Plane (mathematics)

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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.

Euclidean geometry

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.

Properties

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 {\vec  p} be the point we wish to lie in the plane, and let {\vec  n} 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 {\vec  r} such that

{\vec  n}\cdot ({\vec  r}-{\vec  p})=0.

If we write {\vec  n}=(a,b,c), {\vec  r}=(x,y,z), and {\vec  n}\cdot {\vec  p}=-d, then the plane is determined by the condition

ax+by+cz+d=0,

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

{\vec  {u}}+s{\vec  {v}}+t{\vec  {w}},

where s and t range over all real numbers, and {\vec  {u}}, {\vec  {v}} and {\vec  {w}} are given vectors defining the plane.

Plane through three points

The plane passing through three points {\vec  p}_{1}=(x_{1},y_{1},z_{1}), {\vec  p}_{2}=(x_{2},y_{2},z_{2}) and {\vec  p}_{3}=(x_{3},y_{3},z_{3}) can be determined by the following determinant equations:

{\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}}={\begin{vmatrix}x-x_{1}&y-y_{1}&z-z_{1}\\x-x_{2}&y-y_{2}&z-z_{2}\\x-x_{3}&y-y_{3}&z-z_{3}\end{vmatrix}}=0.

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 {\vec  n}=({\vec  p}_{2}-{\vec  p}_{1})\times ({\vec  p}_{3}-{\vec  p}_{1}), and the point {\vec  p} can be taken to be {\vec  p}_{1}.

The distance from a point to a plane

For a plane ax+by+cz+d=0 and a point {\vec  p}_{1}=(x_{1},y_{1},z_{1}) not necessarily lying on the plane, the distance from {\vec  p}_{1} to the plane is

D={\frac  {\left|ax_{1}+by_{1}+cz_{1}+d\right|}{{\sqrt  {a^{2}+b^{2}+c^{2}}}}}.

The line of intersection between two planes

Given intersecting planes described by {\vec  n}_{1}\cdot {\vec  r}=h_{1} and {\vec  n}_{2}\cdot {\vec  r}=h_{2}, the line of intersection is perpendicular to both {\vec  n}_{1} and {\vec  n}_{2} and thus parallel to

{\vec  n}_{1}\times {\vec  n}_{2}.

If we further assume that {\vec  n}_{1} and {\vec  n}_{2} are orthonormal then the closest point on the line of intersection to the origin is

{\vec  r}_{0}=h_{1}{\vec  n}_{1}+h_{2}{\vec  n}_{2}.

The dihedral angle

Given two intersecting planes described by 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, the dihedral angle between them is defined to be the angle \alpha between their normal directions:

\cos \alpha ={\hat  n}_{1}\cdot {\hat  n}_{2}={\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}}}}}.

See also

External links

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