“Sphere”的概念、定义、习题解题方法及翻译-数学辞典

单词

Sphere

释义

Sphere

A sphere is defined as the set of all points in which are a distance (the ``Radius'') from a givenpoint (the ``Center''). Twice the Radius is called the Diameter, and pairs of points on opposite sidesof a Diameter are called Antipodes. The term ``sphere'' technically refers to the outer surface ofa ``Bubble,'' which is denoted . However, in common usage, the word sphere is also used to mean theUnion of a sphere and its Interior (a ``solid sphere''), where the Interior is called a Ball. TheSurface Area of the sphere and Volume of the Ball of Radius are given by

			(1)
			(2)

(Beyer 1987, p. 130). In On the Sphere and Cylinder (ca. 225 BC

), Archimedes

became the first toderive these equations (although he expressed

in terms of the sphere's circular cross-section). The fact that

(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 tothe sphere, a point). The size of the Circle is maximized when the Plane defining the cross-section passesthrough a Diameter.

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

(4)

which is a special case of the Ellipsoid

(5)

and Spheroid

(6)

A sphere may also be specified in Spherical Coordinates by

			(7)
			(8)
			(9)

where

is an azimuthal coordinate running from 0 to

(Longitude),

is a polar coordinate runningfrom 0 to

(Colatitude), and

is the Radius. Note that there are several other notationssometimes 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 FundamentalForms

			(10)
			(11)
			(12)

second Fundamental Forms

			(13)
			(14)
			(15)

Area Element

(16)

Gaussian Curvature

(17)

and Mean Curvature

(18)

A sphere may also be represented parametrically by letting

, so

			(19)
			(20)
			(21)

where

runs from 0 to

and

runs from

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, thesphere containing them is given by the beautiful Determinant equation

(23)

(Beyer 1987, p. 210).

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

(24)

The distribution of Angles for random rotation of a sphere is

(25)

giving a Mean of

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)

Another easy way to pick a random point on a Sphere is to generate three gaussian random variables

, and

. Then the distribution of the vectors

(29)

is uniform over the surface

. Another method is to pick

from a Uniform Distribution over

and

from a Uniform Distribution over

. Then the points

(30)

are uniformly distributed over

Pick four points on a sphere. What is the probability that the Tetrahedron having these points asVertices contains the Center of the sphere? In the 1-D case, the probability that asecond 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 aTriangle containing the Center, it must lie in the quadrant bisected by a Line Segment passing throughthe center of the Circle and the bisector of the two points. This happens for one Quadrant, so the probabilityis 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 ofgenerality. The second point can be given the coordinates with since all pointswith the same are rotationally identical. The distance between the two points is then

(31)

Because the surface Area element is

(32)

the probability that two points are a distance

apart is


			(33)

The Delta Function contributes when

(34)

(35)


			(36)

However, we need

(37)

and


			(38)

(39)

and

(40)

for

. Somewhat surprisingly, the largest distances are the most common, contrary to most people's intuition. Aplot of 15 random lines is shown below.

The Moments about zero are

(41)

giving the first few as

			(42)
			(43)
			(44)
			(45)

Moments about the Mean are

			(46)
			(47)
			(48)
			(49)

so the Skewness and Kurtosis are

			(50)
			(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, ProlateSpheroid, 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/.

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