solid geometry

can someone provide the equation for the intersection of a cone with a solid parabolic dome? the axis of both are parallel. i'm interested in the view that one would see when viewing perpendicular to the axis and parallel to the axis. this is probably an ellipse in both cases.

If they have the same axis, then the intersection is (obviously) a circle (or pair of circles), but when they have different parallel axes then the intersection(s) are certainly not ellipses. The curvature is sharper at the 'lower' end of the upper intersection and at the 'upper' end of the lower intersection. In certain circumstances the two intersections can join together.

I've created some images and short animations in a trustworthy 3d modelling package, but I can't post them here because this isn't a binary newsgroup.

thank you. i was looking for a formula/general expression.

If you intersect the cone x^2 + y^2 = a^2 z^2 with the parabolic dome z = b -(x-c)^2 - y^2 you get the equation x^2 + y^2 - a^2 (b - (x-c)^2 - y^2)^2 = 0

It's a polynomial of degree 4, and doesn't factor in general, so solutions are not going to be simple: certainly you shouldn't expect ellipses.

Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2