Cartesian coordinate system
In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes of the system. The point where the axes meet is called the origin and has as coordinates. The axes directions represent an orthogonal basis. The combination of origin and basis forms a coordinate frame called the Cartesian frame.
Similarly, the position of any point in three-dimensional space can be specified by three Cartesian coordinates, which are the signed distances from the point to three mutually perpendicular planes. More generally, Cartesian coordinates specify the point in an -dimensional Euclidean space for any dimension. These coordinates are the signed distances from the point to mutually perpendicular fixed hyperplanes.
Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of algebra and calculus. Using the Cartesian coordinate system, geometric shapes can be described by equations involving the coordinates of points of the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates and satisfy the equation ; the area, the perimeter and the tangent line at any point can be computed from this equation by using integrals and derivatives, in a way that can be applied to any curve.
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. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.
History
The adjective Cartesian refers to the French mathematician and philosopher René Descartes, who published this idea in 1637 while he was resident in the Netherlands. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat.Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes's work.
The development of the Cartesian coordinate system played a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two-coordinate description of the plane was later generalized into the concept of vector spaces.
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.
Description
One dimension
An affine line with a chosen Cartesian coordinate system is called a number line. Every point on the line has a real-number coordinate, and every real number represents some point on the line.There are two degrees of freedom in the choice of Cartesian coordinate system for a line, which can be specified by choosing two distinct points along the line and assigning them to two distinct real numbers. Other points can then be uniquely assigned to numbers by linear interpolation. Equivalently, one point can be assigned to a specific real number, for instance an origin point corresponding to zero, and an oriented length along the line can be chosen as a unit, with the orientation indicating the correspondence between directions along the line and positive or negative numbers. Each point corresponds to its signed distance from the origin.
A geometric transformation of the line can be represented by a function of a real variable, for example translation of the line corresponds to addition, and scaling the line corresponds to multiplication. Any two Cartesian coordinate systems on the line can be related to each-other by a linear function (function of the form taking a specific point's coordinate in one system to its coordinate in the other system. Choosing a coordinate system for each of two different lines establishes an affine map from one line to the other taking each point on one line to the point on the other line with the same coordinate.
Two dimensions
A Cartesian coordinate system in two dimensions is defined by an ordered pair of perpendicular lines, a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning each axis into a number line. For any point P, a line is drawn through P perpendicular to each axis, and the position where it meets the axis is interpreted as a number. The two numbers, in that chosen order, are the Cartesian coordinates of P. The reverse construction allows one to determine the point P given its coordinates.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. The coordinates are usually written as two numbers in parentheses, in that order, separated by a comma, as in. Thus the origin has coordinates, and the points on the positive half-axes, one unit away from the origin, have coordinates and.
In mathematics, physics, and engineering, the first axis is usually defined or depicted as horizontal and oriented to the right, and the second axis is vertical and oriented upwards. The origin is often labeled O, and the two coordinates are often denoted by the letters X and Y, or x and y. The axes may then be referred to as the X-axis and Y-axis. The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values.
A Euclidean plane with a chosen Cartesian coordinate system is called a . In a Cartesian plane, one can define canonical representatives of certain geometric figures, such as the unit circle, the unit square, the unit hyperbola, and so on.
The two axes divide the plane into four right angles, called quadrants. The quadrants may be named or numbered in various ways, but the quadrant where all coordinates are positive is usually called the first quadrant.
If the coordinates of a point are, then its distances from the X-axis and from the Y-axis are and, respectively; where denotes the absolute value of a number.
Three dimensions
A Cartesian coordinate system for a three-dimensional space consists of an ordered triplet of lines that go through a common point, and are pair-wise perpendicular; an orientation for each axis; and a single unit of length for all three axes. As in the two-dimensional case, each axis becomes a number line. For any point P of space, one considers a plane through P perpendicular to each coordinate axis, and interprets the point where that plane cuts the axis as a number. The Cartesian coordinates of P are those three numbers, in the chosen order. The reverse construction determines the point P given its three coordinates.Alternatively, each coordinate of a point P can be taken as the distance from P to the plane defined by the other two axes, with the sign determined by the orientation of the corresponding axis.
Each pair of axes defines a coordinate plane. These planes divide space into eight octants. The octants are:
The coordinates are usually written as three numbers surrounded by parentheses and separated by commas, as in or. Thus, the origin has coordinates, and the unit points on the three axes are,, and.
Standard names for the coordinates in the three axes are abscissa, ordinate and applicate. The coordinates are often denoted by the letters x, y, and z. The axes may then be referred to as the x-axis, y-axis, and z-axis, respectively. Then the coordinate planes can be referred to as the xy-plane, yz-plane, and xz-plane.
In mathematics, physics, and engineering contexts, the first two axes are often defined or depicted as horizontal, with the third axis pointing up. In that case the third coordinate may be called height or altitude. The orientation is usually chosen so that the 90-degree angle from the first axis to the second axis looks counter-clockwise when seen from the point ; a convention that is commonly called the right-hand rule.