---
tag: W3DCQ22JA3
kind: concept
name: Dandelin spheres
msc: [51M04, 51N20]
sources:
  - url: https://en.wikipedia.org/wiki/Dandelin_spheres
same_as:
  wikidata: Q676413
  wikipedia: Dandelin spheres
---

## Summary

Cut a right circular cone by a plane.
The Dandelin spheres of this plane section are the spheres that touch the cone along a circle and also touch the cutting plane.
For an ellipse there are two, and their points of contact with the plane are the two foci of the ellipse.
They give a short proof that the plane sections of a cone are the curves defined by foci and distances.
So the two classical definitions of the [[conic sections|00HJCMC2CD]] agree.
They are named after Germinal Pierre Dandelin.

## Explanations

### L1 human

Slice an ice-cream cone with a tilted plane: the cut is an ellipse.
Now drop one ball into the cone above the plane and one below it.
Each ball sits snugly against the cone along a circle and just touches the plane.
The two balls touch the plane at two points, and these points are exactly the foci of the ellipse.
The reason is that the two tangent segments from any point to a ball have the same length.
So the sum of distances to the foci equals the distance between the two circles along the cone.
This distance is the same for every point of the ellipse.