“Distance”的概念、定义、习题解题方法及翻译-数学辞典

单词

Distance

释义

Distance

Let be a smooth curve in a Manifold from to with and . Then where is the Tangent Space of at . The Lengthof with respect to the Riemannian structure is given by

(1)

and the distance

between

and

is the shortest distance between

and

given by

(2)

In order to specify the relative distances of points in the plane, coordinates are needed, since thefirst can always be taken as (0, 0) and the second as , which defines the x-Axis. Theremaining points need two coordinates each. However, the total number of distances is

(3)

where

is a Binomial Coefficient. The distances between

points are therefore subject to

relationships,where

(4)

For

, 2, ..., this gives 0, 0, 0, 1, 3, 6, 10, 15, 21, 28, ... (Sloane's A000217) relationships, and the number ofrelationships between

points is the Triangular Number

Although there are no relationships for and points, for (a Quadrilateral), there is one (Weinberg 1972):







			(5)

This equation can be derived by writing

(6)

and eliminating

and

from the equations for

, and

See also Arc Length, Cube Point Picking, Expansive, Length (Curve), Metric, PlanarDistance, Point-Line Distance--2-D, Point-Line Distance--3-D, Point-Plane Distance, Point-Point Distance--1-D, Point-PointDistance--2-D, Point-Point Distance--3-D, SpaceDistance, Sphere

References

Gray, A. ``The Intuitive Idea of Distance on a Surface.'' §13.1 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 251-255, 1993.

Sloane, N. J. A. SequenceA000217/M2535in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972.

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