Dandelin Spheres for the Ellipse
- Sisto Baldo
Dandelin Spheres and the focal property of the ellipse. Short movie: https://www.youtube.com/watch?v=-Yoq-Rv4cLc
The Dandelin spheres of the ellipse are the two spheres, which are tangent both to the cone and the cutting plane. The points where the spheres touch the cutting plane are the foci of the ellipse. Consider a point on the ellipse and the segment of the generatrix line through , bounded by the two (red) circles where the spheres touch the cone. The part of this segment pictured in green is equal to the green segment from to one of the foci (since both segments are tangent to the upper sphere and issue from the same point). Likewise, the portion of the generatrix segment pictured in red, is equal to the red segment from to the other focus. It follows that the sum of the distances from to the foci is constant and equal to the segment of generatrix between the red circles!