A plane is a 2-D Surface spanned by two linearly independent vectors. The generalization of the plane to higher 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) |

(5) | |||

(6) | |||

(7) |

and lies at a Distance

(8) |

The plane through and parallel to
and
is

(9) |

(10) |

(11) |

The Distance from a point
to a plane

(12) |

(13) |

(14) | |||

(15) |

is

(16) |

In order to specify the relative distances of points in the plane,
coordinates are needed, since the
first 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) |

(18) |

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, there
exists a point in the plane having the property such that *every* straight line through has at least 1/3
of 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.

**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.

© 1996-9

1999-05-25