In mathematics, a plane is a fundamental two-dimensional object. Intuitively, it may be visualized as a flat infinite sheet of paper. There are several definitions of the plane, equivalent in the sense of Euclidean geometry, but which can be extended in different ways to define object in other areas of mathematics.
In some areas of mathematics, such as plane geometry or 2D computer graphics, the whole space in which the work is carried out is a single plane. In such situations the definite article is used: the plane. Many fundamental tasks in geometry, trigonometry, and graphing are performed in the two dimensional space, or in other words, in the plane.
A plane is a surface such that, given any two points on the surface, the surface also contains the straight line that passes through the two points. One can introduce a Cartesian coordinate system on a given plane in order to label every point on it uniquely with two numbers, the point's coordinates.
Within any Euclidean space, 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
- two different lines which intersect
- two different lines which are parallel
Planes embedded in R3
This section is specifically concerned with planes embedded in three dimensions: specifically, in R3.
In three-dimensional space, we may exploit the following facts that do not hold in higher dimensions:
- Two planes are either parallel or they intersect in a line.
- A line is either parallel to a plane or they intersect at a single point.
- Two lines normal to the same plane must be parallel to each other.
- Two planes normal to the same line must be parallel to each other.
Point and a normal vector
In a three-dimensional ambient space, there is another important way of defining a plane:
- a point and a line, which is normal (perpendicular) to the plane
We can explicitly describe the resulting plane; let be the point we wish to lie in the plane, and let be a nonzero vector parallel to the line we wish to be normal to the plane. The desired plane is the set of all points such that
If we write , , and , then the plane is determined by the condition
where a, b, c and d could be any 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
where s and t range over all real numbers, and , and are given vectors defining the plane.
Plane through three points
The plane passing through three points , and can be determined by the following determinant equations:
This plane can also be described by the "point and a normal vector" prescription above. A suitable normal vector is given by the cross product and the point can be taken to be .
The distance from a point to a plane
For a plane and a point not necessarily lying on the plane, the distance from to the plane is
The line of intersection between two planes
Given intersecting planes described by and , the line of intersection is perpendicular to both and and thus parallel to
If we further assume that and are orthonormal then the closest point on the line of intersection to the origin is
The dihedral angle
Given two intersecting planes described by and , the dihedral angle between them is defined to be the angle between their normal directions:
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