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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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:
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