Plane (geometry)

planeplanarplanesthe planeEuclidean planegeometric planeplane geometryflatinfinite planeplane equation
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.wikipedia
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Dimension

dimensionsdimensionalone-dimensional
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere.

Two-dimensional space

Euclidean planetwo-dimensional2D
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.
Two-dimensional space can be seen as a projection of the physical universe onto a plane.

Surface (topology)

surfaceclosed surfacesurfaces
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far.
In mathematics, a surface is a geometrical shape that resembles a deformed plane.

Three-dimensional space

three-dimensional3Dthree dimensions
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.
In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane).

Cartesian coordinate system

Cartesian coordinatesCartesian coordinateCartesian
In this way the Euclidean plane is not quite the same as the Cartesian plane.
A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length.

Line (geometry)

linestraight linelines
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.
In two dimensions, i.e., the Euclidean plane, two lines which do not intersect are called parallel.

Ruled surface

doubly ruled surfaceruledconical bilinear complexes
A plane is a ruled surface.
Examples include the plane, the curved surface of a cylinder or cone, a conical surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space.

Parallel (geometry)

parallelparallel linesparallelism
In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.

Point (geometry)

pointpointslocation
A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space.
Similar constructions exist that define the plane, line segment and other related concepts.

Normal (geometry)

normalnormal vectorsurface normal
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". Let the hyperplane have equation, where the \mathbf{n} is a normal vector and is a position vector to a point in the hyperplane.
The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles).

Perpendicular

perpendicularlyPerpendicularitynormal
can be perpendicular, but cannot be parallel.
A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects.

Hyperplane

hyperplanesaffine hyperplanehyper-plane
Let the hyperplane have equation, where the \mathbf{n} is a normal vector and is a position vector to a point in the hyperplane.
If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines.

Geometry

geometricgeometricalgeometries
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.
These include the concepts of point, line, plane, distance, angle, surface, and curve, as well as the more advanced notions of topology and manifold.

Cartesian product

productCartesian squareCartesian power
This section is solely concerned with planes embedded in three dimensions: specifically, in R 3.
An example is the 2-dimensional plane R 2 = R × R where R is the set of real numbers: R 2 is the set of all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).

Planar graph

planarplanar graphsmaximal planar graph
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.
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints.

Hesse normal form

Another vector form for the equation of a plane, known as the Hesse normal form relies on the parameter D.
The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in or a plane in Euclidean space or a hyperplane in higher dimensions.

Manifold

manifoldsboundarymanifold with boundary
Differential geometry views a plane as a 2-dimensional real manifold, a topological plane which is provided with a differential structure.
Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.

Euclidean vector

vectorvectorsvector addition
are given linearly independent vectors defining the plane, and
On a two-dimensional diagram, sometimes a vector perpendicular to the plane of the diagram is desired.

Curvature

curvednegative curvatureextrinsic curvature
In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have.
Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.

Stereographic projection

stereographicLittle planet effectstereographically projected
The plane may be given a spherical geometry by using the stereographic projection. 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.
In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

Cross product

vector cross productvector productcross-product
A suitable normal vector is given by the cross product
In computational geometry of the plane, the cross product is used to determine the sign of the acute angle defined by three points and.

Topology

topologicaltopologicallytopologist
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.
Examples include the plane, the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane, which cannot (that is, all their realizations are surfaces that are not manifolds).

Linear equation

linearlinear equationsslope-intercept form
This is just a linear equation
In the case of three variable, this hyperplane is a plane.

Determinant

determinantsdetmatrix determinant
can be described as the set of all points (x,y,z) that satisfy the following determinant equations:
For example, given two linearly independent vectors v 1, v 2 in R 3, a third vector v 3 lies in the plane spanned by the former two vectors exactly if the determinant of the 3 × 3 matrix consisting of the three vectors is zero.

Sphere

sphericalhemisphereglobose
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.
This is analogous to the situation in the plane, where the terms "circle" and "disk" can also be confounded.