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From: wdh@faron.UUCP (Dale Hall)
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <444@faron.UUCP>
Date: Thu, 23-Jan-86 14:10:41 EST
Article-I.D.: faron.444
Posted: Thu Jan 23 14:10:41 1986
Date-Received: Sat, 25-Jan-86 03:46:52 EST
References: <532@well.UUCP>
Reply-To: wdh@faron.UUCP (Dale Hall)
Organization: The MITRE Coporation, Bedford, MA
Lines: 88
Keywords: conic sections, quadratic equations
Summary: the answer is yes

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.

	So what? Well, take a look at this equation "at infinity", z=0:
	   2		 2
	a x  + b xy + c y  = 0,

	and solve for x in terms of y:
			    _____________
		     +	   /  2		 
		-b   -   \/  b  - 4 a c
	x =    ----------------------------  ,
			2 a


	using the quadratic formula.  Notice that the "discriminant"
	determines whether the intersection consists of zero, one, or two
	points. In fact:

		zero : ellipse  -- curve doesn't reach infinity in x or y

		one  : parabola -- curve just barely reaches infinity
					(it's tangent to the line at infinity)

		two  : hyperbola - curve intersects the line at infinity
		                   at two points (these will turn out
				   to yield the slopes of the asymptotes)

	To discriminate circles from ellipses, note that a circle will
	always have equal coefficients for the x**2, y**2 terms, and that
	there will be no xy term at all, while an ellipse will have either
	different coefficients for  x**2 and y**2, or a non-zero xy term
	or both. To determine whether the conic is degenerate, you need
	to know three things: 

	(i) if the quadratic factors into two linear factors (redundant: if
	it factors, not too much else can happen), then the curve is the 
	union of two lines (either parallel or not):
			  2	    2
	example:	2x  + xy - y  + 2x - y = 0

	(ii) if the gradient of the polynomial has a zero along the curve,
	then there can be a singularity (such as an isolated point, or the
	crossing of two lines):
			 2    2
	example:	x  + y  = 0

	(iii) it is possible to have an empty point set for the solution set.
			 2    2
	example:	x  + y  + 1 = 0.

	If you've established that (i) the quadratic doesn't factor,
				   (ii) the quadratic doesn't have any
					solutions in common with zeroes
					of both its partial derivatives
				   (iii)there is at least one solution,

	then you're home free, and have a real-live conic section you can
	call your own.

	by now you've quit reading this, so I'll go quietly.

					Dale Hall