Plane equation in normal form

In Euclidean geometry, a plane is a flat two-dimensional surface that extends indefinitely. Planes often arise as subspaces of three-dimensional space, as with one of a room's walls, infinitely extended.

Background

Euclid set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry.^[1] He selected a small core of undefined terms (called common notions) and postulates (or axioms) which he then used to prove various geometrical statements. Although the plane in its modern sense is not directly given a definition anywhere in the Elements, it may be thought of as part of the common notions.^[2] Euclid never used numbers to measure length, angle, or area. The Euclidean plane equipped with a chosen Cartesian coordinate system is called a Cartesian plane; a non-Cartesian Euclidean plane equipped with a polar coordinate system would be called a polar plane.

Three parallel planes.

A plane is a ruled surface.

Representation

This section is solely concerned with planes embedded in three dimensions: specifically, in R³.

Determination by contained points and lines

In a Euclidean space of any number of dimensions, a plane is uniquely determined by any of the following:

• Three non-collinear points (points not on a single line).
• A line and a point not on that line.
• Two distinct but intersecting lines.
• Two distinct but parallel lines.

Properties

The following statements hold in three-dimensional Euclidean space but not in higher dimensions, though they have higher-dimensional analogues:

• Two distinct planes are either parallel or they intersect in a line.
• A line is either parallel to a plane, intersects it at a single point, or is contained in the plane.
• Two distinct lines perpendicular to the same plane must be parallel to each other.
• Two distinct planes perpendicular to the same line must be parallel to each other.

Point–normal form and general form of the equation of a plane

In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination".

Specifically, let r₀ be the position vector of some point P₀ = (x₀, y₀, z₀), and let n = (a, b, c) be a nonzero vector. The plane determined by the point P₀ and the vector n consists of those points P, with position vector r, such that the vector drawn from P₀ to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the desired plane can be described as the set of all points r such that

$${\displaystyle {\boldsymbol {n))\cdot ({\boldsymbol {r))-{\boldsymbol {r))_{0})=0.}$$

$${\displaystyle a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0,}$$

$${\displaystyle ax+by+cz+d=0,}$$

$${\displaystyle d=-(ax_{0}+by_{0}+cz_{0}),}$$

${\displaystyle -{\boldsymbol {n))\cdot {\boldsymbol {r))_{0}.}$

In mathematics it is a common convention to express the normal as a unit vector, but the above argument holds for a normal vector of any non-zero length.

Conversely, it is easily shown that if a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation

$${\displaystyle ax+by+cz+d=0,}$$

Thus for example a regression equation of the form y = d + ax + cz (with b = −1) establishes a best-fit plane in three-dimensional space when there are two explanatory variables.

Describing a plane with a point and two vectors lying on it

Alternatively, a plane may be described parametrically as the set of all points of the form

$${\displaystyle {\boldsymbol {r))={\boldsymbol {r))_{0}+s{\boldsymbol {v))+t{\boldsymbol {w)),}$$

Vector description of a plane

where s and t range over all real numbers, v and w are given linearly independent vectors defining the plane, and r₀ is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors v and w can be visualized as vectors starting at r₀ and pointing in different directions along the plane. The vectors v and w can be perpendicular, but cannot be parallel.

Describing a plane through three points

Let p₁ = (x₁, y₁, z₁), p₂ = (x₂, y₂, z₂), and p₃ = (x₃, y₃, z₃) be non-collinear points.

Method 1

The plane passing through p₁, p₂, and p₃ can be described as the set of all points (x,y,z) that satisfy the following determinant equations:

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

Method 2

To describe the plane by an equation of the form ${\displaystyle ax+by+cz+d=0}$, solve the following system of equations:

$${\displaystyle ax_{1}+by_{1}+cz_{1}+d=0}$$

$${\displaystyle ax_{2}+by_{2}+cz_{2}+d=0}$$

$${\displaystyle ax_{3}+by_{3}+cz_{3}+d=0.}$$

This system can be solved using Cramer's rule and basic matrix manipulations. Let

$${\displaystyle D={\begin{vmatrix}x_{1}&y_{1}&z_{1}\\x_{2}&y_{2}&z_{2}\\x_{3}&y_{3}&z_{3}\end{vmatrix)).}$$

If D is non-zero (so for planes not through the origin) the values for a, b and c can be calculated as follows:

$${\displaystyle a={\frac {-d}{D)){\begin{vmatrix}1&y_{1}&z_{1}\\1&y_{2}&z_{2}\\1&y_{3}&z_{3}\end{vmatrix))}$$

$${\displaystyle b={\frac {-d}{D)){\begin{vmatrix}x_{1}&1&z_{1}\\x_{2}&1&z_{2}\\x_{3}&1&z_{3}\end{vmatrix))}$$

$${\displaystyle c={\frac {-d}{D)){\begin{vmatrix}x_{1}&y_{1}&1\\x_{2}&y_{2}&1\\x_{3}&y_{3}&1\end{vmatrix)).}$$

These equations are parametric in d. Setting d equal to any non-zero number and substituting it into these equations will yield one solution set.

Method 3

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

$${\displaystyle {\boldsymbol {n))=({\boldsymbol {p))_{2}-{\boldsymbol {p))_{1})\times ({\boldsymbol {p))_{3}-{\boldsymbol {p))_{1}),}$$

Operations

Distance from a point to a plane

In Euclidean space, the distance from a point to a plane is the distance between a given point and its orthogonal projection on the plane, the perpendicular distance to the nearest point on the plane.

It can be found starting with a change of variables that moves the origin to coincide with the given point then finding the point on the shifted plane ${\displaystyle ax+by+cz=d}$ that is closest to the origin. The resulting point has Cartesian coordinates ${\displaystyle (x,y,z)}$:

${\displaystyle \displaystyle x={\frac {ad}{a^{2}+b^{2}+c^{2))},\quad \quad \displaystyle y={\frac {bd}{a^{2}+b^{2}+c^{2))},\quad \quad \displaystyle z={\frac {cd}{a^{2}+b^{2}+c^{2))))$.

The distance between the origin and the point ${\displaystyle (x,y,z)}$ is ${\displaystyle {\sqrt {x^{2}+y^{2}+z^{2))))$.

Line–plane intersection

The three possible plane-line relationships in three dimensions. (Shown in each case is only a portion of the plane, which extends infinitely far.)

In analytic geometry, the intersection of a line and a plane in three-dimensional space can be the empty set, a point, or a line. It is the entire line if that line is embedded in the plane, and is the empty set if the line is parallel to the plane but outside it. Otherwise, the line cuts through the plane at a single point.

Distinguishing these cases, and determining equations for the point and line in the latter cases, have use in computer graphics, motion planning, and collision detection.

Line of intersection between two planes

Two intersecting planes in three-dimensional space

In analytic geometry, the intersection of two planes in three-dimensional space is a line.

Sphere–plane intersection