[net.math] Foci of cylindical sections

leeper@mtgzz.UUCP (m.r.leeper) (01/20/86)

This is kind of a nice, simple problem that I posed for myself and
solved.  The solution is kind of pleasing too.  In a three-space
(coordinates x, y and z) you have a cylinder whose equation is x^2 +
z^2 = 1.  Now if you take planes through the z-axis, each (but for the
one containing the y-axis) will intersect the cylinder in an ellipse.
Describe the locus of points that are foci of those elipses.  What
kind of a curve in three-space will the foci trace out as the angle of
the plane changes?  Express the solution in each of rectangular and
polar-coordinates, each is interesting.