Dandelin's sphere

A Dandelin sphere (after Germinal Pierre Dandelin ) is a geometric aid to prove that the plane section of a rotary cone is a regular conic section , provided that the section plane does not go through the tip.

If a cone of rotation is intersected by a plane , the sectional figure is a conic section. Depending on the position of the plane, you can then find one or two spheres that touch both the cutting plane (at one point) and the cone (in a circumferential circular line from the inside). This is shown using an example in the figure below. C and C 'are the two circles of contact between the cone and one of the balls. F and F 'are the points of contact between the cutting plane e and one of the two balls.

This allows the following geometrical consideration: Let P be any point on the conic section. Let m be the surface line drawn from the apex S through P. m meets the two contact circles at points Q and Q '. Both and are segments that lie on tangents to the lower sphere. Since the tangent sections from a point to a sphere are all the same length, is . It also follows that it must be. So is . Since the distance between the two contact circles (measured on a surface line) is C and C ', this sum is the same for any point P of the conic section. Hence it follows: is constant. ${\ displaystyle {\ overline {PF '}}}$ ${\ displaystyle {\ overline {PQ '}}}$ ${\ displaystyle {\ overline {PF '}} = {\ overline {PQ'}}}$ ${\ displaystyle {\ overline {PF}} = {\ overline {PQ}}}$ ${\ displaystyle {\ overline {PF}} + {\ overline {PF '}} = {\ overline {PQ}} + {\ overline {PQ'}}}$ ${\ displaystyle {\ overline {PQ}} + {\ overline {PQ '}}}$ ${\ displaystyle {\ overline {PF}} + {\ overline {PF '}}}$

The set of all points on a plane which have the same distance sum from two fixed points F and F 'is an ellipse . This corresponds exactly to the definition of an ellipse, where F and F 'are the two focal points of the ellipse.

This proves: The conic section is an ellipse and the Dandelin spheres touch the plane of the section at the focal points of this ellipse.

Similar considerations can also be made for the other types of conic sections ( parabola , hyperbola ).

Cylinder: Dandelin balls

If you let the tip of the cone wander into infinity, the cone becomes a straight circular cylinder and the two spheres have the same radius. The proof that a plane section with a plane not parallel to the cylinder axis is an ellipse can be taken from the cone case (see picture).

literature

Fucke, Kirch, Nickel: Descriptive Geometry. Fachbuch-Verlag, Leipzig 1998, ISBN 3-446-00778-4 , pp. 69.75.
Graf, Barner: Descriptive Geometry. Quelle & Meyer, Heidelberg 1961, ISBN 3-494-00488-9 , pp. 115, 169.

Web links

Commons : Dandelin spheres - collection of images, videos and audio files