In mathematics, a **plane** is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.

- Euclidean geometry
- Representation
- Determination by contained points and lines
- Properties
- Point–normal form and general form of the equation of a plane
- Describing a plane with a point and two vectors lying on it
- Describing a plane through three points
- Operations
- Distance from a point to a plane
- Line–plane intersection
- Line of intersection between two planes
- Planes in various areas of mathematics
- Topological and differential geometric notions
- See also
- Notes
- References
- External links

When working exclusively in two-dimensional Euclidean space, the definite article is used, so *the* plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional space, or, in other words, in the plane.

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. In this way the Euclidean plane is not quite the same as the Cartesian plane.

A plane is a ruled surface.

This section is solely concerned with planes embedded in three dimensions: specifically, in **R**^{3}.

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

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.

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 **r**_{0} be the position vector of some point *P*_{0} = (*x*_{0}, *y*_{0}, *z*_{0}), and let * n* = (

The dot here means a dot (scalar) product.

Expanded this becomes

which is the *point–normal* form of the equation of a plane.^{ [3] } This is just a linear equation

where

- ,

which is the expanded form of

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

is a plane having the vector * n* = (

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.

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, * v* and

Let **p**_{1}=(x_{1}, y_{1}, z_{1}), **p**_{2}=(x_{2}, y_{2}, z_{2}), and **p**_{3}=(x_{3}, y_{3}, z_{3}) be non-collinear points.

The plane passing through **p**_{1}, **p**_{2}, and **p**_{3} can be described as the set of all points (x,y,z) that satisfy the following determinant equations:

To describe the plane by an equation of the form , solve the following system of equations:

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

- .

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

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.

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 **r**_{0} can be taken to be any of the given points **p**_{1}, **p**_{2} or **p**_{3}^{ [6] } (or any other point in the plane).

For a plane and a point not necessarily lying on the plane, the shortest distance from to the plane is

It follows that lies in the plane if and only if *D=0*.

If meaning that *a*, *b*, and *c* are normalized^{ [7] } then the equation becomes

Another vector form for the equation of a plane, known as the Hesse normal form relies on the parameter *D*. This form is:^{ [5] }

where is a unit normal vector to the plane, a position vector of a point of the plane and *D*_{0} the distance of the plane from the origin.

The general formula for higher dimensions can be quickly arrived at using vector notation. Let the hyperplane have equation , where the is a normal vector and is a position vector to a point in the hyperplane. We desire the perpendicular distance to the point . The hyperplane may also be represented by the scalar equation , for constants . Likewise, a corresponding may be represented as . We desire the scalar projection of the vector in the direction of . Noting that (as satisfies the equation of the hyperplane) we have

- .

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 has been suggested that this section be split out into another articletitled Plane–plane intersection . (Discuss) (July 2020) |

The line of intersection between two planes and where are normalized is given by

where

This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product (this cross product is zero if and only if the planes are parallel, and are therefore non-intersecting or entirely coincident).

The remainder of the expression is arrived at by finding an arbitrary point on the line. To do so, consider that any point in space may be written as , since is a basis. We wish to find a point which is on both planes (i.e. on their intersection), so insert this equation into each of the equations of the planes to get two simultaneous equations which can be solved for and .

If we further assume that and are orthonormal then the closest point on the line of intersection to the origin is . If that is not the case, then a more complex procedure must be used.^{ [8] }

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

In addition to its familiar geometric structure, with isomorphisms that are isometries with respect to the usual inner product, the plane may be viewed at various other levels of abstraction. Each level of abstraction corresponds to a specific category.

At one extreme, all geometrical and metric concepts may be dropped to leave the topological plane, which may be thought of as an idealized homotopically trivial infinite rubber sheet, which retains a notion of proximity, but has no distances. The topological plane has a concept of a linear path, but no concept of a straight line. The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous bijections. The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem.

The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps. From this viewpoint there are no distances, but collinearity and ratios of distances on any line are preserved.

Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure. Again in this case, there is no notion of distance, but there is now a concept of smoothness of maps, for example a differentiable or smooth path (depending on the type of differential structure applied). The isomorphisms in this case are bijections with the chosen degree of differentiability.

In the opposite direction of abstraction, we may apply a compatible field structure to the geometric plane, giving rise to the complex plane and the major area of complex analysis. The complex field has only two isomorphisms that leave the real line fixed, the identity and conjugation.

In the same way as in the real case, the plane may also be viewed as the simplest, one-dimensional (over the complex numbers) complex manifold, sometimes called the complex line. However, this viewpoint contrasts sharply with the case of the plane as a 2-dimensional real manifold. The isomorphisms are all conformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation.

In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere on the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point). This is one of the projections that may be used in making a flat map of part of the Earth's surface. The resulting geometry has constant positive curvature.

Alternatively, the plane can also be given a metric which gives it constant negative curvature giving the hyperbolic plane. The latter possibility finds an application in the theory of special relativity in the simplified case where there are two spatial dimensions and one time dimension. (The hyperbolic plane is a timelike hypersurface in three-dimensional Minkowski space.)

The one-point compactification of the plane is homeomorphic to a sphere (see stereographic projection); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is a manifold referred to as the Riemann sphere or the complex projective line. The projection from the Euclidean plane to a sphere without a point is a diffeomorphism and even a conformal map.

The plane itself is homeomorphic (and diffeomorphic) to an open disk. For the hyperbolic plane such diffeomorphism is conformal, but for the Euclidean plane it is not.

- ↑ Eves 1963 , p. 19
- ↑ Joyce, D.E. (1996),
*Euclid's Elements, Book I, Definition 7*, Clark University, retrieved 8 August 2009 - ↑ Anton 1994 , p. 155
- ↑ Anton 1994 , p. 156
- 1 2 Weisstein, Eric W. (2009), "Plane",
*MathWorld--A Wolfram Web Resource*, retrieved 8 August 2009 - ↑ Dawkins, Paul, "Equations of Planes",
*Calculus III* - ↑ To normalize arbitrary coefficients, divide each of
*a*,*b*,*c*and*d*by (which can not be 0). The "new" coefficients are now normalized and the following formula is valid for the "new" coefficients. - ↑ Plane-Plane Intersection - from Wolfram MathWorld. Mathworld.wolfram.com. Retrieved on 2013-08-20.

In classical mathematics, **analytic geometry**, also known as **coordinate geometry** or **Cartesian geometry**, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.

In vector calculus, the **curl** is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.

In mathematics, a **linear equation** is an equation that may be put in the form

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In physics, the **Navier–Stokes equations** are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

In mathematics, **curvature** is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

In physics, **angular velocity**, also known as **angular frequency vector**, is a vector measure of rotation rate, that refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time.

In mathematics, the **cross product** or **vector product** is a binary operation on two vectors in three-dimensional space , and is denoted by the symbol . Given two linearly independent vectors **a** and **b**, the cross product, **a** × **b**, is a vector that is perpendicular to both **a** and **b**, and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.

The **moment of inertia**, otherwise known as the **mass moment of inertia**, **angular mass** or **rotational inertia**, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation.

In geometry, a **normal** is an object such as a line, ray, or vector that is perpendicular to a given object. For example, in two dimensions, the **normal line** to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

The **stream function** is defined for incompressible (divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional **Lagrange stream function** was introduced by Joseph Louis Lagrange in 1781. The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes.

In mathematics, specifically linear algebra, the **Cauchy–Binet formula**, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes. It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with the entries from any commutative ring.

In geometry, a **barycentric coordinate system** is a coordinate system in which the location of a point is specified by reference to a simplex. The **barycentric coordinates** of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex.

**Three-dimensional space** is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

In geometry, the **trilinear coordinates***x:y:z* of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio *x:y* is the ratio of the perpendicular distances from the point to the sides opposite vertices *A* and *B* respectively; the ratio *y:z* is the ratio of the perpendicular distances from the point to the sidelines opposite vertices *B* and *C* respectively; and likewise for *z:x* and vertices *C* and *A*.

In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or a line. Distinguishing these cases and finding the intersection point have use, for example, in computer graphics, motion planning, and collision detection.

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 or the nearest point on the plane.

Geometric algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a **function of several real variables** or **real multivariate function** is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article.

- Anton, Howard (1994),
*Elementary Linear Algebra*(7th ed.), John Wiley & Sons, ISBN 0-471-58742-7 - Eves, Howard (1963),
*A Survey of Geometry*,**I**, Boston: Allyn and Bacon, Inc.

Wikimedia Commons has media related to . Euclidean planes |

- "Plane",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Weisstein, Eric W. "Plane".
*MathWorld*. - "Easing the Difficulty of Arithmetic and Planar Geometry" is an Arabic manuscript, from the 15th century, that serves as a tutorial about plane geometry and arithmetic.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.