Path: utzoo!utgpu!jarvis.csri.toronto.edu!cs.utexas.edu!natinst!rpp386!woody
From: woody@rpp386.cactus.org (Woodrow Baker)
Newsgroups: comp.lang.postscript
Subject: Re: Evaluation of Splines
Summary: evaluating splines
Message-ID: <17739@rpp386.cactus.org>
Date: 23 Jan 90 23:28:29 GMT
Organization: River Parishes Programming, Plano, TX
Lines: 76

I recently had a conversation with a very mathematically oriented freind in 
California, in which he mentioned that it would be possible to evaluate a 
Bezier spline with *NO* multiplies using a "diffrence table"
Can anyone shed some light on exactly what he is talking about, and point 
me to some reasonably clear explaination of how to accomplish the 
computation of the table?  I currently have a very simple minded Bexier 
drawing routine, written in 'C' that I'n not altogether happy with.  I've 
distilled it down to 3 multiplies, but would like to get rid of them.
 
code follows:
/*
	compute the coefficents for the 3rd order
	cubic parametric equations that make up a Bezier
	cubic spline
*/
computcof()
{
	xd=x[1]-x[2];     /* factor out common expressions for */
	yd=y[1]-y[2];     /* more speed and effeciency */
	xd1=3*x[0];
	yd1=3*y[0];
	coeff[ax]=3*xd+x[3]-x[0];
	coeff[ay]=3*yd+y[3]-y[0];
	coeff[bx]=-3*(x[1]+xd)+xd1;
	coeff[by]=-3*(y[1]+yd)+yd1;
	coeff[cx]=3*(x[1]-x[0]);
	coeff[cy]=3*(y[1]-y[0]);
}
/*
	sets up and draws the spline.  f is a flag that
	determines whether to erase or draw the spline
*/
dospline(x0,y0,x1,y1,x2,y2,x3,y3,f)
int x0,y0,x1,y1,x2,y2,x3,y3,f;
{
	x[0]=x0;
	y[0]=y0;
	x[1]=x1;
	y[1]=y1;
	x[2]=x2;
	y[2]=y2;
	x[3]=x3;
	y[3]=y3;
	computcof();
	drawspline(f);
}
/*
	draws splines
*/
drawspline(f)
int f;
	{
	float t;
	int x1,y1,p;
	for (t=0;t<1;t=t+.01)
		{
		x1=(((coeff[ax]*t)+coeff[bx])*t+coeff[cx])*t+x[0];
		y1=(((coeff[ay]*t)+coeff[by])*t+coeff[cy])*t+y[0];
		if(f==0)
			{
			SETPIX(x1,y1,1,0); /* put dot down */
			}
		else
			{
			SETPIX(x1,y1,1,3); /* erase dot */
			}
		};
	}
 
I'm sure that this can be speeded up dramatically, if the multiplies
in drawspline can be removed, and the entire thing can be integerized.
Any help would be appreciated:
 
Cheers
Woody