## Dandelin Spheres (*)

### Introduction

Given a cone and a plane, not passing through the vertex, there are always one or two spheres which are tangent simultaneously to the cone and to the plane: they are called Dandelin's spheres. The intersection curve of the plane with the cone is a conic and the points of tangency of the spheres (could be one!) with the plane are the foci of this conic.
The analysis of this Dandelin sphere (respectively, these Dandelin spheres) allows to easily demonstrate that the conics, seen as the intersection of a plane with a cone, are the same curves as the ones defined in the usual manner, in terms of the known metric properties involving their foci.

The following applets enable the observation of the variation of the Dandelin's spheres as a function of various parameters.

1. Given a conic and a Dandelin sphere of radius $$r$$, the applet constructs the cone (which divides the conical plane produced by the tangent conical to the sphere) and the other Dandelin sphere, in case it exists.
• Case 1: the conic is an ellipse given by its major semi-axis and the semi-focal distance.
• Case 2: the conic is a parabola given by the distance between its vertex and focus.
• Case 3: the conic is a hyperbola given by the distance between its center and the vertices and by the distance between its center and focus.
2. Case 4: given a cone with variable opening angle and a plane with variable slope and height, the applet constructs the section produced by the plane in the cone. The cone consists of flaps grid holes that open and close, allowing or not the visualization into the interior of the cone.

Translated for Atractor by a CMUC team, from its original version in Portuguese. Atractor is grateful for this cooperation.

(*) The applets were built in the context of the European project PENCIL in which Atractor and Ciência Viva were involved.

Difficulty level: Upper Secondary, University