A sphere is defined as the set of all points in which are a distance (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 . 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 are given by

(1) | |||

(2) |

(Beyer 1987, p. 130). In

(3) |

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 is given in Cartesian Coordinates by

(4) |

(5) |

(6) |

(7) | |||

(8) | |||

(9) |

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

(10) | |||

(11) | |||

(12) |

second Fundamental Forms

(13) | |||

(14) | |||

(15) |

Area Element

(16) |

(17) |

(18) |

(19) | |||

(20) | |||

(21) |

where runs from 0 to and runs from to .

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
and
lying on a Diameter is given by

(22) |

Four points are sufficient to uniquely define a sphere. Given the points
with , 2, 3, and 4, the
sphere containing them is given by the beautiful Determinant equation

(23) |

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

(24) |

The distribution of Angles for random rotation of a sphere is

(25) |

To pick a random point on the surface of a sphere, let and be random variates on . Then

(26) | |||

(27) |

This works since the Solid Angle is

(28) |

(29) |

(30) |

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
with
since all points
with the same are rotationally identical. The distance between the two points is then

(31) |

(32) |

(33) |

The Delta Function contributes when

(34) |

(35) |

(36) |

However, we need

(37) |

(38) |

so

(39) |

(40) |

The Moments about zero are

(41) |

(42) | |||

(43) | |||

(44) | |||

(45) |

Moments about the Mean are

(46) | |||

(47) | |||

(48) | |||

(49) |

so the Skewness and Kurtosis are

(50) | |||

(51) |

**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/.

© 1996-9

1999-05-26