Cone-Sphere Intersection

Let a cone of opening parameter and vertex at intersect a sphere of radius centered at , with the cone oriented such that its axis does not pass through the center of the sphere. Then the equations of the curve of intersection are

			(1)
			(2)

Combining (1) and (2) gives

(3)

(4)

Therefore, and are connected by a complicated quartic equation, and , , and by a quadratic equation.

If the cone-sphere intersection is on-axis so that a cone of opening parameter and vertex at is oriented with its axis along a radial of the sphere of radius centered at , then the equations of the curve of intersection are

			(5)
			(6)

Combining (5) and (6) gives

(7)

(8)

(9)

Using the quadratic equation gives

			(10)
			(11)

So the curve of intersection is planar. Plugging (11) into (◇) shows that the curve is actually a circle, with radius given by

(12)