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Dandelin Spheres

Suppose we have an infinite circular cone in vertical position with angle between its axis and its generator. Consider a plane at angle with the vertical axis. The plane intersects the cone in the light blue curve, a conic section. This applet demonstrates that the curve is either an ellipse, a hyperbola, or a parabola. To show this we construct two spheres tangent to the cone and to the plane and we show that the red tangent points of the plane and the spheres ( and ) are the focci of the conic section. This construction was made first by the French mathematician Germinal Pierre Dandelin in 1822. In the construction below the light blue point is a point on the curve. An orange and a red segments connect this point to and . A generator (the red line) through the blue point on the curve is tangent to each sphere at some point on the respective circles along which the cone touches the spheres. We can see the properties of the conics numerically when we drag the blue point or press the “Animate Point” button. Using the property of the tangents and the fact that the distance between the tangent circles is a constant it is easy to prove that :
  • When > , the sum of the distances (orange and red) to the tangent points and is constant. The conic section is an ellipse.
  • When <, the difference of the distances (orange and red) to the tangent points and is constant. The conic section is a hyperbola.
  • When = , one of the spheres disappears. The conic section is a parabola. Click on the "Set = " button to see this case. In this last case, click on the “Show additional constructions” checkbox for help with the proof that the directrix is the intersection of the plane with the plane through the circle in which the sphere is touching the cone. It is easy to see that the distance from the point on the conic section to the tangent point equals the distance to the directrix.
All this demonstrates that the intersection of a plane and a circular cone is either a parabola, an ellipse, or a hyperbola. Note : In order to rotate the coordinate system you can click and drag anywhere in the 3D Graphics View.