Cubinder

A cubinder is a prism based on a cylinder. As such, it is the Cartesian product of a circle and a square, and the limit of n,4-duoprisms as n goes to infinity. It consists of a ring of 4 cylinders joined at their circles, joined to a lateral surface similar to a square torus.

It can roll on its square torus surcell; it rolls like a circle and covers the space of a line.

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 edges of a cubinder are all points (x,y,z,w) such that


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

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


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

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


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

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