Xref: utzoo sci.math:6940 sci.math.num-analysis:19 comp.graphics:6043
Path: utzoo!utgpu!jarvis.csri.toronto.edu!rutgers!iuvax!watmath!watcgl!rhbartels
From: rhbartels@watcgl.waterloo.edu (Richard Bartels)
Newsgroups: sci.math,sci.math.num-analysis,comp.graphics
Subject: Re: Computing Bezier Control Points for an Arc of a Circle
Keywords: bezier, arc, graphics
Message-ID: <10150@watcgl.waterloo.edu>
Date: 6 Jun 89 22:35:54 GMT
References: <108121@sun.Eng.Sun.COM>
Reply-To: rhbartels@watcgl.waterloo.edu (Richard Bartels)
Organization: U. of Waterloo, Ontario
Lines: 38

In article <108121@sun.Eng.Sun.COM>
stephenj@deblil.Sun.COM (Stephen Johnson) writes:
>
>How do you compute the control points for a fourth order (third
>degree) Bezier if the user gives you the following data:
>
>	x, y	- center of circle
>	rad	- radius of circle
>	ang1	- starting angle for arc
>	ang2	- ending angle for arc
>
>Once you have the control points, its a simple task to render the
>Bezier.  So, how do you compute the control points?  Can it be done
>exactly for any angle between 0 and 360?

This is one of the Golden Oldies.  You can't do a circle exactly
as a Bezier curve (at least as an integral Bezier curve, you can do
conic sections as rational Bezier curves).

For those interested in how close you can get, however, the following
was presented in April at a conference at Oberwolfach:

	"Almost best Approximations of Circles
	 by Curvature-continuous Bezier curves"
			by
	 Tor Dokken, Morten Daehlen, Tom Lyche, Knut Morken

The userids of the 2nd - 4th authors are:

	mortend@ifi.uio.no
	tom@ifi.uio.no
	knut@ifi.uio.no

I hope they will forgive me for giving this pointer to their work,
but the world is waiting; they promised me a reprint at least a month ago,
and nothing has been heard since.  I am tired waiting.  This is my revenge.

-Richard