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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a plane is a Euclidean ( flat), two-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
. 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 two-dimensional Euclidean geometry. Sometimes the word ''plane'' is used more generally to describe a two-dimensional surface, for example the
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
and elliptic plane. When working exclusively in two-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the definite article is used, so ''the'' plane refers to the whole space. Many fundamental tasks in mathematics,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
,
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, and graphing are performed in a two-dimensional space, often in the plane.


Euclidean geometry

Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
set forth the first great landmark of mathematical thought, an axiomatic treatment of geometry. 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. 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''. A plane is a ruled surface.


Representation

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


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 Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
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 In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
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 be the position vector of some point , and let be a nonzero vector. The plane determined by the point and the vector consists of those points , with position vector , such that the vector drawn from to is perpendicular to . 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 such that \boldsymbol \cdot (\boldsymbol-\boldsymbol_0)=0. The dot here means a dot (scalar) product.
Expanded this becomes a (x-x_0) + b(y-y_0) + c(z-z_0) = 0, which is the ''point–normal'' form of the equation of a plane. This is just a linear equation ax + by + cz + d = 0, where d = -(ax_0 + by_0 + cz_0), which is the expanded form of - \boldsymbol \cdot \boldsymbol_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 , , , and are constants and , , and are not all zero, then the graph of the equation ax + by + cz + d = 0, is a plane having the vector as a normal. This familiar equation for a plane is called the ''general form'' of the equation of the plane. Thus for example a regression equation of the form (with ) 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 \boldsymbol = \boldsymbol_0 + s \boldsymbol + t \boldsymbol, where and range over all real numbers, and are given linearly independent vectors defining the plane, and is the vector representing the position of an arbitrary (but fixed) point on the plane. The vectors and can be visualized as vectors starting at and pointing in different directions along the plane. The vectors and can be
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
, but cannot be parallel.


Describing a plane through three points

Let , , and be non-collinear points.


Method 1

The plane passing through , , and can be described as the set of all points (''x'',''y'',''z'') that satisfy the following
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
equations: \begin 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 = \begin 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 = 0.


Method 2

To describe the plane by an equation of the form ax + by + cz + d = 0 , solve the following system of equations: ax_1 + by_1 + cz_1 + d = 0 ax_2 + by_2 + cz_2 + d = 0 ax_3 + by_3 + cz_3 + d = 0. This system can be solved using Cramer's rule and basic matrix manipulations. Let D = \begin x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end. If is non-zero (so for planes not through the origin) the values for , and can be calculated as follows: a = \frac \begin 1 & y_1 & z_1 \\ 1 & y_2 & z_2 \\ 1 & y_3 & z_3 \end b = \frac \begin x_1 & 1 & z_1 \\ x_2 & 1 & z_2 \\ x_3 & 1 & z_3 \end c = \frac \begin x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end. 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 \boldsymbol n = ( \boldsymbol p_2 - \boldsymbol p_1 ) \times ( \boldsymbol p_3 - \boldsymbol p_1 ), and the point can be taken to be any of the given points , or (or any other point in the plane).


Operations


Distance from a point to a plane

For a plane \Pi : ax + by + cz + d = 0 and a point \boldsymbol p_1 = (x_1,y_1,z_1) not necessarily lying on the plane, the shortest distance from \boldsymbol p_1 to the plane is : D = \frac. It follows that \boldsymbol p_1 lies in the plane
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
''D'' = 0. If a^2+b^2+c^2=1, meaning that ''a'', ''b'', and ''c'' are normalized, then the equation becomes : D = \left, a x_1 + b y_1 + c z_1+d \. Another vector form for the equation of a plane, known as the Hesse normal form relies on the parameter ''D''. This form is: :\boldsymbol \cdot \boldsymbol - D_0 = 0, where \boldsymbol is a unit normal vector to the plane, \boldsymbol 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 \boldsymbol \cdot (\boldsymbol - \boldsymbol_0) = 0 , where the \boldsymbol is a normal vector and \boldsymbol_0 = (x_, x_, \dots, x_) is a position vector to a point in the hyperplane. We desire the perpendicular distance to the point \boldsymbol_1 = (x_, x_, \dots, x_). The hyperplane may also be represented by the scalar equation \sum_^N a_i x_i = -a_0, for constants \. Likewise, a corresponding \boldsymbol may be represented as (a_1,a_2, \dots, a_N). We desire the scalar projection of the vector \boldsymbol_1 - \boldsymbol_0 in the direction of \boldsymbol. Noting that \boldsymbol \cdot \boldsymbol_0 = \boldsymbol_0 \cdot \boldsymbol = -a_0 (as \boldsymbol_0 satisfies the equation of the hyperplane) we have :\begin D &= \frac \\ &= \frac \\ &= \frac \\ &= \frac. \end


Line–plane intersection

In analytic geometry, the intersection of a line and a plane in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
can be the empty set, a point, or a line.


Line of intersection between two planes

The line of intersection between two planes \Pi_1 : \boldsymbol _1 \cdot \boldsymbol r = h_1 and \Pi_2 : \boldsymbol _2 \cdot \boldsymbol r = h_2 where \boldsymbol _i are normalized is given by : \boldsymbol = (c_1 \boldsymbol _1 + c_2 \boldsymbol _2) + \lambda (\boldsymbol _1 \times \boldsymbol _2) where : c_1 = \frac : c_2 = \frac. This is found by noticing that the line must be perpendicular to both plane normals, and so parallel to their cross product \boldsymbol _1 \times \boldsymbol _2 (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 \boldsymbol r = c_1\boldsymbol _1 + c_2\boldsymbol _2 + \lambda(\boldsymbol _1 \times \boldsymbol _2), 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 c_1 and c_2. If we further assume that \boldsymbol _1 and \boldsymbol _2 are
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
then the closest point on the line of intersection to the origin is \boldsymbol r_0 = h_1\boldsymbol _1 + h_2\boldsymbol _2. If that is not the case, then a more complex procedure must be used.Plane-Plane Intersection - from Wolfram MathWorld
Mathworld.wolfram.com. Retrieved 2013-08-20.


Dihedral angle

Given two intersecting planes described by \Pi_1 : a_1 x + b_1 y + c_1 z + d_1 = 0 and \Pi_2 : 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 = \frac = \frac.


Planes in various areas of mathematics

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 Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
bijections. The topological plane is the natural context for the branch of
graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
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 Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change ...
. 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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
. The latter possibility finds an application in the theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
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.)


Topological and differential geometric notions

The one-point compactification of the plane is homeomorphic to a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
(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 In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...
such diffeomorphism is conformal, but for the Euclidean plane it is not.


See also

* Face (geometry) * Flat (geometry) * Half-plane * Hyperplane *
Line–plane intersection 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 pla ...
* Plane coordinates * Plane of incidence *
Plane of rotation In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space. In three dimensions it is an alternative to the axis of rotation, but unlike the axis of rotation it can be used in other dimensions, such as ...
* Point on plane closest to origin *
Polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
* Projective plane


Notes


References

* *


External links

* *
"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. {{DEFAULTSORT:Plane (Geometry) Euclidean plane geometry Mathematical concepts *