Sphere

A sphere is defined as the set of all points in $\Bbb{R}^3$ which are a distance $r$ (the ``Radius'') from a given point (the ``Center''). Twice the Radius is called the Diameter, and pairs of points on opposite sides of a Diameter are called Antipodes. The term ``sphere'' technically refers to the outer surface of a ``Bubble,'' which is denoted ${\Bbb{S}}^2$. However, in common usage, the word sphere is also used to mean the Union of a sphere and its Interior (a ``solid sphere''), where the Interior is called a Ball. The Surface Area of the sphere and Volume of the Ball of Radius $r$ are given by

$$S = 4\pi r^2$$ (1)

$$V = {\textstyle{4\over 3}}\pi r^3$$ (2)

(Beyer 1987, p. 130). In On the Sphere and Cylinder (ca. 225 BC ), Archimedes became the first to derive these equations (although he expressed $\pi$ in terms of the sphere's circular cross-section). The fact that

$${V_{\rm sphere}\over V_{\rm circumscribed\ cylinder}-V_{\rm ... ...3}}} = {{\textstyle{4\over 3}}\over{\textstyle{2\over 3}}} = 2$$ (3)

was also known to Archimedes.

Any cross-section through a sphere is a Circle (or, in the degenerate case where the slicing Plane is tangent to the sphere, a point). The size of the Circle is maximized when the Plane defining the cross-section passes through a Diameter.

The equation of a sphere of Radius $r$ is given in Cartesian Coordinates by

$$x^2+y^2+z^2=r^2,$$ (4)

which is a special case of the Ellipsoid

$${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2}=1$$ (5)

and Spheroid

$${x^2+y^2\over a^2}+{z^2\over c^2}=1.$$ (6)

A sphere may also be specified in Spherical Coordinates by

$$x = \rho\cos\theta\sin\phi$$ (7)

$$y = \rho\sin\theta\sin\phi$$ (8)

$$z = \rho\cos\phi,$$ (9)

where $\theta$ is an azimuthal coordinate running from 0 to $2\pi$ (Longitude), $\phi$ is a polar coordinate running from 0 to $\pi$ (Colatitude), and $\rho$ is the Radius. Note that there are several other notations sometimes used in which the symbols for $\theta$ and $\phi$ are interchanged or where $r$ is used instead of $\rho$. If $\rho$ is allowed to run from 0 to a given Radius $r$, then a solid Ball is obtained. Converting to ``standard'' parametric variables $a=\rho$, $u=\theta$, and $v=\phi$ gives the first Fundamental Forms

$$E = a\sin^2 v$$ (10)

$$F = 0$$ (11)

$$G = a,$$ (12)

second Fundamental Forms

$$e = a^2\sin^2 v$$ (13)

$$f = 0$$ (14)

$$g = a^2,$$ (15)

Area Element

$$dA=a\sin v,$$ (16)

Gaussian Curvature

$$K={1\over a^2},$$ (17)

and Mean Curvature

$$H={1\over a}.$$ (18)

A sphere may also be represented parametrically by letting $u\equiv r\cos\phi$, so

$$x = \sqrt{r^2-u^2}\,\cos\theta$$ (19)

$$y = \sqrt{r^2-u^2}\,\sin\theta$$ (20)

$$z = u,$$ (21)

where $\theta$ runs from 0 to $2\pi$ and $u$ runs from $-r$ to $r$.

Given two points on a sphere, the shortest path on the surface of the sphere which connects them (the Sphere Geodesic) is an Arc of a Circle known as a Great Circle. The equation of the sphere with points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ lying on a Diameter is given by

$$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)+(z-z_1)(z-z_2)=0.$$ (22)

Four points are sufficient to uniquely define a sphere. Given the points $(x_i, y_i, z_i)$ with $i=1$, 2, 3, and 4, the sphere containing them is given by the beautiful Determinant equation

$$\left\vert\matrix{ {x_{}}^2+{y_{}}^2+{z_{}}^2 & x_{} & y_{} ... ...{4}}^2+{z_{4}}^2 & x_{4} & y_{4} & z_{4} & 1\cr }\right\vert=0$$ (23)

(Beyer 1987, p. 210).

The generalization of a sphere in $n$ dimensions is called a Hypersphere. An $n$-D Hypersphere can be specified by the equation

$${x_1}^2+{x_2}^2+\ldots+{x_n}^2=r^2.$$ (24)

The distribution of Angles for random rotation of a sphere is

$$P(\theta)={2\over \pi}\sin^2({\textstyle{1\over 2}}\theta),$$ (25)

giving a Mean of $\pi/2+2/\pi$.

To pick a random point on the surface of a sphere, let $u$ and $v$ be random variates on $[0,1]$. Then

$$\theta = 2\pi u$$ (26)

$$\phi = \cos^{-1} (2v-1).$$ (27)

This works since the Solid Angle is

$$d\Omega=\sin\phi\,d\theta\,d\phi=d\theta\,d(\cos\phi).$$ (28)

Another easy way to pick a random point on a Sphere is to generate three gaussian random variables $x$, $y$, and $z$. Then the distribution of the vectors

$${1\over\sqrt{x^2+y^2+z^2}}\left[{\matrix{x\cr y\cr z\cr}}\right]$$ (29)

is uniform over the surface ${\Bbb{S}}^2$. Another method is to pick $z$ from a Uniform Distribution over $[-1,1]$ and $\theta$ from a Uniform Distribution over $[0, 2\pi)$. Then the points

$$\left[{\matrix{\sqrt{1-z^2}\cos\theta\cr \sqrt{1-z^2}\sin\theta\cr z\cr}}\right]$$ (30)

are uniformly distributed over ${\Bbb{S}}^2$.

Pick four points on a sphere. What is the probability that the Tetrahedron having these points as Vertices contains the Center of the sphere? In the 1-D case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the 2-D case, pick two points. In order for the third to form a Triangle containing the Center, it must lie in the quadrant bisected by a Line Segment passing through the center of the Circle and the bisector of the two points. This happens for one Quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one Octant, or 1/8.

Pick two points at random on a unit sphere. The first one can be assigned the coordinate (0, 0, 1) without loss of generality. The second point can be given the coordinates $(\sin\phi, 0 \cos\phi)$ with $\theta\equiv 0$ since all points with the same $\phi$ are rotationally identical. The distance between the two points is then

$$r=\sqrt{\sin^2\phi+(1-\cos\phi)^2}=\sqrt{2-\cos\phi}=2\sin({\textstyle{1\over 2}}\phi).$$ (31)

Because the surface Area element is

$$d\Omega=\sin\phi\,d\theta\,d\phi,$$ (32)

the probability that two points are a distance $r$ apart is

$$P_\phi(r) = {\int_0^\pi \delta(\phi-r)\sin\phi\,d\phi\over \int_0^\pi \sin\phi\,d\phi}$$

$$= {\textstyle{1\over 2}}\int_0^\pi \delta[r-2\sin({\textstyle{1\over 2}}\phi)]\sin\phi\,d\phi.$$ (33)

The Delta Function contributes when

$${\textstyle{1\over 2}}r=\sin({\textstyle{1\over 2}}\phi)$$ (34)

$$\phi=2\sin^{-1}({\textstyle{1\over 2}}r),$$ (35)

so

$$P_\phi(r) = {\textstyle{1\over 2}}\sin[2\sin^{-1}({\textstyle{1\over 2}}r)] = \sin[\sin^{-1}({\textstyle{1\over 2}}r)]\cos[\sin^{-1}({\textstyle{1\over 2}}r)]$$

$$= {\textstyle{1\over 2}}r\sqrt{1-({\textstyle{1\over 2}}r)^2} = {\textstyle{1\over 4}}r\sqrt{4-r^2}.$$ (36)

However, we need

$$P_r(r)\,dr = P_\phi(r){d\phi\over dr}\,dr,$$ (37)

and

$${\textstyle{1\over 2}}dr = {\textstyle{1\over 2}}\cos({\textstyle{1\over 2}}\phi)\,d\phi={\textstyle{1\over 2}}\sqrt{1-\sin^2({\textstyle{1\over 2}}\phi)}\,d\phi$$

$$= {\textstyle{1\over 2}}\sqrt{1-({\textstyle{1\over 2}}r)^2}\,d\phi={\textstyle{1\over 4}}\sqrt{4-r^2}\,d\phi$$ (38)

so

$${d\phi\over dr}={2\over\sqrt{4-r^2}},$$ (39)

and

$$P_r(r)={\textstyle{1\over 4}}r\sqrt{4-r^2}{2\over\sqrt{4-r^2}}={\textstyle{1\over 2}}r$$ (40)

for $r\in[0,2]$. Somewhat surprisingly, the largest distances are the most common, contrary to most people's intuition. A plot of 15 random lines is shown below.

The Moments about zero are

$$\mu_n'=\left\langle{r^n}\right\rangle{}=\int_0^2 r^n\,dr = {2^{n+1}\over 2+n},$$ (41)

giving the first few as

$$\mu_1' = {\textstyle{4\over 3}}$$ (42)

$$\mu_2' = 2$$ (43)

$$\mu_3' = {\textstyle{16\over 5}}$$ (44)

$$\mu_4' = {\textstyle{16\over 3}}.$$ (45)

Moments about the Mean are

$$\mu = {\textstyle{4\over 3}}$$ (46)

$$\mu_2 = \sigma^2={\textstyle{2\over 9}}$$ (47)

$$\mu_3 = -{\textstyle{8\over 135}}$$ (48)

$$\mu_4 = {\textstyle{16\over 135}},$$ (49)

so the Skewness and Kurtosis are

$$\gamma_1 = {\textstyle{4\over 5}}\sqrt{2}$$ (50)

$$\gamma_2 = -{\textstyle{5\over 3}}.$$ (51)

See also Ball, Bing's Theorem, Bubble, Circle, Dandelin Spheres, Diameter, Ellipsoid, Exotic Sphere, Fejes Tóth's Problem, Hypersphere, Liebmann's Theorem, Liouville's Sphere-Preserving Theorem, Mikusinski's Problem, Noise Sphere, Oblate Spheroid, Osculating Sphere, Parallelizable, Prolate Spheroid, Radius, Space Division, Sphere Packing, Tennis Ball Theorem

References

Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.

Eppstein, D. ``Circles and Spheres.'' http://www.ics.uci.edu/~eppstein/junkyard/sphere.html.

Geometry Center. ``The Sphere.'' http://www.geom.umn.edu/zoo/toptype/sphere/.