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Plane (mathematics)

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 \vec{n} = (a, b, c), the plane equation is

ax + by + cz = ax0 + by0 + cz0

for the plane passing through the point P0 and perpendicular to the vector \vec{n}.

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:

\begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0

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:

D = \frac{\left | a x_1 + b y_1 + c z_1+d \right |}{\sqrt{a^2+b^2+c^2}}

The angle between two planes

The angle between the plains a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is following

cos \alpha = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}.

Last updated: 02-10-2005 21:08:10
Last updated: 05-03-2005 17:50:55