Cartesian Coordinate System

A Cartesian Coordinate System is a coordinate system based on fixed perpendicular directed lines.

• AKA: Cartesian Coordinates, Rectangular Coordinates
• Context:
  • It can range from being a Two-Dimensional Cartesian Coordinate System, to being a Three-Dimensional Cartesian Coordinate System, to being a Two-Dimensional Cartesian Coordinate System.
    • In the two-dimensional Cartesian coordinates, conventionally, the x-axis corresponds to the axis in horizontal direction going left to right and y-axis to the vertical axis going up. Both are axis are linear and perpendicular to each other. A 2-dimensional vector in Cartesian coordinates is represented as [math]\displaystyle{ (x,y) }[/math]
    • The three-dimensional Cartesian coordinates is an extension of the two-dimensional one formed by the addition of a third axis mutually perpendicular to the x- and y-axes defined above. This new axis is conventionally referred to as the z-axis. A 3-dimensional vector in Cartesian coordinates is represented as [math]\displaystyle{ (x,y,z) }[/math]
    • In the n-dimensional Cartesian coordinates each coordinate specify a point in an n-dimensional Euclidean space, where the [math]\displaystyle{ n }[/math] dimensions are mutually perpendicular hyperplanes.
  • …
• Example(s):
  • Three-Dimensional Space.
• Counter-Example(s):
  • Polar Coordinate System.
• See: Euclidean Space, Computational Geometry, Point (Geometry), Plane (Geometry), Perpendicular, Unit Length, Origin (Mathematics), Orthogonal Projection, Space (Mathematics).

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References

• http://mathworld.wolfram.com/CartesianCoordinates.html

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Cartesian_coordinate_system Retrieved:2015-11-7.
  • 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. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

    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). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

    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. 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. 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 x and y satisfy the equation .

    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.

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