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From: markb@sdcrdcf.UUCP (Mark Biggar)
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <2597@sdcrdcf.UUCP>
Date: Fri, 24-Jan-86 14:22:47 EST
Article-I.D.: sdcrdcf.2597
Posted: Fri Jan 24 14:22:47 1986
Date-Received: Sun, 26-Jan-86 04:36:26 EST
References: <532@well.UUCP> <144@linus.UUCP>
Reply-To: markb@sdcrdcf.UUCP (Mark Biggar)
Distribution: net
Organization: System Development Corporation R&D, Santa Monica
Lines: 33

In article <144@linus.UUCP> bs@linus.UUCP (Robert D. Silverman) 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?
>> 
>This is a simple high school analytic geometry problem. A general quadratic
>form in two variables is:
>
>		   2     2
>	F(x,y) = Ax  + Cy  + Dx + Ey + F
> 
>F(x,y) = 0 is the general equation representing a conic section. The following
>conditions hold:
>
>If A or C is zero the conic is a parabola or in special cases two parallel
>lines which may be distinct, coincident or imaginary.
>
>If A and C have the same sign we have an ellipse or in special cases it can
>degenerate into a single point or imaginary ellipse. It is a circle when
>A = C.
>
>If A and C have different signs we have a hyperbola or in degenerate cases
>two intersecting lines.

This only produces thoses ellipses, parabolas and hyperbolas that have an
axis parallel to the X and Y axis.  If you want rotated versions you have to
add in the Bxy term giving the equation:

      Ax**2 + Bxy + Cy**2 + Dx + Ey + F = 0

Mark Biggar
{allegra,burdvax,cbosgd,hplabs,ihnp4,akgua,sdcsvax}!sdcrdcf!markb