Dandelin spheres

Dandelin spheres. Image by Hop David, used with permission

If a cone is sliced through by a plane, the two spheres that just fit inside the cone, one on each side of the plane and both tangent to it and touching the cone, are known as Dandelin spheres. They are named after the Belgian mathematician and military engineer Germinal Pierre Dandelin (1794–1847) who gave an elegant proof that the two spheres touch the conic section at its foci. In 1826, Dandelin showed that the same result applies to the plane sections of a hyperboloid of revolution.

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