From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Intersection of cones
Date: 21 Mar 1995 18:02:38 GMT

In article <D5K1C8.KG3@zeno.fit.edu> you write:
>A friend wanted to know what equation describes the intersection of two cones
>since he thought it could be useful in tracking model rockets.  This curve
>would be a parabola under certain simple conditions, so let's specify:
>
>The equation describing cone 1 is z = sqrt(x^2 + y^2).       
>The equation describing cone 2 is z = 2*sqrt[x^2 + (y-1)^2].
>What equation describes the intersection?
>
>This is simple but not too simple.
>(Parametric?)

Eliminate  z  from the two equations to get  x^2+(y-4/3)^2 = (2/3)^2, that is,
the shadow of the intersection lies on a circle. You can parameterize the
shadow as  x = (2/3) cos(a), y = (4/3) + (2/3) sin(a)  for any  a. Then
z = sqrt(x^2+y^2) = (2/3) sqrt( cos^2(a) + [2 + sin(a)]^2) 
= (2/3) sqrt(5 + 4 sin a). In this case it is clear then that the intersection
forms a closed curve (actually two closed curves: z>0 and z<0).

The curve is not planar: if  z = Ax + B y + C  were the equation of the
curve, then  z^2 would be a quadratic in  x  and  y; eliminating  a,
we see the quadratics would have to be of the form
(8y-4) + D (x^2 + (y-2)^2) for some constant  D; none of these contain
an xy term, from which it is easy to see that the purported linear
formula for  z  cannot hold.

dave