Plane (mathematics)
From Freepedia
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
- <math>ax + by + cz + d = 0</math>,
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 non-collinear points (not lying on the same line)
- a line and a point not on the line
- a point and a line, which is normal (perpendicular) to the plane
- two lines which intersect
- two lines which 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.
Contents |
Plane determined by a point and a normal vector
For a point <math> P_0 = (x_0,y_0,z_0) </math> and a vector <math>\vec{n} = (a, b, c) </math>, the plane equation is
- <math> ax + by + cz = a x_0 + b y_0 + c z_0 </math>
for the plane passing through the point <math> P_0 </math> and perpendicular to the vector <math>\vec{n}</math>.
Plane through three points
The equation for the plane passing through three points <math> P_1 = (x_1,y_1,z_1) </math>, <math> P_2 = (x_2,y_2,z_2) </math> and <math> P_3 = (x_3,y_3,z_3) </math> can be represented by the following determinant:
- <math> \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 </math>
The distance from a point to a plane
For a point <math> P_1 = (x_1,y_1,z_1) </math> and a plane <math>ax + by + cz + d = 0</math>, the distance from <math> P_1 </math> to the plane is:
- <math> D = \frac{\left | a x_1 + b y_1 + c z_1+d \right |}{\sqrt{a^2+b^2+c^2}} </math>
The angle between two planes
The angle between two intersecting planes, called the dihedral angle, defined using plane equations <math>a_1 x + b_1 y + c_1 z + d_1 = 0</math> and <math>a_2 x + b_2 y + c_2 z + d_2 = 0</math>, is the following:
- <math> \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}} </math>.
