Ellipse as Section of a Cylinder, Dandelin Spheres

Ellipse as Section of a Cylinder, Dandelin Spheres

A cylinder yields an ellipse when it is sliced by a plane that is not parallel to its center line. The proof using Danelin spheres is straight forward and has been discussed in detail by Hilbert & Cohn-Vossen (1932, p. 7) and others. In reality, it takes a bit of imagination to see why the boundary of the intersection consists of points whose combined distances to the two foci remain a constant.

The proof starts with "... take a sphere that just fits into the cylinder, and move it within the cylinder until it touches the intersecting plane.... take another such sphere and do the same thing with it on the other side of the plane." (p. 7). It is indeed a brilliant idea!

The present design is a physical model that demonstrates the sphere-based proof. It may be helpful for those who would like to see and touch the big ideas behind the argument. The same is true of conic sections.

References

1. https://www.matematicasvisuales.com/english/html/geometry/elipses/dandelincylinder.html

2. Hilbert, D., & Cohn-Vossen, S. (1999). Geometry and the imagination (2nd ed.). Chelsea. (Original work published 1932)