Intersection of a Cone and a Sphere

This Demonstration explores the shape of the difference between a right elliptic cone and a sphere.

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DETAILS

The parametric equation of a right elliptic cone of height and an elliptical base with semi-axes and ( is the distance of the cone's apex to the center of the sphere) is
,
,
,
where and are parameters.
The parametric equation of a sphere with radius is
,
,
,
where and are parameters.
The intersection curve of the two surfaces can be obtained by solving the system of three equations
for three of the four parameters .
In this Demonstration, solving for , , and gives the parametric equations for the intersection curve with parameter . The curve consists of four parts of similar form, depending on the sign of some parts of the equations:
,
where
,
,
,
,
.
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