Spherinder

A spherinder is a prism based on a ball. It has two solid balls for bases connected by a surface sometimes referred to as a spherical hose. It is the most direct 4D analog of a 3D cylinder.

It is a rotatope, thus it is also a toratope, a tapertope, and a bracketope.

Coordinates
Where r is the radius of the base and h is the height:

Points on the faces of a spherinder are all points (x,y,z,w) such that


 * $$x^2+y^2+z^2=r^2 \quad\text{and}\quad w^2=\left(\tfrac{h}{2}\right)^2.$$

Points on the surcell of a spherinder are all points (x,y,z,w) such that


 * $$x^2+y^2+z^2<r^2 \quad\text{and}\quad w^2=\left(\tfrac{h}{2}\right)^2,$$ (balls)
 * $$x^2+y^2+z^2=r^2 \quad\text{and}\quad w^2<\left(\tfrac{h}{2}\right)^2.$$ (spherical hose)

Points in the interior of a spherinder are all points (x,y,z,w) such that
 * $$x^2+y^2+z^2<r^2 \quad\text{and}\quad w^2<\left(\tfrac{h}{2}\right)^2.$$