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 (or solid 2-sphere) of radius r1 and a line segment of length 2r2:

D

=

{

(

x

,

y

,

z

,

w

)

|

x

2

+

y

2

+

z

2

≤

r

1

2

,

w

2

≤

r

2

2

}

{\displaystyle D=\{(x,y,z,w)|x^{2}+y^{2}+z^{2}\leq r_{1}^{2},\ w^{2}\leq r_{2}^{2}\}}

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 (cubic prism) 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 (r, θ, φ, w), consisting of spherical coordinates with an extra coordinate w. This is analogous to how cylindrical coordinates are defined: r and φ being polar coordinates with an elevation coordinate z. Spherindrical coordinates can be converted to Cartesian coordinates using the formulas

x

=

r

cos

⁡

φ

sin

⁡

θ

y

=

r

sin

⁡

φ

sin

⁡

θ

z

=

r

cos

⁡

θ

w

=

w

{\displaystyle {\begin{aligned}x&=r\cos \varphi \sin \theta \\y&=r\sin \varphi \sin \theta \\z&=r\cos \theta \\w&=w\end{aligned}}}

where r is the radius, θ is the zenith angle, φ is the azimuthal angle, and w is the height. Cartesian coordinates can be converted to spherindrical coordinates using the formulas

r

=

x

2

+

y

2

+

z

2

φ

=

arctan

⁡

y

x

θ

=

arccot

⁡

z

x

2

+

y

2

w

=

w

{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\varphi &=\arctan {\frac {y}{x}}\\\theta &=\operatorname {arccot} {\frac {z}{\sqrt {x^{2}+y^{2}}}}\\w&=w\end{aligned}}}

The hypervolume element for spherindrical coordinates is

d

H

=

r

2

sin

⁡

θ

d

r

d

θ

d

φ

d

w

,

{\displaystyle \mathrm {d} H=r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w,}

which can be derived by computing the Jacobian.

Measurements

Hypervolume

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

H

=

4

3

π

r

3

h

{\displaystyle H={\frac {4}{3}}\pi r^{3}h}

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:

4

3

π

r

3

{\textstyle {\frac {4}{3}}\pi r^{3}}

the volume of the bottom base:

4

3

π

r

3

{\textstyle {\frac {4}{3}}\pi r^{3}}

the volume of the lateral 3D surface:

4

π

r

2

h

{\textstyle 4\pi r^{2}h}

, which is the surface area of the spherical base times the height

Therefore, the total surface volume is

S

V

=

8

3

π

r

3

+

4

π

r

2

h

{\displaystyle SV={\frac {8}{3}}\pi r^{3}+4\pi r^{2}h}

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

H

=

⨌

D

d

H

{\displaystyle H=\iiiint \limits _{D}\mathrm {d} H}

The hypervolume of the spherinder can be integrated over spherindrical coordinates.

H

s

p

h

e

r

i

n

d

e

r

=

⨌

D

d

H

=

∫

0

h

∫

0

2

π

∫

0

π

∫

0

R

r

2

sin

⁡

θ

d

r

d

θ

d

φ

d

w

=

4

3

π

R

3

h

{\displaystyle H_{\mathrm {spherinder} }=\iiiint \limits _{D}\mathrm {d} H=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{R}r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi \,\mathrm {d} w={\frac {4}{3}}\pi R^{3}h}

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.