Re: drinking the Dandelin-whines: Yes, indeed. But I haven't been able to "see" how a circle transforms into a Dandelin ellipse merely by rotating it somehow in 3-space. If one takes a slightly different Dandelin configuration, wherein equal spheres are embedded in a circular cylindrical tube, then the inner spheres touch an oblique plane/ellipse at the foci. But I'm still having trouble finding a cool relationship between the locations of the foci and the angle of the plane. Perhaps Dandelin spheres in 4-space ? At 09:29 AM 11/11/2013, 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.)