In
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
The dot here means a
dot (scalar) product.
Expanded this becomes
which is the ''point–normal'' form of the equation of a plane. 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 , , , and are constants and , , and are not all zero, then the graph of the equation
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
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:
Method 2
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 is non-zero (so for planes not through the origin) the values for , and 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.
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
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
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
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
, meaning that ''a'', ''b'', and ''c'' are normalized, 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:
[
:
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
:
]
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 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
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 . 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
and
, the
dihedral angle between them is defined to be the angle
between their normal directions:
:
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
*