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

Its points satisfy The intersection is the collection of points satisfying both equations.

, the sphere lies entirely in the interior of the cylinder.

The intersection is the empty set.

, the sphere lies in the interior of the cylinder except for one point.

If the center of the sphere lies on the axis of the cylinder,

In that case, the intersection consists of two circles of radius

These circles lie in the planes If

, the intersection is a single circle in the plane

, the projection of the intersection onto the xz-plane is the section of an orthogonal parabola; it is only a section due to the fact that

The vertex of the parabola lies at point

cuts the parabola into two segments.

In this case, the intersection of sphere and cylinder consists of two closed curves, which are mirror images of each other.

Their projection in the xy-plane are circles of radius

Each part of the intersection can be parametrized by an angle

, the intersection of sphere and cylinder consists of a single closed curve.

It can be described by the same parameter equation as in the previous section, but the angle

The curve contains the following extreme points: In the case

, the cylinder and sphere are tangential to each other at point

Its parameter representation is The volume of the intersection of the two bodies, sometimes called Viviani's volume, is[1] [2] [3]