Sphere–cylinder intersection

In the theory of analytic geometry for real three-dimensional space, the curve formed from the intersection between a sphere and a cylinder can be a circle, a point, the empty set, or a special type of curve.

For the analysis of this situation, assume (without loss of generality) that the axis of the cylinder coincides with the z-axis; points on the cylinder (with radius ) satisfy

We also assume that the sphere, with radius is centered at a point on the positive x-axis, at point . Its points satisfy

The intersection is the collection of points satisfying both equations.