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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <445@faron.UUCP>
Date: Thu, 23-Jan-86 15:06:09 EST
Article-I.D.: faron.445
Posted: Thu Jan 23 15:06:09 1986
Date-Received: Sat, 25-Jan-86 03:47:07 EST
References: <532@well.UUCP> <444@faron.UUCP>
Organization: The MITRE Coporation, Bedford, MA
Lines: 39

> In article <532@well.UUCP> rab@well.UUCP (Bob Bickford) writes:
> >
> >	Is there a formula which describes *all* conic sections, which
> >will generate a particular class of same (e.g., circles, hyperbolas)
> >when certain coefficients are plugged into it?
> >
> 
> 	The general quadratic equation in two real variables:
> 
> 	   2		 2
> 	a x  + b xy + c y  + d x + e y + f = 0
> 
> 	yields (possibly degenerate) conic sections as solution sets.
> 	The way to determine which of the three types (ellipse, parabola,
> 	hyperbola) emerges with a given set of coefficients is to turn
> 	the equation into a homogeneous equation in three variables:
> 
> 	   2		 2		      2
> 	a x  + b xy + c y  + d xz + e yz + f z  = 0.
> 
> 	This new equation represents a curve in the (real) projective plane,
> 	essentially a completion of the affine plane which preserves limits
> 	of curves at infinity. To be strictly honest, points are triples:
> 	[x,y,z], in which [x,y,z] ~ [tx,ty,tz] for any non-zero number t;
> 	the subspace of points for which z is non-zero (equivalently z = 1)
> 	is the standard (affine) plane, and the "line at infinity" is the
> 	set of points for which z = 0.
 
etc. etc.

You can remove the  xy , yz , and xz terms via rotation without affecting
any properties of the curve. Rotating the curve through an angle theta
where:

	tan(2 theta) = B/(A-C)

will remove the Bxy term. 
 
Bob Silverman