Definition:Dandelin Spheres

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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:


Dandelin-spheres-ellipse.png


Hyperbola

Let $\PP$ be a plane which intersects $\CC$ to form a hyperbola.


The Dandelin spheres $\SS$ and $\SS'$ appear as follows:


Dandelin-spheres-hyperbola.png


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 $\SS$.


Dandelin-spheres-parabola.png


Also see

  • Results about Dandelin spheres can be found here.


Source of Name

This entry was named for Germinal Pierre Dandelin.


Sources