Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Download
Faster access than browser!
 

Plane (geometry)

+ Save concept

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. [1]

86 relations: Abstraction (mathematics), Affine space, Alexandroff extension, Axiom, Basis (linear algebra), Bijection, Cartesian coordinate system, Cartesian product, Category (mathematics), Collinearity, Complex analysis, Complex conjugate, Complex manifold, Complex number, Complex plane, Conformal map, Continuous function, Cramer's rule, Cross product, Curvature, Determinant, Diffeomorphism, Differentiable function, Differential geometry, Differential structure, Dihedral angle, Dimension, Disk (mathematics), Distance from a point to a plane, Dot product, Euclid, Euclidean geometry, Euclidean space, Euclidean vector, Face (geometry), Flat (geometry), Four color theorem, Geometry, Graph of a function, Graph theory, Half-space (geometry), Hesse normal form, Homotopy, Hyperbolic geometry, Hyperplane, Hypersurface, If and only if, Isometry, Isomorphism, Line (geometry), ..., Linear equation, Linear independence, Line–plane intersection, Low-dimensional topology, Manifold, Mathematics, Metric (mathematics), Minkowski space, Normal (geometry), Orthonormality, Parallel (geometry), Perpendicular, Planar graph, Plane of incidence, Plane of rotation, Point (geometry), Position (vector), Projective line, Projective plane, Regression analysis, Riemann sphere, Ruled surface, Scalar projection, Skew lines, Smoothness, Spacetime, Special relativity, Sphere, Spherical geometry, Stereographic projection, Surface (topology), Three-dimensional space, Topology, Trigonometry, Two-dimensional space, Vector notation. Expand index (36 more) »

Abstraction (mathematics)

Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.

New!!: Plane (geometry) and Abstraction (mathematics) · See more »

Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

New!!: Plane (geometry) and Affine space · See more »

Alexandroff extension

In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.

New!!: Plane (geometry) and Alexandroff extension · See more »

Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

New!!: Plane (geometry) and Axiom · See more »

Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

New!!: Plane (geometry) and Basis (linear algebra) · See more »

Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

New!!: Plane (geometry) and Bijection · See more »

Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

New!!: Plane (geometry) and Cartesian coordinate system · See more »

Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

New!!: Plane (geometry) and Cartesian product · See more »

Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

New!!: Plane (geometry) and Category (mathematics) · See more »

Collinearity

In geometry, collinearity of a set of points is the property of their lying on a single line.

New!!: Plane (geometry) and Collinearity · See more »

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

New!!: Plane (geometry) and Complex analysis · See more »

Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

New!!: Plane (geometry) and Complex conjugate · See more »

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

New!!: Plane (geometry) and Complex manifold · See more »

Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

New!!: Plane (geometry) and Complex number · See more »

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.

New!!: Plane (geometry) and Complex plane · See more »

Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

New!!: Plane (geometry) and Conformal map · See more »

Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

New!!: Plane (geometry) and Continuous function · See more »

Cramer's rule

In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.

New!!: Plane (geometry) and Cramer's rule · See more »

Cross product

In mathematics and vector algebra, the cross product or vector product (occasionally directed area product to emphasize the geometric significance) is a binary operation on two vectors in three-dimensional space \left(\mathbb^3\right) and is denoted by the symbol \times.

New!!: Plane (geometry) and Cross product · See more »

Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

New!!: Plane (geometry) and Curvature · See more »

Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

New!!: Plane (geometry) and Determinant · See more »

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

New!!: Plane (geometry) and Diffeomorphism · See more »

Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

New!!: Plane (geometry) and Differentiable function · See more »

Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

New!!: Plane (geometry) and Differential geometry · See more »

Differential structure

In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.

New!!: Plane (geometry) and Differential structure · See more »

Dihedral angle

A dihedral angle is the angle between two intersecting planes.

New!!: Plane (geometry) and Dihedral angle · See more »

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.

New!!: Plane (geometry) and Dimension · See more »

Disk (mathematics)

In geometry, a disk (also spelled disc).

New!!: Plane (geometry) and Disk (mathematics) · See more »

Distance from a point to a plane

In Euclidean space, the point on a plane ax + by + cz.

New!!: Plane (geometry) and Distance from a point to a plane · See more »

Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

New!!: Plane (geometry) and Dot product · See more »

Euclid

Euclid (Εὐκλείδης Eukleidēs; fl. 300 BC), sometimes given the name Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry".

New!!: Plane (geometry) and Euclid · See more »

Euclidean geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.

New!!: Plane (geometry) and Euclidean geometry · See more »

Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

New!!: Plane (geometry) and Euclidean space · See more »

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

New!!: Plane (geometry) and Euclidean vector · See more »

Face (geometry)

In solid geometry, a face is a flat (planar) surface that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by flat faces is a polyhedron.

New!!: Plane (geometry) and Face (geometry) · See more »

Flat (geometry)

In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension.

New!!: Plane (geometry) and Flat (geometry) · See more »

Four color theorem

In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

New!!: Plane (geometry) and Four color theorem · See more »

Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

New!!: Plane (geometry) and Geometry · See more »

Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

New!!: Plane (geometry) and Graph of a function · See more »

Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

New!!: Plane (geometry) and Graph theory · See more »

Half-space (geometry)

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.

New!!: Plane (geometry) and Half-space (geometry) · See more »

Hesse normal form

The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line in \mathbb^2 or a plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.

New!!: Plane (geometry) and Hesse normal form · See more »

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

New!!: Plane (geometry) and Homotopy · See more »

Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

New!!: Plane (geometry) and Hyperbolic geometry · See more »

Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

New!!: Plane (geometry) and Hyperplane · See more »

Hypersurface

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.

New!!: Plane (geometry) and Hypersurface · See more »

If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

New!!: Plane (geometry) and If and only if · See more »

Isometry

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

New!!: Plane (geometry) and Isometry · See more »

Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

New!!: Plane (geometry) and Isomorphism · See more »

Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

New!!: Plane (geometry) and Line (geometry) · See more »

Linear equation

In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.

New!!: Plane (geometry) and Linear equation · See more »

Linear independence

In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.

New!!: Plane (geometry) and Linear independence · See more »

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.

New!!: Plane (geometry) and Line–plane intersection · See more »

Low-dimensional topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions.

New!!: Plane (geometry) and Low-dimensional topology · See more »

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

New!!: Plane (geometry) and Manifold · See more »

Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

New!!: Plane (geometry) and Mathematics · See more »

Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

New!!: Plane (geometry) and Metric (mathematics) · See more »

Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

New!!: Plane (geometry) and Minkowski space · See more »

Normal (geometry)

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object.

New!!: Plane (geometry) and Normal (geometry) · See more »

Orthonormality

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.

New!!: Plane (geometry) and Orthonormality · See more »

Parallel (geometry)

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.

New!!: Plane (geometry) and Parallel (geometry) · See more »

Perpendicular

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).

New!!: Plane (geometry) and Perpendicular · See more »

Planar graph

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.

New!!: Plane (geometry) and Planar graph · See more »

Plane of incidence

In describing reflection and refraction in optics, the plane of incidence (also called the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation.

New!!: Plane (geometry) and Plane of incidence · See more »

Plane of rotation

In geometry, a plane of rotation is an abstract object used to describe or visualize rotations in space.

New!!: Plane (geometry) and Plane of rotation · See more »

Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space.

New!!: Plane (geometry) and Point (geometry) · See more »

Position (vector)

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight-line from O to P. The term "position vector" is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus.

New!!: Plane (geometry) and Position (vector) · See more »

Projective line

In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.

New!!: Plane (geometry) and Projective line · See more »

Projective plane

In mathematics, a projective plane is a geometric structure that extends the concept of a plane.

New!!: Plane (geometry) and Projective plane · See more »

Regression analysis

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships among variables.

New!!: Plane (geometry) and Regression analysis · See more »

Riemann sphere

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.

New!!: Plane (geometry) and Riemann sphere · See more »

Ruled surface

In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. 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.

New!!: Plane (geometry) and Ruled surface · See more »

Scalar projection

In mathematics, the scalar projection of a vector \mathbf on (or onto) a vector \mathbf, also known as the scalar resolute of \mathbf in the direction of \mathbf, is given by: where the operator \cdot denotes a dot product, \hat is the unit vector in the direction of \mathbf, |\mathbf| is the length of \mathbf, and \theta is the angle between \mathbf and \mathbf.

New!!: Plane (geometry) and Scalar projection · See more »

Skew lines

In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel.

New!!: Plane (geometry) and Skew lines · See more »

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

New!!: Plane (geometry) and Smoothness · See more »

Spacetime

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

New!!: Plane (geometry) and Spacetime · See more »

Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

New!!: Plane (geometry) and Special relativity · See more »

Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

New!!: Plane (geometry) and Sphere · See more »

Spherical geometry

Spherical geometry is the geometry of the two-dimensional surface of a sphere.

New!!: Plane (geometry) and Spherical geometry · See more »

Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

New!!: Plane (geometry) and Stereographic projection · See more »

Surface (topology)

In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space.

New!!: Plane (geometry) and Surface (topology) · See more »

Three-dimensional space

Three-dimensional space (also: 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).

New!!: Plane (geometry) and Three-dimensional space · See more »

Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

New!!: Plane (geometry) and Topology · See more »

Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.

New!!: Plane (geometry) and Trigonometry · See more »

Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

New!!: Plane (geometry) and Two-dimensional space · See more »

Vector notation

Vector notation is a commonly used mathematical notation for working with mathematical vectors, which may be geometric vectors or members of vector spaces.

New!!: Plane (geometry) and Vector notation · See more »

Redirects here:

2D plane, Infinite Plane, Infinite plane, Intersection of two planes, Mathematical plane, Plane (Mathematics), Plane (mathematics), Plane (physics), Plane equation, The plane.

References

[1] https://en.wikipedia.org/wiki/Plane_(geometry)

OutgoingIncoming
Hey! We are on Facebook now! »