Coordinates turn geometry into ordered lists of numbers
A coordinate system pins down each point of a space with one or more numbers, turning geometry problems into arithmetic and back again, the foundation of analytic geometry. The simplest is the number line; the Cartesian plane measures from two lines at right angles, and polar coordinates use a distance and an angle.
On a number line, an origin is chosen and each point gets its signed distance from it, positive on one side and negative on the other, so every real number marks exactly one point. The Cartesian plane measures signed distances to two perpendicular lines, and in three dimensions to three mutually perpendicular planes, which can be arranged as a right-handed or left-handed system depending on axis order and direction. The same idea extends to n coordinates in n-dimensional space. Coordinates are ordered, not interchangeable, and while they are usually real numbers they can be complex or come from more abstract rings.
Polar coordinates fix a pole and a ray called the polar axis, then locate a point by an angle measured counterclockwise and a signed distance along that direction. Each pair names one point, but a point has many names: adding a full turn to the angle, or flipping the sign of r while adding half a turn, lands in the same place, and the pole itself is zero with any angle. Cylindrical coordinates bolt a height z onto the polar pair, and spherical coordinates replace r and z with a distance and a second angle.
Homogeneous coordinates describe a plane point with three numbers whose ratios x/z and y/z give its Cartesian position. The redundant third number pays off by covering the whole projective plane without appeal to infinity. Plücker coordinates use six such numbers to fix a line in space, and coordinates can locate planes, circles or spheres as well as points.
Many other systems serve special tasks. Log-polar coordinates use the logarithm of distance, barycentric and trilinear systems handle triangles and ternary plots, and mechanics uses generalized coordinates for Lagrangian and canonical ones for Hamiltonian treatments. Curves can even be described without coordinates, as in the Whewell equation linking arc length to tangent angle and the Cesàro equation linking it to curvature.
Source: Coordinate system
