Plane (geometry)
Definition
In geometry, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.
Description
A plane is defined by three non-collinear points, or equivalently, by a point and a line not containing that point, or a line and a point not on that line. It contains infinitely many lines. If a plane contains two distinct points, it contains the unique line containing those points. If two planes intersect, their intersection is a line. If two lines intersect, their intersection is a point.
Properties
Euclidean planes
In Euclidean geometry, the following properties are satisfied:
- Two distinct points lie on a unique plane.
- Three non-collinear points lie on a unique plane.
- A line and a point not on the line lie on a unique plane.
- Two intersecting lines lie on a unique plane.
- Two parallel lines lie on a unique plane.
Non-Euclidean planes
In non-Euclidean geometry, planes may have different properties. For example, in spherical geometry, planes are represented as great circles on a sphere, and lines within these planes are represented as smaller circles.
Equations of a plane
In three-dimensional space, a plane can be defined by a point and a normal vector to the plane. The equation of a plane is given by: Ax + By + Cz = D where A, B, and C are the coefficients of the normal vector, and D is a constant.
Applications
Planes are used in various fields such as mathematics, physics, engineering, and computer graphics. They are fundamental in the study of geometry and are used to describe the space in which we live.
