Intersection of a Cone and a Sphere

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

Contributed by: Erik Mahieu (April 2014)
Open content licensed under CC BY-NC-SA


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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:

,

where

,

,

,

,

.



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