Yes. What a beautiful and ingenious way to prove the counterintuitive fact that the intersection of a plane and a cone is an ellipse (generically)! --Dan On 2013-11-11, at 9:29 AM, rkg wrote: Does everyone know about the Dandelin spheres? A plane intersects a cone in a conic section. The spheres inscribed in the cone and touching the plane do so at the foci. For proof, note that the tangents to a sphere from a point are equal in length, and use the `pins and string' construction for the ellipse (or hyperbola, or parabola -- focus-directrix for this last.) R. On Mon, 11 Nov 2013, Henry Baker wrote: Take a circle & look at it along its axis; it appears as a circle. Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse. Consider the foci of the ellipse. Is there anything interesting and/or cool about the relationship of the foci and the angle alpha? (I don't know any interesting answer; I'm just curious.) Aha. http://en.wikipedia.org/wiki/Dandelin_spheres . Thank you Richard. Here's some recent mail to some youngsters complaining that http://en.wikipedia.org/wiki/Hyperbola doesn't seem to clearly connect the conic section definition to the distances-to-foci definition. http://www.tweedledum.com/rwg/hyperb.GIF is an old Macsyma picture I made. At greater length, jam two (possibly unequal) spheres into opposite branches of a cone, tangent to it in rings of distance r1 and r2 from the apex. A plane tangent to both spheres cuts the cone in a hyperbola whose foci are those tangencies.
From the diagram, the distance of any point on the hyperbola from the farther focus, minus the distance from the nearer focus, will always be r1+r2. --Bill Just saw a thm by Bruce Reznick: If a triangle's vertices are (square) gridpoints, but its sides intersect no other, and its interior contains just one, then that point is the centroid. I wonder if this could be true for "If the sides and interior of a triangle each contain one gridpoint,..."
(Quickly shot down by Josh Pollock.) In a message vaporized by GMail, I also mentioned (reconstruction:) Here's one not even Coxeter knew (until he stomped off to the U. Toronto library and found http://archive.org/stream/atreatiseonanal00rogegoog#page/n169/mode/2up .) Thm: The locus of the apex ("vertex") of a cone containing a fixed ellipse (in 3-space) is a hyperbola through the foci of that ellipse, and the locus of the apex of the cone through that hyperbola is the original ellipse, which pierces the foci of the hyperbola. http://gosper.org/conethm.png is a fairly lame illustration--a cross section showing two cones from each family. With translucency, it might make a nice Mma animation, with the two conics fixed in space and the two conesweeps alternating. We could smoothly switch branches of the hyperbola when the cone thru the ellipse goes planar. --rwg