Definition:Dandelin Spheres
Definition
Let $\CC$ be a double napped right circular cone with apex $O$.
Let $\PP$ be a plane which intersects $\CC$ such that:
- $\PP$ does not pass through $O$
- $\PP$ is not parallel to a generatrix of $\CC$
- $\PP$ is not perpendicular to the axis of $\CC$.
Hence, by construction, the resulting conic section $\EE$ is either an ellipse or a hyperbola, and is not degenerate.
Let two spheres $\SS$ and $\SS'$ be constructed so that they have ring-contact with $\CC$ such that $\PP$ is tangent to both $\SS$ and $\SS'$.
Then $\SS$ and $\SS'$ are known as Dandelin spheres.
Ellipse
Let $\PP$ be a plane which intersects $\CC$ to form an ellipse.
The Dandelin spheres $\SS$ and $\SS'$ appear as follows:
Hyperbola
Let $\PP$ be a plane which intersects $\CC$ to form a hyperbola.
The Dandelin spheres $\SS$ and $\SS'$ appear as follows:
Parabola
Let $\PP$ be a plane which intersects $\CC$ to form a parabola.
Then it is possible to construct one sphere $\SS$ so that it has ring-contact with $\CC$ such that $\PP$ is tangent to $\SS$.
Also see
- Dandelin's Theorem: $\PP$ touches $\SS$ and $\SS'$ at the foci of $\EE$.
- Results about Dandelin spheres can be found here.
Source of Name
This entry was named for Germinal Pierre Dandelin.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $1 \text a$. Focal properties
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Dandelin sphere
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Dandelin sphere
- Weisstein, Eric W. "Dandelin Spheres." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DandelinSpheres.html