; t<0.5; t=t+delt)
		{
		subdiv(3,coeffx,weight,t,bleftx,brightx,wleft,wright);
		subdiv(3,coeffy,weight,t,blefty,brighty,wleft,wright);
		h=curvature_0(brightx,brighty,wright,3);

		/* to print file, we'll have to use the GLOBAL
		   parameter u!   */

		u=knot[i]+t*diff;
		fprintf(outfile,"%2.3f  %f\n",u,h);
		}

		for(t=0.5; t<0.999; t=t+delt)
		{
		subdiv(3,coeffx,weight,t,bleftx,brightx,wleft,wright);
		subdiv(3,coeffy,weight,t,blefty,brighty,wleft,wright);

		h=curvature_0(bleftx,blefty,wleft,3);
		h=0.0-h;
		/* minus sign since order of polygon 
		   traversal is reversed!
		*/
		u=knot[i]+t*diff;
		fprintf(outfile,"%2.3f    %f\n",u,h);
		}
	}
	/* print very last kappa: */
	t=1.0;
	h=curvature_1(coeffx,coeffy,weight,3);
	fprintf(outfile,"%2.3f    %f\n",knot[l],h);


}
::::::::::::::
c2_spline.c
::::::::::::::
void c2_spline(knot,l,data_x,data_y,bspl_x,bspl_y)
/* Finds the C2 cubic spline interpolant to
the data points in data_x, data_y.
Input:	knot:	the knot sequence  knot[0], ..., knot[l]
	l:	the number of intervals
	data_x, data_y: the data points data_x[0], ...,
		data[l+2]. Attention:
		data_x[1] and data_x[l+1] are filled
		by  Bessel end conditions and are
		thus ignored on input. Same for data_y.
Output:	bspl_x, bspl_y: the B-spline control polygon of
		the interpolant. Dimensions:
		bspl_x[0], ..., bspl_x[l+2]. Same for
		bspl_y.
*/
	float knot[],data_x[],data_y[],bspl_x[],bspl_y[];
	int l;
{
	float alpha[100], beta[100], gamma[100], up[100], low[100];
	set_up_system(knot,l,alpha,beta,gamma);

	l_u_system(alpha,beta,gamma,l,up,low);

	bessel_ends(data_x,knot,l);
	bessel_ends(data_y,knot,l);

	solve_system(up,low,gamma,l,data_x,bspl_x);
	solve_system(up,low,gamma,l,data_y,bspl_y);

}
::::::::::::::
check_flat.c
::::::::::::::
#include <math.h>

int check_flat(bx,by,degree,tol)
/* Checks if a polygon is flat. If all points
   are closer  than tol to the connection of the 
   two endpoints, then it is flat. Crashes if the endpoints  
   are identical.

Input:	bx,by, degree: the Bezier curve
	tol:            tolerance

*/
	float bx[],by[];
	int degree; float tol;
{
	float dist;
	int i;
	float height();

	for (i=1; i<degree; i++)
	{
		dist = height(bx[i],by[i],bx[0],by[0],
			bx[degree],by[degree]);
		if (fabs(dist) > tol)  return(0);
	}
	
	return(1);
}
::::::::::::::
conic_weight.c
::::::::::::::
#include <math.h>
float conic_weight(b0,b1,b2,p)
/* 
Input:	b0,b1,b2:	conic control polygon
	p:		point on conic
Output:	weight of b1 (assuming standard form).

For method, check Farin  14.5, 14.6.
*/

	float b0[],b1[],b2[],p[];
{
	float area(),length_2();
	float tau0,  tau2 , t,t1, d, dist, weight;
	float b11[2], b10[2];

	tau0=area(b1,p,b2);
	tau2=area(b1,b0,p);
	return(area(b0,b2,p)/(2.0*sqrt(tau0*tau2))); /* weight of b1         */

}
::::::::::::::
curvature_0.c
::::::::::::::
#include <math.h>
float curvature_0(bez_x,bez_y,weight,degree)
/* computes curvature of rational  Bezier curve at t=0 */
	float bez_x[], bez_y[], weight[];
	int degree;
{
	float b0[2],b1[2],b2[2];
	float dist;
	float area();

	b0[0]=bez_x[0]; b1[0]=bez_x[1]; b2[0]=bez_x[2];
	b0[1]=bez_y[0]; b1[1]=bez_y[1]; b2[1]=bez_y[2];

	dist = sqrt( (b1[0]-b0[0])*(b1[0]-b0[0])+
	             (b1[1]-b0[1])*(b1[1]-b0[1])  );


	return (2.0*(degree-1)*weight[0]*weight[2]*area(b0,b1,b2)
	       /(degree*weight[1]*weight[1]*dist*dist*dist));
}
::::::::::::::
deboor.c
::::::::::::::
float deboor(degree,coeff,knot,u,i)
/*
	uses de Boor algorithm to compute one
	coordinate on B-spline curve for param. value u in interval i.
input:  degree:	polynomial degree of each piece of curve
	coeff:	B-spline control points
	knot:	knot sequence
	u:	evaluation abscissa
	i:	u's interval: u[i]<= u < u[i+1]
output:	coordinate value. 
*/

	float coeff[],knot[];
	float u;
	int degree,i;
{
	int k,j;
	float t1,t2;
	float coeffa[30];  /* might need adjustment! */
	
	for (j=i-degree+1; j<=i+1; j++)coeffa[j]=coeff[j];

	for (k=1; k<= degree; k++)
	for ( j=i+1 ;j>=i-degree+k+1; j--)
	{
		t1= (knot[j+degree-k] - u )/(knot[j+degree-k]-knot[j-1]);
		t2= 1.0-t1;

		coeffa[j]=t1* coeffa[j-1]+t2* coeffa[j];
	}	
	return (coeffa[i+1]);
}



::::::::::::::
deboormain.c
::::::::::::::
#include <stdio.h>
#include <math.h>


	FILE *file, *outfile, *psfile;

main()
{

	int i,j,l,polygon,buttno,l2,dense,degree,l21,l2n;
	int index[2],numpts;
	float points_x[1000],points_y[1000];
	float mmbox[4], del_x, del_y;
	float bspl_x[200],bspl_y[200],knot[200];
  	float dum,x,y,xx,yy,tight,scale_x,scale_y;
	/* ----------------------------- */

	file = fopen("bspl.dat","r");
	outfile= fopen("dch.dat","w");
	psfile= fopen("db.ps","w");

	fscanf(file,"%d", &l);
	fscanf(file,"%d", &degree);
	fscanf(file,"%d", &polygon);
	fscanf(file,"%d", &dense);
	fscanf(file,"\n");
	
	l2=l+degree-1;
	for(i=0; i <= l2 ; i++)
	{
	        fscanf(file,"%f",&dum);
		bspl_x[i]= dum;
	        fscanf(file,"%f",&dum);
		bspl_y[i]= dum;	
 	}	 
	fscanf(file,"\n");

/*	for(i=0;i<=l2;i++)
		{
		printf("(%f,%f),(%d,%d) ",
		bspl_x[i],bspl_y[i],l,dense);
		printf("\n"); 
		}
*/

	l2n = l + 2*degree - 2;
	for(i=0; i <= l2n ; i++)
	{
	        fscanf(file,"%f",&dum);
		knot[i]= dum;	    
 	}	

	fclose(file);
	fclose(outfile);

	minmax(bspl_x,bspl_y,l2,mmbox);
	del_x=mmbox[1]-mmbox[0]; del_y=mmbox[3]-mmbox[2];
	if(del_x > del_y) mmbox[3]=mmbox[2]+del_x;
	else mmbox[1]=mmbox[0]+del_y;

	scale_x=350.0/(mmbox[1]-mmbox[0]);
	scale_y=350.0/(mmbox[3]-mmbox[2]);



	bspl_to_points(degree,l,bspl_x,knot,dense,points_x,&numpts);
	bspl_to_points(degree,l,bspl_y,knot,dense,points_y,&numpts);

/*************** ps output ********************************/


	/* macro to mark B-spline vertex:*/
		fprintf(psfile,"/blackbox\n");
		fprintf(psfile,"{0 setgray\n");
		fprintf(psfile,"-3.5 -3.5 rlineto\n");
		fprintf(psfile,"7 0 rlineto\n");
		fprintf(psfile,"0 7 rlineto\n");
		fprintf(psfile,"-7 0 rlineto\n");
		fprintf(psfile,"0 -7 rlineto\n");
		fprintf(psfile,"closepath fill} def\n");
	

		fprintf(psfile,"/whitebox\n");
		fprintf(psfile,"{1 setgray\n");
		fprintf(psfile,"-3 -3 rlineto\n");
		fprintf(psfile,"6 0 rlineto\n");
		fprintf(psfile,"0 6 rlineto\n");
		fprintf(psfile,"-6 0 rlineto\n");
		fprintf(psfile,"0 -6 rlineto\n");
		fprintf(psfile,"closepath fill} def\n");
	

/* plot curve */

	fprintf(psfile,"3 setlinewidth \n");
	fprintf(psfile,"0 setgray\n");

	fprintf(psfile,"200 400 translate\n");

		fprintf(psfile,"newpath \n");

		fprintf(psfile,"%f %f moveto\n",
			scale_x*(points_x[0]-mmbox[0]),
			scale_y*(points_y[0]-mmbox[2]));
		
		for( j=1; j<numpts ; j++)
		{
			fprintf(psfile,"%f %f lineto\n",
			scale_x*(points_x[j]-mmbox[0]),
			scale_y*(points_y[j]-mmbox[2]));
 		fprintf(psfile,"stroke \n");


	
	/**** fill in junction points  black: **********/
	fprintf(psfile,"1 setlinewidth \n");


	for(j=0; j<= numpts; j=j+dense)
			{
			fprintf(psfile,"%f %f %f %f %f arc fill\n",
			scale_x*(points_x[j]-mmbox[0]), 
			scale_y*(points_y[j]-mmbox[2]),
		 	3.0, 0.0, 360.0);
			j++; /* there in points twice */
			}

if (polygon == 0)
	{
		fprintf(psfile,"showpage \n");
		 return;
	}


if (polygon>0) /* plot B-spline polygon: */
	{
		fprintf(psfile,"0.0 setgray\n ");
		fprintf(psfile,"1 setlinewidth\n ");

		fprintf(psfile,"newpath \n");
		fprintf(psfile,"%f %f moveto\n",
			scale_x*(bspl_x[0]-mmbox[0]), 
			scale_y*(bspl_y[0]-mmbox[2]));

		for(j=1; j<=  l2; j++)
			{
			fprintf(psfile,"%f %f lineto\n",
			scale_x*(bspl_x[j]-mmbox[0]), 
			scale_y*(bspl_y[j]-mmbox[2]));
			}
		fprintf(psfile,"stroke \n");



	/***** plot  control vertices: ************/


	for(j=0; j<=l2; j++)
	{
		fprintf(psfile,"%f %f moveto\n",
		scale_x*(bspl_x[j]-mmbox[0]), 
		scale_y*(bspl_y[j]-mmbox[2]));
		fprintf(psfile,"blackbox \n");   

		if (j!=0 && j!=l2)
		{
			fprintf(psfile,"%f %f moveto\n",
			scale_x*(bspl_x[j]-mmbox[0]), 
			scale_y*(bspl_y[j]-mmbox[2]));
			fprintf(psfile,"whitebox \n");  
		}
	}

	}

		
		fprintf(psfile,"showpage \n");

	fclose(psfile);

/***********************************************************/


}

::::::::::::::
decas.c
::::::::::::::
float decas(degree,coeff,t)
/*
     uses  de Casteljau to compute one coordinate
     value of a  Bezier curve. Has to be called 
     for each coordinate  (x,y, and/or z) of a control polygon.
     Input:   degree: degree of curve.
              coeff:  array with coefficients of curve.
              t:      parameter value.
      Output: coordinate value.
*/

	float coeff[];
	float t;
	int degree;
{
	int r,i;
	float t1;
	float coeffa[30];  /* an auxiliary array. Change dim. if too small*/

	t1 = 1.0 - t;
	for(i=0; i<=degree; i++) coeffa[i]=coeff[i]; /* save input */

	for (r=1; r<= degree; r++)
		for (i=0; i<= degree - r; i++)
		{
			coeffa[i]= t1* coeffa[i]  +   t * coeffa[i+1] ;
		}

	return (coeffa[0]);
}
::::::::::::::
decas.ps
::::::::::::::
300 350  translate
1.3 2.0 scale
0.1 setlinewidth
newpath
-131.250000 125.000000 moveto
-131.250000 125.000000 lineto
87.500000 75.000000 lineto
87.500000 25.000000 lineto
0.000000 25.000000 lineto
stroke
newpath
-131.250000 125.000000 moveto
-131.250000 125.000000 lineto
87.500000 75.000000 lineto
87.500000 25.000000 lineto
stroke
newpath
-131.250000 125.000000 moveto
-131.250000 125.000000 lineto
87.500000 75.000000 lineto
stroke
newpath
-131.250000 125.000000 moveto
-131.250000 125.000000 lineto
stroke
newpath
-131.250000 125.000000 moveto
-125.781250 123.749995 lineto
87.500000 73.750001 lineto
85.312502 25.000000 lineto
0.000000 22.499999 lineto
stroke
newpath
-125.781250 123.749995 moveto
-120.449217 122.499990 lineto
87.445311 72.531253 lineto
83.179688 24.937500 lineto
stroke
newpath
-120.449217 122.499990 moveto
-115.251856 121.250772 lineto
87.338672 71.341407 lineto
stroke
newpath
-115.251856 121.250772 moveto
-110.187091 120.003033 lineto
stroke
newpath
-131.250000 125.000000 moveto
-120.312500 122.500002 lineto
87.500000 72.500002 lineto
83.124999 25.000000 lineto
0.000000 20.000000 lineto
stroke
newpath
-120.312500 122.500002 moveto
-109.921877 120.000005 lineto
87.281250 70.125002 lineto
78.968748 24.750000 lineto
stroke
newpath
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showpage
::::::::::::::
decasmain.c
::::::::::::::
#include <stdio.h>
#include <math.h>

/* to plot just one bez. curve */

	FILE *file, *outfile, *psfile;

main()
{

	int i,j,jj,l,pol,buttno,dense,degree,l21,l2n;
	int index[2],numpts;
	float points_x[1000],points_y[1000];
	float mmbox[4], del_x, del_y;
	float bez_x[200],bez_y[200],weight[20];
	float cez_x[200],cez_y[200];
	float coeff_x[200],coeff_y[200];
  	float x,y,scale_x,scale_y;
	float cx[100],cy[100],cw[100];
	float xx,yy,dum;
	/* ----------------------------- */

	file = fopen("bez.dat","r");
	outfile= fopen("dch.dat","w");
	psfile= fopen("decas.ps","w");

	fscanf(file,"%d", &degree);
	fscanf(file,"%d", &pol);
	fscanf(file,"%d", &dense);
	fscanf(file,"\n");
	
	for(i=0; i <= degree ; i++)
	{
	        fscanf(file,"%f",&dum);
		bez_x[i]= dum;
	        fscanf(file,"%f",&dum);
		bez_y[i]= dum;	   
		fscanf(file,"%f", &dum);
		weight[i]= dum;
 	}	 
	fscanf(file,"\n");
/*
	for(i=0;i<=degree;i++)
		{xx=bez_x[i]; yy=bez_y[i];
		printf("(%f,%f),(%d,%d) ",
		xx,yy,degree,dense);
		printf("\n"); 
		}
*/


	fclose(file);
	fclose(outfile);

	minmax(bez_x,bez_y,degree,mmbox);
	del_x=mmbox[1]-mmbox[0]; del_y=mmbox[3]-mmbox[2];
	if(del_x > del_y) mmbox[3]=mmbox[2]+del_x;
	else mmbox[1]=mmbox[0]+del_y;

	scale_x=350.0/(mmbox[1]-mmbox[0]);
	scale_y=350.0/(mmbox[3]-mmbox[2]);


	ratbez_to_points(degree,50,bez_x,weight,points_x,1);
	ratbez_to_points(degree,50,bez_y,weight,points_y,1);
/*************** ps output ********************************/

	
	fprintf(psfile,"200 400 translate\n");	
	/***** plot  bezier control polygon if pol=1: ************/
if (pol == 1)
	{
		fprintf(psfile,"0.0 setgray\n ");
		fprintf(psfile,"1  setlinewidth\n ");

		fprintf(psfile,"newpath \n");
		fprintf(psfile,"%f %f moveto\n",
			scale_x*(bez_x[0]-mmbox[0]), scale_y*(bez_y[0]-mmbox[2]));

		for(j=0; j<=  degree; j++)
			{
			fprintf(psfile,"%f %f lineto\n",
			scale_x*(bez_x[j]-mmbox[0]), scale_y*(bez_y[j]-mmbox[2]));
			}
		fprintf(psfile,"stroke \n");



	/***** plot  control vertices: ************/

	fprintf(psfile,"0.0 setgray\n ");

	for(j=1; j< degree ; j++)
		{
			xx=scale_x*(bez_x[j]-mmbox[0]);
		 	yy=scale_y*(bez_y[j]-mmbox[2]);
			fprintf(psfile,"%f %f %f %f %f arc stroke\n",
		 	xx,yy,3.0, 0.0, 360.0);
		}
	fprintf(psfile,"1.0 setgray\n ");

	for(j=1; j< degree; j++)
		{
			xx=scale_x*(bez_x[j]-mmbox[0]);
		 	yy=scale_y*(bez_y[j]-mmbox[2]);
			fprintf(psfile,"%f %f %f %f %f arc fill\n",
		 	xx,yy,2.80, 0.0, 360.0);
		}

	}	

	

/* plot curve */

	fprintf(psfile,"3 setlinewidth \n");
	fprintf(psfile,"0 setgray\n");

		fprintf(psfile,"newpath \n");
		xx = scale_x*(points_x[0]-mmbox[0]);
		yy = scale_y*(points_y[0]-mmbox[2]);

		fprintf(psfile,"%f %f moveto\n", xx,yy);

		for( j=1; j<=50 ; j++)
		{
			xx = scale_x*(points_x[j]-mmbox[0]);
			yy = scale_y*(points_y[j]-mmbox[2]);
			fprintf(psfile,"%f %f lineto\n", xx,yy);
		}
 		fprintf(psfile,"stroke \n");
	

/**** fill in junction points  black: **********/
	fprintf(psfile,"1 setlinewidth \n");

	fprintf(psfile,"0 setgray\n");
	for(j=0; j<= dense; j=j+dense)
		{
			xx = scale_x*(points_x[j]-mmbox[0]);
			yy = scale_y*(points_y[j]-mmbox[2]);
			fprintf(psfile,"%f %f %f %f %f arc fill\n",
			xx,yy, 3.0, 0.0, 360.0);
		}




		

	fprintf(psfile,"showpage\n");
	return;

}		
::::::::::::::
differences.c
::::::::::::::
void differences(degree,coeff,diffs)
/*
     Computes all forward differences Delta^i(b_0).
     Has to be called 
     for each coordinate  (x,y, and/or z) of a control polygon.
     Input:   degree: length (from 0) of coeff.
              coeff:  array of coefficients.
      Output: diffs:  diffs[i]= Delta^i(coeff[0]).
*/
     float coeff[],diffs[];
     int degree; 
     {
     int r,i;
	float diffs1[20];

	for(i=0; i<=degree; i++) diffs1[degree-i]=coeff[i]; 

	for (r=1; r<= degree; r++)
	for (i=0; i<= degree - r; i++)
		diffs1[i]=  diffs1[i] - diffs1[i+1] ;  

	for(i=0;i<=degree;i++) diffs[i]=diffs1[degree-i];   

     }

::::::::::::::
dist.c
::::::::::::::
#include <math.h>

float dist(a,b)
/* finds distance between 2D points a   and b.

Input: a,b:	points;
*/
	float a[], b[];
{
	return(sqrt((a[0]-b[0])*(a[0]-b[0])+
	            (a[1]-b[1])*(a[1]-b[1]) ) );
}
::::::::::::::
fair1.ps
::::::::::::::
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::::::::::::::
fair_bspline.c
::::::::::::::
void fair_kr_bspline(bspl,knot,l,from,to)
/* Fairs a cubic rational B-spline curve by knot removal/reinsertion.
 
Input:	bspl:      cubic B-spline control polygon (one coord.)
	knot:      knot sequence
	l:         no. of intervals
	from, to:  from where to where to fair
Output:	same as input, but hopefully fairer.
*/


	float bspl[], knot[];
	int l,from,to;
{
	int i, i3,  l2;
	float left,right, middle;
	float Knot[100]; /*to simulate multiple knots at ends */

	Knot[0]=Knot[1]=knot[0];

	for(i=0; i<=l; i++)Knot[i+2]=knot[i];
	Knot[l+3]=Knot[l+4]=knot[l];

	/* With new knot vector, bspl[i] is associated with Knot[i+1] */
	
	l2=l+2; 

	if(from <= 2) from = 2;
	if(to >=l) to=l;

	for (i=from; i<=to; i++) 
	{
		left=((Knot[i+2]-Knot[i-2])*bspl[i-1]
                       -(Knot[i+2]-Knot[i+1]  )*bspl[i-2]  )
                       /(Knot[i+1]-Knot[i-2] );

		
		right=((Knot[i+4]-Knot[i])*bspl[i+1]
                       -(Knot[i+1]-Knot[i]  )*bspl[i+2] )
                       /(Knot[i+4]-Knot[i+1] );

		
		bspl[i]=((Knot[i+3]-Knot[i+1])*left
		          +(Knot[i+1]-Knot[i-1])*right )
		          /(Knot[i+3]-Knot[i-1]) ;

	
	
	}
   
}
 
::::::::::::::
fair_surf.c
::::::::::::::
void fair_surf(bspl,lu,lv,knot_u, knot_v)
/* 	Fairs B-spline control net.
Input:	bspl:   B-spline control net (one coordinate only)
	lu,lv:  no. of intervals in u- and v-direction
	knots_u, knots_v:   knot vectors in u- and v-direction
Output:	as input

*/
	float bspl[20][20],knot_u[],knot_v[];
	int  lu,lv;
{
	int  i,j,lu2,lv2;
	float b[20];

	lu2=lu+2; lv2=lv+2; 
 

	for(i=0;  i<= lu2; i++)
	{

		for(j=0; j<=lv2; j++)  b[j]=bspl[i][j]; /* grab i-th row*/
		fair_bspline(b,knot_v,lv,0,lv);         /* fair row     */
		for(j=0; j<=lv2; j++)  bspl[i][j]=b[j]; /* put row back */
	}

	for(j=0;  j<= lv2; j++)
	{

		for(i=0; i<=lu2; i++)  b[i]=bspl[i][j]; /* grab j-th col*/
		fair_bspline(b,knot_u,lu,0,lu);         /* fair col     */
		for(i=0; i<=lu2; i++)  bspl[i][j]=b[i]; /* put col back */
	}


}
::::::::::::::
gamma_spline.c
::::::::::::::
void  gamma_spline(bspl,l,gamma,bez)
/*From B-spline polygon and gammas, find interior
  Bezier points b3i+-1.
Input:	bspl:	control polygon (one coordinate only)
	l:	number of cubic pieces
	gamma:	gamma_i's
Output:	bez:	piecewise Bezier polygon,  but without the 
		junction points b3i
*/
	int l;
	float bspl[],gamma[],bez[];
{
	int i,i3,l3;

	bez[0]=bspl[0];
	bez[1]=bspl[1];
	if(l>1)
	{
	bez[2]=(gamma[2]*bspl[1]
		+bspl[2]    )/(gamma[2]+1.0);
	}   
	if(l>2)
	{
	bez[4]=( (1.0+gamma[3])*bspl[2]
	        +gamma[2]*bspl[3]  )/(1.0+gamma[2]+gamma[3]);
	}
/* Now the main part:   */

	for(i=2; i<l-1; i++)
	{		
	i3=3*i;
	bez[i3-1]=( gamma[i+1]*bspl[i]
	         +  (gamma[i]+1.0)*bspl[i+1])/(1.0+gamma[i]+gamma[i+1]);

	bez[i3+1]=( (gamma[i+2]+1.0)*bspl[i+1]
	         +  gamma[i+1]*bspl[i+2]  )/(1.0+gamma[i+1]+gamma[i+2]);

	}
	
/* The last points need special attention:  */

	l3=l*3;
	if(l>2)
	{
	bez[l3-4]=( gamma[l]*bspl[l-1]
	         +(1.0+gamma[l-1])*bspl[l]  )/(1.0+gamma[l-1]+gamma[l]);
	}
	if(l>1)
	{
	bez[l3-2]=( bspl[l]
		 +  gamma[l]*bspl[l+1]  )/(gamma[l]+1.0);
	}
	bez[l3-1]= bspl[l+1];
	bez[l3]  = bspl[l+2]; 

}
::::::::::::::
height.c
::::::::::::::
#include <stdio.h>
#include <math.h>


float height(px,py,ax,ay,bx,by)
/* find the height of point (px,py) over
   straight line thru (ax,ay) and (bx,by). 
*/
	float px,py,ax,ay,bx,by;
{
	float det,length;
	det = px*ay + ax*by + bx*py -ay*bx - by*px - py*ax;

	length = sqrt( (bx-ax)*(bx-ax) + (by-ay)*(by-ay) );

	if(length < 0.000001)   /* height doesn't make sense, hence: */
		return( sqrt( (px-ax)*(px-ax) + (py-ay)*(py-ay) ) );
	else
	return (det/length);
}

::::::::::::::
hornbez.c
::::::::::::::
float hornbez(degree,coeff,t)
/*
     uses  Horner's scheme to compute one coordinate
     value of a  Bezier curve. Has to be called 
     for each coordinate  (x,y, and/or z) of a control polygon.
     Input:   degree: degree of curve.
              coeff:  array with coefficients of curve.
              t:      parameter value.
      Output: coordinate value.
*/
	int degree;
	float coeff[];
	float t;
{
	int i;
	int n_choose_i;          /* shouldn't be too large! */
	float fact,t1,aux;

	t1=1.0 - t;  fact=1.0;
	n_choose_i=1;

	aux=coeff[0]*t1;          /* starting the evaluation
                                  loop */
	for(i=1; i<degree; i++)
	{
		fact=fact*t;
		n_choose_i=n_choose_i*(degree-i+1)/i;  /* always int! */
		aux=(aux + fact*n_choose_i*coeff[i])*t1;
	}
	aux = aux + fact*t*coeff[degree];

	return aux;
}

::::::::::::::
horner.c
::::::::::::::
float horner(degree,coeff,t)
/*
     uses  Horner's scheme to compute one coordinate
     value of a curve in power form. Has to be called 
     for each coordinate  (x,y, and/or z) of a control polygon.
     Input:   degree: degree of curve.
              coeff:  array with coefficients of curve.
              t:      parameter value.
      Output: coordinate value.
*/
	int degree;
	float coeff[];
	float t;
{
	int i;
	float aux;

	aux=coeff[degree];      

	for(i=degree; i>0; i--)
		aux= t*aux + coeff[i-1];


	return(aux);
}

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:::::::::::::::::
intersect_lines.c
:::::::::::::::::

void intersect_lines(a,b,v,w,p)
/* finds the intersection p of the two straight lines a + tv and
	b+tw. No check for infinite p is made.
*/
	float a[2],b[2],v[2],w[2],p[2];

{
	float D,t;
	float diff[2];

	diff[0]=a[0]-b[0];  diff[1]=a[1]-b[1];

	D= w[1]*v[0] - w[0]*v[1]; 


	t= (diff[1]*w[0] - diff[0]*w[1]  )/D;

	p[0]=a[0]+t*v[0];  p[1]=a[1]+t*v[1];

}

::::::::::::::::::
intersect_points.c
::::::::::::::::::

void intersect_points(a,b,c,d,p)
/* finds the intersection p of the two straight lines [a,b] and
	[c,d]. No check for infinite p is made.
*/
	float a[2],b[2],c[2],d[2],p[2];

{
	float diff1[2],diff2[2];

	diff1[0]=b[0]-a[0]; diff1[1]=b[1]-a[1];
	diff2[0]=d[0]-c[0]; diff2[1]=d[1]-c[1];

	intersect_lines(a,d,diff1,diff2,p);

}


::::::::::::::
intersect.c
::::::::::::::
#include <stdio.h>
#include <math.h>
extern FILE *outfile, *psfile;

void intersect(bx,by,w,degree,tol)
/* Intersects bezier curve with x-axis  by  adaptive subdivision. 
   Subdivision is controlled by tolerance tol. There is
   no check for stack depth! Intersection points are not found  in
   `natural' order.  Results are written into file outfile.
	Input: bx,by,w:	rational Bezier curve
		degree:	its degree
		tol: 	accuracy for results
 */

	int degree; float tol;
	float bx[],by[],w[];
{
	float a[2],b[2],c[2],d[2],p[2];
	float bxleft[20],byleft[20],bxright[20],
		byright[20],wleft[20],wright[20];
	float box[4],xx,yy;
	int j;
	int check_flat();
	
	/********* does minmaxbox contain x-axis? **********/

	minmax(bx,by,degree,box);



	if (box[2]*box[3] > 0.0) return;

	/********* is polygon flat enough? ********/

	if( check_flat(bx,by,degree,tol)== 1 )
		{ 
		a[0]=bx[0];      a[1]=by[0];        /* intersect curve       */
		b[0]=bx[degree]; b[1]=by[degree];   /* endpoints with        */
		c[0]=0.0;        c[1]=0.0;          /* x- axis. Intersection */
		d[0]=1.0;	 d[1]=0.0;          /* is in p[0], p[1].     */
		intersect_points(a,b,c,d,p);

		fprintf(outfile,"%f  %f\n",p[0],p[1]);
		}
	/******** if not flat enough, subdivide: ********/
	else
		{

		subdiv(degree,bx,w,0.5,bxleft,bxright,wleft,wright);
		subdiv(degree,by,w,0.5,byleft,byright,wleft,wright);

	/******** ... and repeat for each half: **********/

		intersect(bxleft, byleft, wleft, degree,tol);
		intersect(bxright,byright,wright,degree,tol);
		}
}
::::::::::::::
l_u_system.c
::::::::::::::
void l_u_system(alpha,beta,gamma,l,up,low)
/*	perform LU decomposition of tridiagonal system with
	lower diagonal alpha, diagonal beta, upper diagonal gamma.

input:	alpha,beta,gamma: the coefficient matrix entries
	l:	matrix size [0,l]x[0,l]
	low:	L-matrix entries
	up:	U-matrix entries
*/
	float alpha[],beta[],gamma[],low[],up[];
	int l;
{
	int i;

	up[0]=beta[0];
	for(i=1; i<=l; i++)
	{
		low[i]=alpha[i]/up[i-1];
		up[i] =beta[i]-low[i]*gamma[i-1];
	}
}
::::::::::::::
length_2.c
::::::::::::::
#include <math.h>
float length_2(a)

/* computes length of vector a
input:	a
output:	length
*/
float a[];

{float x;
	x=sqrt(a[0]*a[0]+a[1]*a[1]);
	return(x);
}
::::::::::::::
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::::::::::::::
minmax.c
::::::::::::::
 float minmax(bx,by,degree,value)
	/* find minmax box of 2D polygon bx,by */

	int degree;
	float bx[],by[],value[];
{
	int i;
	float xmin,xmax,ymin,ymax;

	xmin= bx[0]; ymin = by[0];
	xmax= bx[0]; ymax = by[0] ;

	for (i=0; i<=degree; i++)
	{
		if (bx[i]<xmin)xmin=bx[i];
		if (bx[i]>xmax)xmax=bx[i];

		if (by[i]<ymin)ymin=by[i];
		if (by[i]>ymax)ymax=by[i];
	
	}
		value[0]=xmin;
		value[1]=xmax;
		value[2]=ymin;
		value[3]=ymax;

}

	
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::::::::::::::
minmax_surf.c
::::::::::::::
 minmax_surf(bx,by,degree_u,degree_v,value)
	/* find minmax box of 2D polygon bx,by */

	int degree_u,degree_v;
	float bx[20][20],by[20][20],value[];
{
	int i,j;
	float xmin,xmax,ymin,ymax;

	xmin= bx[0][0]; ymin = by[0][0];
	xmax= bx[0][0]; ymax = by[0][0] ;

	for (i=0; i<=degree_u; i++)
	for (j=0; j<=degree_v; j++)
	{
		if (bx[i][j]<xmin)xmin=bx[i][j];
		if (bx[i][j]>xmax)xmax=bx[i][j];

		if (by[i][j]<ymin)ymin=by[i][j];
		if (by[i][j]>ymax)ymax=by[i][j];
	
	}
		value[0]=xmin;
		value[1]=xmax;
		value[2]=ymin;
		value[3]=ymax;

}

	
::::::::::::::
netcoons.c
::::::::::::::
void netcoons(net,rows,columns)
/* Uses bilinear Coons blending to complete a control
net of which only the four boundary polygons are given.
Works for one coordinate only.
Input:	net:	control net - only boundaries are considered as input.
	rows, columns: dimensions of net.
Output:	net:	the completed net, with the old boundaries.
*/
	float net[20][20]; 
	int rows, columns; /*If either of these is >20, adjust dimensions*/
                           /*in previous statement!                      */
{
	int i,j;
	float u,v,u1,v1;

	for(i=0; i<=rows;    i++)
	for(j=0; j<=columns; j++)
	{

		u=((float)i)/rows;    u1=1.0-u;
		v=((float)j)/columns; v1=1.0-v;

		net[i][j]=u1*net[0][j] + u*net[rows][j]
			 +v1*net[i][0] + v*net[i][columns]
			 -u1*v1*net[0][0]    - u1*v*net[0][columns]
			 -u *v1*net[rows][0] - u *v*net[rows][columns];

	}
}
::::::::::::::
nurb_circ.dat
::::::::::::::

-1 0 
-1 0.5  
0 2     
1 0.5   
1 0 

1 0.66666 0.33333 0.66666 1
0 1 2 


% a semicircle, written as a C^2 NURB with
%two cubic segments over the uniform knot sequence
%0,1,2. First: the 5 control points. Next: their weights.
::::::::::::::
parameters.c
::::::::::::::
#include <math.h>

void parameters(data_x,data_y,l,knot)
/*	Finds a centripetal parametrization for a given set
	of 2D data points. 
Input:	data_x, data_y:	input points, numbered from 0 to l+2. 
	l:		number of intervals.
Output:	knot:		knot sequence. Note: not (knot[l]=1.0)!
Note:	data_x[1], data_x[l+1] are not used! Same for data_y.
*/
	float data_x[], data_y[],knot[];
	int l;

{
	int i,  l1,l2;     /* In the following, special care must be    */
	float delta;       /* at the ends because of the data structure */
	                   /* used. See note above.                     */
	l1= l-1; l2 = l+2;
	knot[0] = 0.0;     /* initialize -- arbitrary */

	delta=sqrt((data_x[2]-data_x[0])*(data_x[2]-data_x[0])
		         +(data_y[2]-data_y[0])*(data_y[2]-data_y[0]));
	knot[1]= sqrt(delta);                /* leave out this sqrt
		                                if you want chord length.*/

	for(i=2; i<l; i++)
	{
	        delta=sqrt((data_x[i+1]-data_x[i])*(data_x[i+1]-data_x[i])
		         +(data_y[i+1]-data_y[i])*(data_y[i+1]-data_y[i]));

		knot[i]=knot[i-1]+sqrt(delta);/*leave out this sqrt
		                                if you want chord length.*/
	}

	delta=sqrt((data_x[l2]-data_x[l])*(data_x[l2]-data_x[l])
		         +(data_y[l2]-data_y[l])*(data_y[l2]-data_y[l]));
	knot[l]= knot[l-1]+sqrt(delta);      /* leave out this sqrt
		                                if you want chord length.*/

}
::::::::::::::
plot_b_kappa.c
::::::::::::::
#include <stdio.h>
#include <math.h>
 

main() /* program to plot curvature of a rational 
	  B-spline curve, expressed in piecewise
 	  rational cubic form. All curvatures are expected
	  in the input file b_kappas.dat. */
{
	FILE *file, *plotfile;

	int i,j,degree,degree1,degree2,istep,tol,buttno,howmany,dense;
	float t,shoulder,width,xmin1,ymin1,xmax1,ymax1,x,y,xx,yy,
		kappa_min, kappa_max, scale_x,scale_y, setline;
	float coeffx[1000],value[4];
	float coeffy[1000];
	float abs();
	char min[20],max[20];

	file = fopen("b_kappas.dat","r");
	plotfile= fopen("psplot.ps","w");
	fscanf(file,"%d", &howmany);
	fscanf(file,"%d", &dense);

	for(i=0; i <= howmany ; i++)
	{
		fscanf(file,"%f",&coeffx[i]);
		fscanf(file,"%f",&coeffy[i]);
 	}	
	minmax(coeffx,coeffy,howmany,value);
	sprintf(min,"%4.2f",value[2]);
	sprintf(max,"%4.2f",value[3]);
	kappa_max=value[3];
	kappa_min=value[2];

	width=value[3]-value[2];  /*to scale kappa printouts  */




	minmax(coeffx,coeffy,howmany,value); /* Setting the window:   */
	if (value[2] > 0.0) value[2]=0.0;  /* make sure kappa=0 is  */
	if (value[3] < 0.0) value[3]=0.0;  /* drawn. Give some room */
	width=(value[3]-value[2])/20.0;    /* in y-direction.       */
	value[2]=value[2]-width;
	value[3]=value[3]+width;


	degree1 = degree+1;






/*********************creating a postscript file for the curvatures: *******************/


		/* rotate, so paper is used sideways */

		fprintf(plotfile,"90 rotate\n");           
		fprintf(plotfile,"70 -602 translate\n");  
		fprintf(plotfile,"10 300 translate\n");

		scale_x=450.0/(value[1]-value[0]);
		scale_y=300.0/(value[3]-value[2]);

	
		fprintf(plotfile,"0.6 setgray \n");
		fprintf(plotfile,"%f setlinewidth\n",0.1);

		fprintf(plotfile,"%f %f translate\n",
			-1.0*scale_x*value[0], -1.0*scale_y*value[2]);

	
		fprintf(plotfile,"newpath \n");
		fprintf(plotfile," %f %f moveto\n",
			0.0,0.0);
		fprintf(plotfile," %3.2f %3.2f lineto\n",
			scale_x*value[1],0.0);
		fprintf(plotfile,"stroke\n");

		fprintf(plotfile,"newpath \n");

		
		fprintf(plotfile," %f %f moveto\n",
			scale_x*value[1],0.0);
		fprintf(plotfile," %f %f lineto\n",
			scale_x*value[0],0.0);
		fprintf(plotfile," %f %f lineto\n",
			scale_x*coeffx[0],scale_y*coeffy[0]);


			for(j=1; j<= howmany; j++)
			{
			fprintf(plotfile," %f %f lineto\n",
			scale_x*coeffx[j], scale_y*coeffy[j]);
			}
		fprintf(plotfile," %f %f lineto\n",
			scale_x*coeffx[howmany],scale_y*coeffy[howmany]);
		fprintf(plotfile,"fill\n");



		/******* plot box : *************/
	
		fprintf(plotfile,"0.0 setgray\n");
		fprintf(plotfile,"0.3 setlinewidth\n");
		fprintf(plotfile,"newpath\n");
		fprintf(plotfile,"%f %f moveto\n",
			scale_x*value[0],scale_y*value[2]);
		fprintf(plotfile,"%f %f lineto\n",
			scale_x*value[0],scale_y*value[3]);
		fprintf(plotfile,"%f %f lineto\n",
			scale_x*value[1],scale_y*value[3]);
		fprintf(plotfile,"%f %f lineto\n",
			scale_x*value[1],scale_y*value[2]);
		fprintf(plotfile,"%f %f lineto\n",
			scale_x*value[0],scale_y*value[2]);
		fprintf(plotfile,"stroke\n");

	/***** plot markers: ************/

	fprintf(plotfile,"0.0 setgray\n ");

	if(howmany > dense)
	for(j=0; j<= howmany; j=j+dense)
			{
			if(j >= 0) /*to keep numbering right */
	/*			    but to skip first point */
			fprintf(plotfile,"%f %f %f %f %f arc stroke\n",
			scale_x*coeffx[j], scale_y*coeffy[j],
		 	3.0, 0.0, 360.0);
			}

	fprintf(plotfile,"1.0 setgray\n ");

	if(howmany > dense)
	for(j=0; j<= howmany; j=j+dense)
			{
			if(j >= 0)  /* see above  */ 
			fprintf(plotfile,"%f %f %f %f %f arc fill\n",
			scale_x*coeffx[j], scale_y*coeffy[j],
		 	2.6, 0.0, 360.0);
			}

	/***** print min-max values: *************/

	fprintf(plotfile,"0.0 setgray\n ");

	fprintf(plotfile,"/Times-Roman findfont 15 scalefont setfont\n");

	fprintf(plotfile,"%f %f moveto\n",
		scale_x*value[1]+10, scale_y*kappa_max);

	fprintf(plotfile,"(max: %3.3f)show\n",kappa_max);

	fprintf(plotfile,"%f %f moveto\n",
		scale_x*value[1]+10, scale_y*kappa_min);

	fprintf(plotfile,"(min: %3.3f)show\n",kappa_min);

		fprintf(plotfile,"showpage");
/*********************************************************************/		
	

	fclose(file);
	fclose(plotfile);
}
::::::::::::::
plot_ratsurf.c
::::::::::::::

extern FILE *psfile;    /* for postscript output */

void plot_ratsurf(bx,by,bw,degree_u,degree_v,u_points,v_points,scale_x,scale_y)
/*	plots v_points isoparametric 
        curves of the  rat Bez surface, each with u_points
        points on it.
	
Input:  bx, by:		 arrays with x- and y- coordinates of
	       		 control net.
        degree_u,degree_v: degrees in u- and v- direction
	u_points,v_points: plot resolution
	scale_x,scale_y:   scale factor for postscript.

Output: postscript file 
*/

	int degree_u,degree_v,u_points,v_points;
	float bx[20][20], by[20][20],bw[20][20],scale_x,scale_y;

{
	int i,j,u_points1;
	float u,v,del_u,del_v;
	float aux_x[5],aux_y[5],aux_w[5], arrax[5],array[5],arraw[5];
	float points_x[300], points_y[300];
	int iv;
	float ratbez(),decas(),hornbez();


	del_v= 1.0/((float)v_points);
	u_points1=u_points+1;
	v=0.0;

	for(iv=0; iv<= v_points; iv++)/* loop over all curves to be drawn*/
	{

		for(i=0;i<=degree_u; i++)/* create aux. arrays*/
		{
		for(j=0; j<= degree_v; j++)
			{
			arraw[j]=bw[i][j];
			arrax[j]=bx[i][j]*arraw[j];
			array[j]=by[i][j]*arraw[j];
			}

		aux_w[i]=hornbez(degree_v,arraw,v);
		aux_x[i]=hornbez(degree_v,arrax,v)/aux_w[i];
		aux_y[i]=hornbez(degree_v,array,v)/aux_w[i];

		}
		
		ratbez_to_points(degree_u,u_points,aux_x,aux_w,points_x);
		ratbez_to_points(degree_u,u_points,aux_y,aux_w,points_y);

	/******************* now ps: *******************************/	
			
		fprintf(psfile,"newpath\n");

		fprintf(psfile,"%f %f moveto\n",
			scale_x*points_x[0], scale_y*points_y[0]);


		for(j=1; j<= u_points; j++)
			fprintf(psfile,"%f %f lineto\n",scale_x*points_x[j], scale_y*points_y[j]);

		fprintf(psfile,"stroke\n");


		v = v+del_v;

	}
}


::::::::::::::
psplot_curve.c
::::::::::::::
#include<stdio.h>
#include <math.h>
extern FILE *psfile;


void psplot_curve(bez_x,bez_y,bez_w,bspl_x,bspl_y,l,degree,value,pol,dense,scale_x,scale_y)
/* plots piecewise rational cubic  Bezier curve, dense points each segment,
  into the postscript-file   psfile. B-spline polygon can also be plotted.
Input:	bez_x, bez_y, bez_w: piecewise control polygons and weights
	bspl_x,bspl_y:B-spline controll polygon 
	l:            number of curve pieces
	degree:       degree of each piece
	value:	      minmax-box of polygon. xmin,xmax,...
	pol:	      which polygon to plot:
		      0: nothing, 1: Bezier, 2: B-spline,
	dense:	      how many points per segment
*/
	float bez_x[],bez_y[],bez_w[],value[],bspl_x[],bspl_y[],scale_x, scale_y;
	int l,degree,pol,dense;
{
	int i,k,j, ldeg,l2,l31,i3;
	float weight[5];

	float points_x[70],points_y[70], q_x[60], q_y[60];
	float aux_x[5], aux_y[5],aux_w[5];
	

	l2 = l+2;

	/* macro to mark B-spline vertex:*/
		fprintf(psfile,"/blackbox\n");
		fprintf(psfile,"{0 setgray\n");
		fprintf(psfile,"-3.5 -3.5 rlineto\n");
		fprintf(psfile,"7 0 rlineto\n");
		fprintf(psfile,"0 7 rlineto\n");
		fprintf(psfile,"-7 0 rlineto\n");
		fprintf(psfile,"0 -7 rlineto\n");
		fprintf(psfile,"closepath fill} def\n");
	

		fprintf(psfile,"/whitebox\n");
		fprintf(psfile,"{1 setgray\n");
		fprintf(psfile,"-3 -3 rlineto\n");
		fprintf(psfile,"6 0 rlineto\n");
		fprintf(psfile,"0 6 rlineto\n");
		fprintf(psfile,"-6 0 rlineto\n");
		fprintf(psfile,"0 -6 rlineto\n");
		fprintf(psfile,"closepath fill} def\n");
	
	/* macro to mark weight point:*/
		fprintf(psfile,"/diamond\n");
		fprintf(psfile,"{0  -5 rlineto\n");
		fprintf(psfile,"5 5 rlineto\n");
		fprintf(psfile,"-5 5 rlineto\n");
		fprintf(psfile,"-5 -5 rlineto\n");
		fprintf(psfile,"5 -5 rlineto\n");
		fprintf(psfile,"closepath} def\n");


/* plot curve */

	fprintf(psfile,"3 setlinewidth \n");
	

	/* for each segment, convert to points  and plot: */
	ldeg=degree*l;
	for(i=0; i< ldeg; i+=degree)
	{

	for(k=0; k<=degree; k++)
			{
			aux_x[k]=bez_x[i+k];
			aux_y[k]=bez_y[i+k];
			aux_w[k]=bez_w[i+k];
			}


		ratbez_to_points(degree,dense,aux_x,aux_w,points_x);
		ratbez_to_points(degree,dense,aux_y,aux_w,points_y);

		fprintf(psfile,"newpath \n");

		fprintf(psfile,"%f %f moveto\n",
			scale_x*(points_x[0]-value[0]),
			scale_y*(points_y[0]-value[2]));
		
		for( j=1; j<=dense ; j++)
		{
			fprintf(psfile,"%f %f lineto\n",
			scale_x*(points_x[j]-value[0]),
			scale_y*(points_y[j]-value[2]));				}
 		fprintf(psfile,"stroke \n");

	}

	



	/**** fill in junction points  black: **********/
	fprintf(psfile,"1 setlinewidth \n");


	for(j=0; j<= ldeg; j=j+degree)
			{
			fprintf(psfile,"%f %f %f %f %f arc fill\n",
			scale_x*(bez_x[j]-value[0]), 
			scale_y*(bez_y[j]-value[2]),
		 	3.0, 0.0, 360.0);
			}


if (pol == 0)
	{
		fprintf(psfile,"showpage \n");
		 return;
	}


if (pol>0) /* plot B-spline polygon: */
	{
		fprintf(psfile,"0.0 setgray\n ");
		fprintf(psfile,"2 setlinewidth\n ");

		fprintf(psfile,"newpath \n");
		fprintf(psfile,"%f %f moveto\n",
			scale_x*(bez_x[0]-value[0]), 
			scale_y*(bez_y[0]-value[2]));

		for(j=1; j<=  l2; j++)
			{
			fprintf(psfile,"%f %f lineto\n",
			scale_x*(bspl_x[j]-value[0]), 
			scale_y*(bspl_y[j]-value[2]));
			}
		fprintf(psfile,"stroke \n");



	/***** plot  control vertices: ************/


	for(j=0; j<=l2; j++)
	{
		fprintf(psfile,"%f %f moveto\n",
		scale_x*bspl_x[j], scale_y*bspl_y[j]);
		fprintf(psfile,"blackbox \n");

		if (j!=0 && j!=l2)
		{
			fprintf(psfile,"%f %f moveto\n",
			scale_x*(bspl_x[j]-value[0]),
			 scale_y*(bspl_y[j]-value[2]));
			fprintf(psfile,"whitebox \n");
		}
	}

	}

		
	
	/***** plot  bezier control polygon if pol=1: ************/
if (pol == 1)
	{
		fprintf(psfile,"0.0 setgray\n ");
		fprintf(psfile,"0.5 setlinewidth\n ");

		fprintf(psfile,"newpath \n");
		fprintf(psfile,"%f %f moveto\n",
			scale_x*(bez_x[0]-value[0]),
			 scale_y*(bez_y[0]-value[2]));

		for(j=0; j<=  ldeg; j++)
			{
			fprintf(psfile,"%f %f lineto\n",
			scale_x*(bez_x[j]-value[0]), 
			scale_y*(bez_y[j]-value[2]));
			}
		fprintf(psfile,"stroke \n");



	/***** plot  control vertices: ************/

	fprintf(psfile,"0.0 setgray\n ");

	for(j=1; j< ldeg ; j++)
			{
		if(j !=(j/degree)*degree)
			fprintf(psfile,"%f %f %f %f %f arc stroke\n",
			scale_x*(bez_x[j]-value[0]), 
			scale_y*(bez_y[j]-value[2]),
		 	3.0, 0.0, 360.0);
			}

	fprintf(psfile,"1.0 setgray\n ");

	for(j=1; j< ldeg; j++)
			{
		if( j!=(j/degree)*degree)	
		fprintf(psfile,"%f %f %f %f %f arc fill\n",
			scale_x*(bez_x[j]-value[0]), 
			scale_y*(bez_y[j]-value[2]),
		 	2.80, 0.0, 360.0);
			}


/*		fprintf(psfile,"showpage\n");  
		return;
*/
	


	/***** plot  weight points: ************

	fprintf(psfile,"0.3 setgray\n ");
	l31 = 3*l -1;

	for (j=0; j<= l31; j++)
	{
		q_x[j]=(bez_w[j]*bez_x[j] + bez_w[j+1]*bez_x[j+1])
			/(bez_w[j] + bez_w[j+1]);

		q_y[j]=(bez_w[j]*bez_y[j] + bez_w[j+1]*bez_y[j+1])
			/(bez_w[j] + bez_w[j+1]);
	}
	for(j=0; j<=l31; j++)
	{
		fprintf(psfile,"%f %f moveto\n",
		scale_x*(q_x[j]-value[0]), scale_y*(q_y[j]-value[2]));
		fprintf(psfile,"diamond fill\n");

	}
	
	************************** weight points plotted  */
	}


	fprintf(psfile,"showpage\n");
	return;

}		
::::::::::::::
psplot_net.c
::::::::::::::
#include <stdio.h>
#include <math.h>

extern FILE *psfile;

void psplot_net(lu,lv,bx,by,step_u,step_v,scale_x,scale_y,value)
/*	plots  control net     	into postscript-file.
Input:	lu,lv:		dimensions of net
	bx,by:		net vertices
	step_u,step_v:	subnet sizes (e.g. both=3 for pw bicubic net)
	scale_x,scale_y:scale factors for ps
	value:		window size in world coords
*/
	int lu,lv,step_u,step_v;
	float bx[20][20], by[20][20],value[4],scale_x,scale_y;

{
	int lv1;
	int i,j;

	
		fprintf(psfile,"0.0 setgray \n");
		fprintf(psfile,"1 setlinewidth\n");


	lv1= lv+1;
	for(i=0; i<= lu; i++)
	{
		fprintf(psfile,"newpath\n");
		fprintf(psfile," %f %f moveto\n",
			scale_x*(bx[i][0]-value[0]),
			scale_y*(by[i][0]-value[2]));
		for(j=1; j<= lv; j++)
			fprintf(psfile,"%f %f lineto\n",
			scale_x*(bx[i][j]-value[0]), 
			scale_y*(by[i][j]-value[2]));
			
		fprintf(psfile,"stroke\n");
	}

	for(j=0;j<= lv;j++)
	{
		fprintf(psfile,"newpath\n");
		fprintf(psfile," %f %f moveto\n",
			scale_x*(bx[0][j]-value[0]),
			scale_y*(by[0][j]-value[2]));

		for(i=1; i<= lu; i++) 
			fprintf(psfile," %f %f lineto\n",
			scale_x*(bx[i][j]-value[0]),
			scale_y*(by[i][j]-value[2]));
		
		fprintf(psfile,"stroke\n");
	}

	/***** now mark control points *****/


	for(i=0; i<= lu; i++)
	{
		for(j=0; j<= lv; j++)
			fprintf(psfile,"%f %f %f %f %f arc stroke\n",
			scale_x*(bx[i][j]-value[0]),
			 scale_y*(by[i][j]-value[2]),
			3.0, 0.0, 360.0);		
	}


	fprintf(psfile,"1 setgray\n"); /* to white out circles */

	for(i=0; i<= lu; i++)
	{
		for(j=0; j<= lv; j++)
			fprintf(psfile,"%f %f %f %f %f arc fill\n",
			scale_x*(bx[i][j]-value[0]),
			 scale_y*(by[i][j]-value[2]),
			2.9, 0.0, 360.0);		
	}


	/* subnet corners in black: */
	fprintf(psfile,"0.0 setgray\n");


	for(i=0; i<= lu; i+= step_u)
	{
		for(j=0; j<= lv; j+= step_v)
			fprintf(psfile,"%f %f %f %f %f arc fill\n",
			scale_x*(bx[i][j]-value[0]), 
			scale_y*(by[i][j]-value[2]),
			2.9, 0.0, 360.0);		
	}
	


}
::::::::::::::
psr.ps
::::::::::::::
% this is referred to as {fig:psr} in chapter {chap:ppint} 
3 setlinewidth 
90 rotate
0 setgray
200 -450 translate
newpath 
18.260871 0.000000 moveto
18.260871 2.129984 lineto
18.260871 4.257262 lineto
18.260871 6.379130 lineto
18.260871 8.492882 lineto
18.260871 10.595814 lineto
18.260871 12.685217 lineto
18.260871 14.758389 lineto
18.260871 16.812624 lineto
18.260871 18.845217 lineto
18.260871 20.853462 lineto
18.260871 22.834653 lineto
18.260871 24.786087 lineto
18.260871 26.705056 lineto
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::::::::::::::
ratbez.c
::::::::::::::
#include <math.h>

float ratbez(degree,coeff,weight,t)
/*
	uses rational de casteljau to compute
	point on ratbez curve for param. value t.
*/
	float coeff[],weight[];
	float t;
	int degree; 
	{
	int r,i;
	float t1,t2,ww1,ww2;
	float coeffa[20], weighta[20];
	t1 = 1.0 - t;

	
	for (i=0;i<=degree; i++)
	{
		coeffa[i]=coeff[i];
		weighta[i]=weight[i];
	}

	for (r=1; r<= degree; r++)
	for (i=0; i<= degree - r; i++)
	{
	/*	t1= (i+r-degree*t)/(float)r;
		t2= 1.0-t1;  ... makes it interpolatory ...
	*/
		t1=1.0-t; t2=t;
		ww1 = weighta[i]; ww2= weighta[i+1];
		weighta[i] = t1*ww1 + t2*ww2;
		coeffa[i]  = ( t1*ww1 * coeffa[i]
			    + t2*ww2 * coeffa[i+1] ) / weighta[i];

	}	
	return (coeffa[0]);
	}
::::::::::::::
ratbez_kappas.
::::::::::::::
#include <stdio.h>
#include <math.h>
extern FILE *outfile;

void ratbez_kappas(bez_x,bez_y,bez_w,l,dense)
/*	writes curvatures of  piecewise cubic rational curve into 
	a file.
input:
	bez_x,bez_y:  2D rat. cubic spline polygon
	bez_w:	the  weights
	dense:	how many curvature values to compute per interval
	l:      no. of intervals
output:		 written into file outfile
Note:		this plots curvatures vs. a uniform u-spacing
*/

	float bez_x[],bez_y[],bez_w[];
	int  dense,l;
{
	float bleftx[4],blefty[4],brightx[4],brighty[4];
	float wleft[4], wright[4],coeffx[4],coeffy[4],weight[4];
	float dist,t,delt,h,u,diff;
	int i,j,i3;
	float curvature_0(),curvature_1(),abs();

	/* Plot kappas for each interval. */
	fprintf(outfile,"%d  %d\n",dense*l,dense);
	for(i=0; i<l; i++)
	{	i3=i*3;
		u= (float)i;  /*starting value for i-th segment*/
		diff=1.0;
		for(j=0;j<=3;j++)
		{       /* create i-th cubic Bezier curve: */ 
			coeffx[j]=bez_x[i3+j]; 
			coeffy[j]=bez_y[i3+j];
			weight[j]=bez_w[i3+j];
		}


		delt=1.0/(float)dense;

		for(t=0.0; t<0.5; t=t+delt)
		{
		subdiv(3,coeffx,weight,t,bleftx,brightx,wleft,wright);
		subdiv(3,coeffy,weight,t,blefty,brighty,wleft,wright);
		h=curvature_0(brightx,brighty,wright,3);
		/* to print file, we'll have to use the GLOBAL
		   parameter u!   */
		u=(float)i+t*diff;
		fprintf(outfile,"%2.3f  %f\n",u,h);
		}

		for(t=0.5; t<0.999; t=t+delt)
		{
		subdiv(3,coeffx,weight,t,bleftx,brightx,wleft,wright);
		subdiv(3,coeffy,weight,t,blefty,brighty,wleft,wright);

		h=curvature_0(bleftx,blefty,wleft,3);
		h=0.0-h;
		/* minus sign since order of polygon 
		   traversal is reversed!
		*/
		u=(float)i+t*diff;
		fprintf(outfile,"%2.3f    %f\n",u,h);
		}
	}
	/* print very last kappa: */
	t=1.0;
	h=curvature_1(coeffx,coeffy,weight,3);
	fprintf(outfile,"%2.3f    %f\n",(float)l,h);


}
:::::::::::::::::
ratbez_to_points.
:::::::::::::::::
void  ratbez_to_points(degree,npoints,coeff,coeff_w,points)
/*	Converts rational Bezier curve into point sequence. Works on
	one coordinate only.
 
	Input:   degree:  degree of curve.
	         npoints: # of coordinates to be generated. (counting
		          from 0!)
	         coeff:   coordinates of control polygon.
	Output:  points:  coordinates of points on curve.

	Remark: For a 2D curve, this routine needs to be called twice,
		once for the x-coordinates and once for y.
*/

	int degree, npoints;
	float coeff[],coeff_w[],points[];

{	
	float t,delt;
	int i;
	float ratbez();

	delt=1.0/(float)npoints;
	t=0.0;
	for(i=0; i<=npoints; i++) 
	{
	points[i]=ratbez(degree,coeff,coeff_w,t);  
	t = t+delt;
	}
}



::::::::::::::
reparam.c
::::::::::::::
void reparam(wold,degree,s,wnew)
/*	reparametrizes ratbez curve: only the weights,
	stored in wold, are changed. New weights are in
	wnew. Parametrization is determined by shoulder
	point s. For s=0.5, nothing changes. Also,
	s should be in (0,1).
*/

	float wold[],wnew[];
	int degree;	float s;
{
	int i;
	float c;

	c = s / (1.0 - s);

	wnew[0] = wold[0];

	for (i=1; i<= degree; i++)
		wnew[i] = c*wnew[i-1]*wold[i]/wold[i-1];
}
::::::::::::::
set_up_system.
::::::::::::::
void set_up_system(knot,l,alpha,beta,gamma)
/*	given the knot sequence, the linear system 
	for clamped end condition B-spline interpolation
	is set up.
input:	knot:	knot sequence
	points:	points to be interpolated
	l:	number of intervals
output:	alpha,beta,gamma: 1-D arrays that constitute
		the elements of the interpolation matrix.

	Note: no data points needed so far! 
*/
	int l;
	float knot[],alpha[],beta[],gamma[];

{
	int i,l1;
	float delta_im2,delta_im1, delta_i, delta_ip1,sum;
	l1=l-1;

	/* some special cases: */
	if(l==1)
	{alpha[0]=0.0; alpha[1]=0.0; beta[0]=1.0; beta[1]=1.0;
	 gamma[0]=0.0; gamma[1]=0.0; return;}

	if(l==2)
	{
		beta[0]=1.0;
		delta_im1=(knot[1]-knot[0]);
		delta_i  =(knot[2]-knot[1]);
		delta_ip1=(knot[3]-knot[2]);
		sum = delta_im1+delta_i;

		alpha[1]=delta_i*delta_i/sum;
		beta[1] =(delta_i*delta_im1)/sum
                + delta_im1*delta_i/ sum;
		gamma[1]=delta_im1*delta_im1/sum;

		alpha[1]=alpha[1]/sum;
		beta[1] =beta[1]/sum;
		gamma[1]=gamma[1]/sum;

		beta[2]=1.0;
		alpha[2]=0.0;
		gamma[2]=0.0;
		return;
	}

	/* the rest does the cases l>2.  */

	delta_im1=(knot[1]-knot[0]);
	delta_i  =(knot[2]-knot[1]);
	delta_ip1=(knot[3]-knot[2]);
	sum = delta_im1+delta_i;

	beta[0]=1.0; gamma[0]=0.0;

	alpha[1]=delta_i*delta_i/sum;
	beta[1] =(delta_i*delta_im1)/sum
                + delta_im1*(delta_i+delta_ip1)
		/ (sum+delta_ip1);
	gamma[1]=delta_im1*delta_im1/(sum+delta_ip1);

	alpha[1]=alpha[1]/sum;
	beta[1] =beta[1]/sum;
	gamma[1]=gamma[1]/sum;

	/*Now for the main loop:   */
	for(i=2; i<l1; i++)
	{
		/* compute delta_i_minus_2,...  */
		delta_im2=(knot[i-1]-knot[i-2]);
		delta_im1=(knot[i]  -knot[i-1]);
		delta_i  =(knot[i+1]-knot[i]);
		delta_ip1=(knot[i+2]-knot[i+1]);
	
		sum = delta_im1+delta_i;

		alpha[i]=delta_i*delta_i/(delta_im2 +sum);
		beta[i] =delta_i*(delta_im2+delta_im1) 
			/(delta_im2 + sum)
			+
			 delta_im1*(delta_i+delta_ip1)
			/(sum + delta_ip1);
		gamma[i]=delta_im1*delta_im1
			/(sum + delta_ip1);
		
		alpha[i]=alpha[i]/sum;
		beta[i] =beta[i]/sum;
		gamma[i]=gamma[i]/sum;
	}

	/*  special care at the end:  */
	delta_im2=knot[l-2]-knot[l-3];
	delta_im1=knot[l1]-knot[l-2];
	delta_i  =knot[l]-knot[l1];
	sum=delta_im1+delta_i;

	alpha[l1]=delta_i*delta_i/(delta_im2+sum);
	beta[l1] =delta_i*(delta_im2+delta_im1)/(delta_im2  + sum)
                 +
		  delta_im1*delta_i / sum;
	gamma[l1]=delta_im1*delta_im1/sum;

	alpha[l1]=alpha[l1]/sum;
	beta[l1] =beta[l1]/sum;
	gamma[l1]=gamma[l1]/sum;


	alpha[l]=0.0; beta[l]=1.0; gamma[l]=0.0;


}
::::::::::::::
shear.ps
::::::::::::::
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::::::::::::::
solve_system.c
::::::::::::::
 void solve_system(up,low,gamma,l,rhs,d)
/*	solve  tridiagonal linear system
	of size (l+1)(l+1) whose LU decompostion has entries up and low,
	and whose right hand side is rhs, and whose original matrix
	had upper diagonal gamma. Solution is d[0],...,d[l+2];
Input: up,low,gamma:  as above.
       l:             size of system: l+1 eqs in l+1 unknowns.
       rhs:           right hand side, i.e, data points with end
                      `tangent Bezier points' in rhs[1] and rhs[l+1].
Output:d:             solution vector.

Note:	Both rhs and d are from 0 to l+2.
*/
	float up[],low[],gamma[],rhs[],d[];
	int l;
{
	int i;
	float aux[100];

	d[0] = rhs[0];
	d[1] = rhs[1];

	/* forward substitution:  */
	aux[0]=rhs[1];
	for(i=1; i<=l; i++) aux[i]=rhs[i+1]-low[i]*aux[i-1];
	
	/* backward substitution:  */
	d[l+1]=aux[l]/up[l];
	for(i=l-1; i>0; i--) d[i+1]=(aux[i]-gamma[i]*d[i+2])/up[i];
	d[l+2]=rhs[l+2];
}
		
::::::::::::::
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::::::::::::::
subdiv.c
::::::::::::::
#include <math.h>

void subdiv(degree,coeff,weight,t,bleft,bright,wleft,wright)
/*
	subdivides ratbez curve at parameter value t.
	Output: left and right polygon with respective weights.
	Ordering of right polygon is reversed.
*/
	float coeff[],weight[],bleft[],
		bright[],wleft[],wright[];
	float t;
	int degree; 
{
	int r,i;
	float t1,ww1,ww2;
	t1 = 1.0 - t;
	
/* 
	first, obtain right subpolygon from rat de Casteljau
*/

	for (i=0;i <= degree; i++) wright[i] = weight[i];
 
	for (i=0;i <= degree; i++) bright[i] = coeff[i];

	for (r=1; r<= degree; r++)
	for (i=0; i<= degree - r; i++)
	{
		ww1 = wright[i]; ww2= wright[i+1];
		wright[i] = t1*wright[i] + t*wright[i+1];
		bright[i]= ( t1*ww1 * bright[i]
			  +  t*ww2 * bright[i+1] ) / wright[i];
	}	

/* 
	use same as above in order to get left half. Idea:
	reverse ordering; then the above yields left half.
 */

	t = 1.0 - t; t1 = 1.0 - t;
	for (i=0;i <= degree; i++) wleft[degree -i] = weight[i];
 
	for (i=0;i <= degree; i++) bleft[degree-i] = coeff[i];

	for (r=1; r<= degree; r++)
	for (i=0; i<= degree - r; i++)
	{
		ww1 = wleft[i]; ww2= wleft[i+1];
		wleft[i] = t1*wleft[i] + t*wleft[i+1];
		bleft[i]= ( t1*ww1 * bleft[i]
			  +  t*ww2 * bleft[i+1] ) / wleft[i];
	}	

}
::::::::::::::
tri_decas.c
::::::::::::::
/*
 * Routine: tri_decast()
 *
 * Function: Triangular de Casteljau algorithm for an n^th
 *           degree triangular Bezier patch. 
 *           Algorithm is applied once for a given (u,v,w) and works on one
 *           coordinate only. 
 *
 * Input:    bpts[i] ................. Bezier points (of one coordinate)
 *                                     as a linear array (see below). 
 *                                     i=0...tri_num  
 *           tri_num ................. Based on the degree of the patch.
 *                                     (n+1)(n+2)/2
 *           ndeg .................... Degree (n) of the patch. 
 *           u[i] .................... Barycentric coordinates (u,v,w) of 
 *                                     evaluation point.  i=0,2
 *           b[i]..................... A working array with dimension >= 
 *                                     to bpts[].
 *
 * Output:   patch_pt ................ One coordinate of the point on 
 *                                     the patch evaluated at (u,v,w).
 *           b[] ..................... Contents have been changed. 
 *
 * Linear array structure: It is assumed that the usual (i,k,j) structure
 *                         of the Bezier net has been put into a linear
 *                         array in the following manner.
 *                         (E.g., for n=3)
 *                         b_(300) --> bpts[0] (u=1)
 *                         b_(030) --> bpts[6] (v=1)
 *                         b_(003) --> bpts[9] (w=1)
 */

#include <stdio.h>
#include <math.h>


tri_decast(bpts, tri_num, ndeg, u, b, patch_pt ) 

    float	bpts[];
    int         tri_num;
    int         ndeg;

    float       u[3];

    float       b[];

    float       *patch_pt;
{

    int         i, j, k, l, m;
    int         r;

    /* ------------------------------ */


    /* 
     * To avoid writing over the original control points, 
     * copy into a working array.
     */

    for (i=0; i<tri_num; i++) 
        b[i] = bpts[i]; 


    /*
     * de Casteljau algorithm applied to the linear array. 
     * The element at the "top" of each mini-triangle in the
     * algorithm is over-written at each step.
     */

    for (r=1; r<=ndeg; r++) {

	m = -1;
	for (i=0; i<=ndeg-r; i++) {

	    for (l=0; l<=i; l++) {

		m = m + 1;
		b[m] = u[0]*b[m] 
                     + u[1]*b[m+1+i] 
                     + u[2]*b[m+2+i]; 
	    }
	}
    } 

    *patch_pt = b[0];

}

::::::::::::::
weights.ps
::::::::::::::
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