| 释义 |
PlaneA plane is a 2-D Surface spanned by two linearly independent vectors. The generalization of the plane tohigher Dimensions is called a Hyperplane.
In intercept form, a plane passing through the points  ,  and  is given by
 | (1) |
The equation of a plane Perpendicular to the Nonzero Vector  through the point is
 | (2) |
 | (3) |
 | (4) |
 -,  -, and  -intercepts at
and lies at a Distance
 | (8) |
The plane through  and parallel to  and  is
 | (9) |
 parallel to direction  is
 | (10) |
 | (11) |
The Distance from a point  to a plane
 | (12) |
 | (13) |
is
 | (16) |
In order to specify the relative distances of  points in the plane,  coordinates are needed, since thefirst can always be placed at (0, 0) and the second at  , where it defines the x-Axis. The remaining  points need two coordinates each. However, the total number of distances is
 | (17) |
 is a Binomial Coefficient, so the distances between points are subject to  relationships, where
 | (18) |
 and  , there are no relationships. However, for a Quadrilateral (with  ), there is one (Weinberg 1972).
It is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996).In 4-D, it is possible for four planes to intersect in exactly one point. For every set of  points in the plane, thereexists a point  in the plane having the property such that every straight line through has at least 1/3of the points on each side of it (Honsberger 1985).
Every Rigid motion of the plane is one of the following types (Singer 1995): - 1. Rotation about a fixed point
 . - 2. Translation in the direction of a line
 . - 3. Reflection across a line .
- 4. Glide-reflections along a line .
Every Rigid motion of the hyperbolic plane is one of the previous types or a - 5. Horocycle rotation.
See also Argand Plane, Complex Plane, Dihedral Angle, Elliptic Plane, Fano Plane,Hyperplane, Moufang Plane, Nirenberg's Conjecture, Normal Section, Point-Plane Distance,Projective Plane
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 208-209, 1987.Eisenberg, B. and Sullivan, R. ``Random Triangles Dimensions.'' Amer. Math. Monthly 103, 308-318, 1996. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 189-191, 1985. Singer, D. A. ``Isometries of the Plane.'' Amer. Math. Monthly 102, 628-631, 1995. Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, p. 7, 1972. |
