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.)
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