Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.3 alpha 4/15/85; site kestrel.ARPA
Path: utzoo!watmath!clyde!burl!ulysses!bellcore!decvax!decwrl!glacier!kestrel!ladkin
From: ladkin@kestrel.ARPA
Newsgroups: net.math
Subject: Re: Question came up at a party
Message-ID: <4203@kestrel.ARPA>
Date: Thu, 23-Jan-86 15:44:14 EST
Article-I.D.: kestrel.4203
Posted: Thu Jan 23 15:44:14 1986
Date-Received: Sat, 25-Jan-86 03:55:36 EST
References: <532@well.UUCP>
Distribution: net
Organization: Kestrel Institute, Palo Alto, CA
Lines: 35

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?
> 

They are all intersections of a plane with a cone.

The equation of a cone with the origin at the point (rather, a double
cone intersecting at the points),with a vertical axis, is

             x**2 + y**2 = a*z for some constant a.

This is because the cross section is a circle, whose radius
varies linearly with height above the origin (the projection
on the y-z plane is straight lines (two of them).

The equation of a plane is the general 3-d linear equation.

Intersect the two. The resulting conic depends on the slope
and position of the plane only, which are determined by the
coefficients of x, y, z, and the constant term, in the plane
equation.

If you need the conic to be elsewhere in 3-space, apply
the appropriate change-of-coordinates function, which is
linear+a-constant, otherwise known as affine, and doesn't
change the power of any of the x,y or z occurrences.

I thought a constuctive-type answer might be more appropriate
than just giving an equation, even though it takes longer

Peter Ladkin
ladkin@kestrel.arpa