In mathematics, a plane is the 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 *x*-*y*-*z* coordinate system, one can define a plane as the set of all solutions of an equation

*a**x* + *b**y* + *c**z* + *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.

### Plane determined by a point and a normal vector

For a point *P*_{0} = (*x*_{0},*y*_{0},*z*_{0}) and a vector , the plane equation is

*a**x* + *b**y* + *c**z* = *a**x*_{0} + *b**y*_{0} + *c**z*_{0}

for the plane passing through the point *P*_{0} and perpendicular to the vector .

### Plane after three points

The equation for the plane passing through three points *P*_{1} = (*x*_{1},*y*_{1},*z*_{1}), *P*_{2} = (*x*_{2},*y*_{2},*z*_{2}) and *P*_{3} = (*x*_{3},*y*_{3},*z*_{3}) can be represented by the following determinant:

### The distance from a point to a plane

For a point *P*_{1} = (*x*_{1},*y*_{1},*z*_{1}) and a plane *a**x* + *b**y* + *c**z* + *d* = 0, the distance from *P*_{1} to the plane is:

### The angle between two planes

The angle between the plains *a*_{1}*x* + *b*_{1}*y* + *c*_{1}*z* + *d*_{1} = 0 and *a*_{2}*x* + *b*_{2}*y* + *c*_{2}*z* + *d*_{2} = 0 is following

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Last updated: 02-10-2005 21:08:10

Last updated: 05-03-2005 17:50:55