Dandelin spheres

Summary

Cut a right circular cone by a plane. The Dandelin spheres of this plane section are the spheres that touch the cone along a circle and also touch the cutting plane. For an ellipse there are two, and their points of contact with the plane are the two foci of the ellipse. They give a short proof that the plane sections of a cone are the curves defined by foci and distances. So the two classical definitions of the conic sections agree. They are named after Germinal Pierre Dandelin.