# 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 ${\displaystyle r}$) satisfy

${\displaystyle x^{2}+y^{2}=r^{2}.}$

We also assume that the sphere, with radius ${\displaystyle R}$ is centered at a point on the positive x-axis, at point ${\displaystyle (a,0,0)}$. Its points satisfy

${\displaystyle (x-a)^{2}+y^{2}+z^{2}=R^{2}.}$

The intersection is the collection of points satisfying both equations.