Spherinder


In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball of radius r1 and a line segment of length 2r2:
Like the duocylinder, it is also analogous to a cylinder in 3-space, which is the Cartesian product of a disk with a line segment. It is a rotatope and a toratope.
It can be seen in 3-dimensional space by stereographic projection as two concentric spheres, in a similar way that a tesseract can be projected as two concentric cubes, and how a circular cylinder can be projected into 2-dimensional space as two concentric circles.

Spherindrical coordinate system

One can define a "spherindrical" coordinate system, consisting of spherical coordinates with an extra coordinate. This is analogous to how cylindrical coordinates are defined: and being polar coordinates with an elevation coordinate. Spherindrical coordinates can be converted to Cartesian coordinates using the formulas where is the radius, is the zenith angle, is the azimuthal angle, and is the height. Cartesian coordinates can be converted to spherindrical coordinates using the formulas The hypervolume element for spherindrical coordinates is which can be derived by computing the Jacobian.

Measurements

Hypervolume

Given a spherinder with a spherical base of radius and a height, the hypervolume of the spherinder is given by

Surface volume

The surface volume of a spherinder, like the surface area of a cylinder, is made up of three parts:
  • the volume of the top base:
  • the volume of the bottom base:
  • the volume of the lateral 3D surface:, which is the surface area of the spherical base times the height
Therefore, the total surface volume is

Proof

The above formulas for hypervolume and surface volume can be proven using integration. The hypervolume of an arbitrary 4D region is given by the quadruple integral
The hypervolume of the spherinder can be integrated over spherindrical coordinates.

Related 4-polytopes

The spherinder is related to the uniform prismatic polychora, which are Cartesian products of a regular or semiregular polyhedron and a line segment. There are eighteen convex uniform prisms based on the Platonic and Archimedean solids, plus an infinite family based on antiprisms, and another infinite family of uniform duoprisms, which are products of two regular polygons.