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From: toddl@rt8.cs.wisc.edu (Todd Lehman)
Newsgroups: sci.math,sci.math.num-analysis,comp.theory
Subject: Cardioid problems
Message-ID: <4404@daffy.cs.wisc.edu>
Date: 6 Mar 90 21:35:23 GMT
Sender: news@daffy.cs.wisc.edu
Organization: U of Wisconsin CS Dept
Lines: 38


Hello, I'm stuck on a problem involving cardioids.  Maybe someone can help me 
out...


    Does anyone know of a easy way to test whether a point (x,y) in the 
Euclidean plane is in the interior of a cardioid?  The cardioid I'm considering 
is

                               r = sin^2 (t/2),

where t is an angle in radians and r is the radius associated with that angle.
Points (x,y) and (r;t) are related by

			         x = r cos(t)
			         y = r sin(t)

as (x,y) are Euclidean coordinates and (r;t) are polar coordinates.

    One way of performing the test is to get the argument t from the arctangent
of y/x, divide this by two, take the sine of that and square it twice, then see
whether the resulting number is less than or equal to x^2+y^2.  But this 
involves four multiplies, two divides, a sine, and an arctangent.   :-(

    In a related problem, to check whether a point (x,y) is inside a circle

                                   r = c,

for some constant c, one need only test whether x^2+y^2 < c^2.  This process
involves only two multiplies (c^2 is a constant).   :-)

    The test for the circle is quite simple.  Surely there must be a similarly 
simple test for the cardioid...?  Any would be much appreciated.

__________

Todd S. Lehman, UW-Madison
toddl%garfield@cs.wisc.edu