Cartesian coordinate system

Cartesian coordinatesCartesian coordinateCartesianCartesian planeaxesx-axisy-axisaxisvertical axiscoordinate axes
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.wikipedia
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Real coordinate space

standard topologyreal spaceR''' 3
In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n.
on Cartesian plane of ordered pairs

Coordinate system

coordinatescoordinateaxis
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.
The prototypical example of a coordinate system is the Cartesian coordinate system.

Plane (geometry)

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

Euclidean space

EuclideanspaceEuclidean vector space
In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n.
An isomorphism from a Euclidean space to \mathbb R^n associates with each point an n-tuple of real numbers, which locate them in the Euclidean space and are called Cartesian coordinates.

René Descartes

DescartesCartesianRene Descartes
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
Descartes' influence in mathematics is equally apparent; the Cartesian coordinate system was named after him.

Number

number systemnumericalnumbers
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.
As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as René Descartes did with negative solutions in a Cartesian coordinate system.

Analytic geometry

analytical geometryCartesian geometrycoordinate geometry
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions.

Equation

equationsmathematical equationunknown
Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape.
When R is chosen to have the value of 2 (R = 2), this equation would be recognized, when sketched in Cartesian coordinates, as the equation for a particular circle with a radius of 2.

Graph of a function

graphgraphsgraphing
A familiar example is the concept of the graph of a function.
are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and thus form a subset of this plane, which is a curve in the case of a continuous function.

Origin (mathematics)

origincoordinate originzero point
Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0).
In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.

Polar coordinate system

polar coordinatespolarpolar coordinate
Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.
The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis.

Vector space

vectorvector spacesvectors
The two-coordinate description of the plane was later generalized into the concept of vector spaces.
The first example above reduces to this one if the arrows are represented by the pair of Cartesian coordinates of their end points.

Unit hyperbola

hyperbolaasymptotes to the unit hyperbolaone hyperbola
In a Cartesian plane one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0, 0) and (1, 1)), the unit hyperbola, and so on.
In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length

Hyperplane

hyperplanesaffine hyperplanehyper-plane
These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
In Cartesian coordinates, such a hyperplane can be described with a single linear equation of the following form (where at least one of the a_i's is non-zero and b is an arbitrary constant):

Two-dimensional space

Euclidean planetwo-dimensional2D
A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.
Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

Dimension

dimensionsdimensionalone-dimensional
In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n.
(See Space and Cartesian coordinate system.)

Euclidean geometry

plane geometryEuclideanEuclidean plane geometry
The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
In this approach, a point on a plane is represented by its Cartesian (x, y) coordinates, a line is represented by its equation, and so on.

Unit circle

circleBase circle (mathematics)base-circle
In a Cartesian plane one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0, 0) and (1, 1)), the unit hyperbola, and so on.
Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane.

Three-dimensional space

three-dimensional3Dthree dimensions
One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).
Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × F is, for F composed of [F x, F y, F z ]:

Sign (mathematics)

positivenon-negativesign
These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
On the Cartesian plane, the rightward and upward directions are usually thought of as positive, with rightward being the positive x-direction, and upward being the positive y-direction.

Abscissa and ordinate

abscissaordinateaxis
The first and second coordinates are called the abscissa and the ordinate of P, respectively; and the point where the axes meet is called the origin of the coordinate system.
Usually these are the horizontal and vertical coordinates of a point in a two-dimensional rectangular Cartesian coordinate system.

Unit square

squaresquare with sides of one unit of lengthsquare with side length 1
In a Cartesian plane one can define canonical representatives of certain geometric figures, such as the unit circle (with radius equal to the length unit, and center at the origin), the unit square (whose diagonal has endpoints at (0, 0) and (1, 1)), the unit hyperbola, and so on.
. Often, "the" unit square refers specifically to the square in the Cartesian plane with corners at the four points

Linear algebra

linearlinear algebraiclinear-algebraic
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more.
There is a strong relationship between linear algebra and geometry, which started with the introduction by René Descartes, in 1637, of Cartesian coordinates.

Calculus

infinitesimal calculusdifferential and integral calculusclassical calculus
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more.
The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis.

Spherical coordinate system

spherical coordinatessphericalspherical polar coordinates
Many other coordinate systems have been developed since Descartes, such as the polar coordinates for the plane, and the spherical and cylindrical coordinates for three-dimensional space.
Just as the two-dimensional Cartesian coordinate system is useful on the plane, a two-dimensional spherical coordinate system is useful on the surface of a sphere.