From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: comp.graphics.algorithms
Subject: Re: start/end-pixels of circle arcs without sin/cos?
Date: 28 Aug 1996 21:45:40 GMT

In article <5018ca$ak@fstgal00.tu-graz.ac.at>,
Mark Piffer <gizmo@sbox.tu-graz.ac.at> wrote:
>A friend of mine needs to draw circle arcs at a given radius from 
>degree x to y. Problem is, that he can't use sin() and cos()

You can parameterize a circle of radius  1  at the origin as the
set of points of the form
	( (2t)/(1+t^2),  (1-t^2)/(1+t^2) )
where  t  ranges over all real numbers. Obviously this can be scaled and
translated to parameterize other circles. One can restrict the ranges
of  t  to obtain circular arcs. (Given the cartesian coordinates of the
arc's endpoints, you can determine their corresponding parameters  t  by
solving one of several quadratic equations.)

This parameterization covers the region near  (-1,0)  very slowly (that is,
one needs to take  t --> oo  or  t --> -oo  to approach  (-1,0). ) If you
prefer, you may compose this parameterization with one which traverses
(most of) the whole real line in only a finite amount of time, such as taking
t:= {u, if  -1 < u < 1;  1/(2-u), if  1 < u < 2; 1/(2+u), if  -2 < u < -1 }.

(Another possibility in the preceding paragraph is to take  t = tan(u/2),
of course  :-)  )

It's computationally faster, I suppose, to use an iterative computation
to march along a circle at constant speed: circular motion has acceleration
parallel to displacement from the center, so one updates the position with
a velocity vector, which in turn is updated proportional to the position.
But of course such a recursive procedure allows for the potential of
accumulation of errors.

dave