# Plane

In mathematics, a plane is a fundamental two-dimensional object. Intuitively, it may be visualized as a flat infinite piece of paper. Most of the fundamental work in geometry, trigonometry, and graphing is performed in two dimensions, or in other words, in a plane.

Given a plane, one can introduce a Cartesian coordinate system on it in order to label every point on the plane uniquely with two numbers, its coordinates. In a three-dimensional coordinate system (x;y;z), one can define a plane as the set of all solutions of an equation

ax + by + cz + d = 0,

where *a*, *b*, *c* and *d* are real numbers such that
not all of *a*, *b*, *c* are zero. Alternatively, a plane may be
described parametrically as the set of all points of the form **u** + *s*
**v** + *t* **w** where *s* and *t* range over all real
numbers, and **u**, **v** and **w** are given vectors defining the
plane.

A plane is uniquely determined by any of the following combinations:

- three points not lying on a line
- a line and a point not lying on the line
- a point and a line, the normal to the plane
- two lines which intersect in a single point or are parallel
- In three-dimensional space, two different planes are either parallel or they intersect in a line. A line which is not parallel to a given plane intersects that plane in a single point.

**See also**

2D | Two-dimensional coordinate system |

3D | Three-dimensional coordinate system |

Angle | Definition of an angle |

Axis | Definition of Cartesian axis |

Cartesian geometry | What is Cartesian geometry? |

Coordinate system | Definition of coordinates |

Curve | Definition of a curve |

Distance | Definition of distance |

Euclidean geometry | What is Euclidean geometry? |

Geometry | Definition of geometry |

Length | Definition of length |

Line | Definition of a line |

Origin | Definition of origin in a Cartesian coordinate system |

Perspective projection | Definition of perspective projection |

Planar homography | Definition of planar homography |

Plane | Definition of a plane |

Point | Definition of a point |

Point (kinematics) | Definition of a point (kinematics) |

Projective geometry | What is projective geometry? |

Segment (kinematics) | Definition of a segment (kinematics) |

Vanishing points | Definition of vanishing points and vanishing lines in perspective projection |

Vector | Definition of a vector |