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
- 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.
Plane determined by a point and a normal vector
For a point P0 = (x0,y0,z0) and a vector , the plane equation is
- ax + by + cz = ax0 + by0 + cz0
for the plane passing through the point P0 and perpendicular to the vector .
Plane after three points
The equation for the plane passing through three points P1 = (x1,y1,z1), P2 = (x2,y2,z2) and P3 = (x3,y3,z3) can be represented by the following determinant:
The distance from a point to a plane
For a point P1 = (x1,y1,z1) and a plane ax + by + cz + d = 0, the distance from P1 to the plane is:
The angle between two planes
The angle between the plains a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is following
Last updated: 02-10-2005 21:08:10
Last updated: 05-03-2005 17:50:55