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From: jp@lanl.ARPA
Newsgroups: net.math
Subject: Re: moRe: Conic sections (new problem)
Message-ID: <613@lanl.ARPA>
Date: Sun, 2-Feb-86 00:54:37 EST
Article-I.D.: lanl.613
Posted: Sun Feb  2 00:54:37 1986
Date-Received: Mon, 3-Feb-86 05:13:43 EST
References: <1165@homxb.UUCP> <557@well.UUCP> <532@well.UUCP> <11591@ucbvax.BERKELEY.EDU> <660@eagle.ukc.ac.uk>
Reply-To: jp@a.UUCP (James Potter)
Organization: Los Alamos National Laboratory
Lines: 28




I am not sure why you care about the rotation of a circle, but I have
implemented an algorithm for plotting circles and ellipses with arbitrary
starting angles.  A good way to see how to do this is to look at the
matrix equations:

x(theta) -x_c        cos(theta)     e sin(theta) |   r cos(phi)
                =  |                             |
y(theta) - y_c     | e^-1 sin(theta)  cos(theta) |   r sin(phi)

This yields essentiall parametric equations for the circle (e=1) or
ellipse whose major axis (if e > 1) makes an angle phi with the x axis.
I choose a small value for theta and by successive multiplications plot
out a circle.  If theta =360/1000 there will be 1000 points on the circle/
ellipse.  I make the size of the angle inversely proportional to r to keep
a reasonable point density.  If the rotation matrix has sufficiently accurate
values for the trig functions and multiplication is reasonably precise the 
figure will hold together for angles much larger than 360 degrees.  As
a test a made a 1000 point circle and went around it 10 times, that's 10000
multiplications and the end point was within 1 half pixel (actually better -
I just don't remember how close) on a 4096 X 3072 display.  Oh,yes the
calculations were done with an 80287 math coprocessor.  
Don't really know if this is in anyway related to your problem and it's not
a very highbrow math problem, but maybe someone is interested.

Jim Potter  jp@lanl.arpa