Intersect a unit-radius cylinder with a plane F inclined at angle A to its axis, and touching a unit sphere inscribed in the cylinder at the focus of your ellipse. Now section by the plane perpendicular to F through the axis. By elementary plane trigonometry, the distance from ellipse centre to focus equals tan A , and the major axis of the ellipse equals sec A . Their ratio (the eccentricity) equals sin A . WFL On 11/11/13, Dan Asimov <dasimov@earthlink.net> wrote:
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.)
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun