Ooh. There is a continuum of cones that have a given ellipse as cross-section. In the limit, with the apex of the cone at infinity, the cone becomes a cylinder. Intuitively, the Dandelin construction must still work with a cylinder. And proving that the cross-section of a cylinder is a stretched circle seems like it should be easy. I bet there is a good proof hiding here. On Fri, Jul 17, 2015 at 5:04 PM, Andy Latto <andy.latto@pobox.com> wrote:
That's a great geometrical proof that "Set of points with a constant distant sum from two foci" is the same as "intersection of a circular cone and a plane". But what's the geometrical argument that either of those is the same as a stretched circle?
Andy
On Fri, Jul 17, 2015 at 3:05 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Is there a proof this short and simple that the ellipse defined by these definitions is related to a circle by an affine transformation?
Yes: https://en.wikipedia.org/wiki/Dandelin_spheres
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