Dandelin spheres

Dandelinsche a ball (after Germinal Pierre Dandelin ) is a geometric tool to prove that the plane section of a cone of rotation is a regular cone section, where the cutting plane does not pass through the top.

If a rotary cone cut by a plane, the result is as intersection a conic. One can then, depending on the position of the plane, find one or two balls touching both the cutting plane and the cone ( from the inside).

This is shown in the figure below with an example. K and K ' are the two Berührkreise between the two spheres and the cone, F and F' are the points of contact between the balls and the sectional plane E.

Thus, you can screw the following geometric consideration: Let P be an arbitrary point on the conic. m is the generating line, which is drawn from the cone apex S by P. m meets the two Berührkreise at the points A and B. Both and are routes that lie on tangents to the lower ball. Since the tangent portions of a point on a sphere are all the same length, is. Similarly, it follows that must be. This is. Since the ( measured on a generating line ) distance between the two Berührkreise K and K ', this sum for any point P of the cone -section is equal. Therefore follows: constant (namely the distance between the circles).

The set of all points from two fixed points F and F ' but have the same distance sum is just the ellipse with foci F and F'.

This proves: the conic section is an ellipse, and the dandelinschen balls touch the cutting plane at the focal points of the ellipse.

A similar consideration can be do for the other types of conic sections ( parabola, hyperbola ).

Leaving the cone tip wander to infinity, then from the cone a right circular cylinder and the balls have the same radius. The proof that a plane section with a non- plane parallel to the cylinder axis is an ellipse can be done by pin fall (see picture).