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Published on Saturday, April 26, 1997 by Gideon Ariel

Two-dimensional coordinate system

The modern Cartesian coordinate system in two dimensions (also called a rectangular coordinate system) is commonly defined by two axes, at right angles to each other, forming a plane (an xy-plane). The horizontal axis is labeled x, and the vertical axis is labeled y. In a three-dimensional coordinate system, another axis, normally labeled z, is added, providing a sense of a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). All the points in a Cartesian coordinate system taken together form a so-called Cartesian plane.

The point of intersection, where the axes meet, is called the origin normally labeled O. With the origin labeled O, we can name the x-axis Ox and the y-axis Oy. The x and y axes define a plane that can be referred to as the xy-plane. Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid. To specify a particular point on a two dimensional coordinate system, you indicate the x unit first (abscissa), followed by the y unit (ordinate) in the form (x,y), an ordered pair. In three dimensions, a third z unit (applicate) is added, (x,y,z).

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.

An example of a point P on the system is indicated in the picture below using the coordinate (5,2).

In the two-dimensional Cartesian coordinate system, a point P in the xy-plane is represent by a tuple of two components (x,y).

  • x is the signed distance from the y-axis to the point P, and
  • y is the signed distance from the x-axis to the point P.

In the three-dimensional Cartesian coordinate system, a point P in the xyz-space is represent by a tuple of three components (x,y,z).

  • x is the signed distance from the yz-plane to the point P,
  • y is the signed distance from the xz-plane to the point P, and
  • z is the signed distance from the xy-plane to the point P.

The arrows on the axes indicate that they extend forever in the same direction (i.e. infinitely). The intersection of the two x-y axes creates four quadrants indicated by the Roman numerals I, II, III, and IV. Conventionally, the quadrants are labeled counter-clockwise starting from the northeast quadrant. In Quadrant I the values are (x,y), and II:(-x,y), III:(-x,-y) and IV:(x,-y).

Quadrant x values y values
I > 0 > 0
II < 0 > 0
III < 0 < 0
IV > 0 < 0

See also

2D Two-dimensional coordinate system
3D Three-dimensional coordinate system
Angle Definition of an angle
Axis Definition of Cartesian axis
Cartesian geometry What is Cartesian geometry?
Coordinate system Definition of coordinates
Curve Definition of a curve
Distance Definition of distance
Euclidean geometry What is Euclidean geometry?
Geometry Definition of geometry
Length Definition of length
Line Definition of a line
Origin Definition of origin in a Cartesian coordinate system
Perspective projection Definition of perspective projection
Planar homography Definition of planar homography
Plane Definition of a plane
Point Definition of a point
Point (kinematics) Definition of a point (kinematics)
Projective geometry What is projective geometry?
Segment (kinematics) Definition of a segment (kinematics)
Vanishing points Definition of vanishing points and vanishing lines in perspective projection
Vector Definition of a vector

Reference: /wizard/manual/concepts/geometry.2d.html
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