# Coordinate System - Systems Commonly Used

Systems Commonly Used

Some coordinate systems are the following:

• The Cartesian coordinate system (also called the "rectangular coordinate system"), which, for two- and three-dimensional spaces, uses two (three) numbers representing distances from the origin in two (three) mutually perpendicular directions.
• Curvilinear coordinates are a generalization of coordinate systems generally; the system is based on the intersection of curves.
• Polar coordinate system represents a point in the plane by a distance from the origin and an angle measured from a reference line intersecting the origin.
• Log-polar coordinate system represents a point in the plane by the logarithm of the distance from the origin and an angle measured from a reference line intersecting the origin.
• Cylindrical coordinate system represents a point in three-space using two perpendicular axes; distance is measured along one axis, while the other axis formes the reference line for a polar coordinate representation of the remaining two components.
• Spherical coordinate system represents a point in three space by the distance from the origin and two angles measured from two reference lines which intersect the origin.
• Plücker coordinates are a way of representing lines in 3D Euclidean space using a six-tuple of numbers as homogeneous coordinates.
• Generalized coordinates are used in the Lagrangian treatment of mechanics.
• Canonical coordinates are used in the Hamiltonian treatment of mechanics.
• Parallel coordinates visualise a point in n-dimensional space as a polyline connecting points on n vertical lines.
• Barycentric coordinates (mathematics) as used for Ternary plot

There are ways of describing curves without coordinates, using intrinsic equations that use invariant quantities such as curvature and arc length. These include:

• Whewell equation relates arc length and tangential angle.
• Cesàro equation relates arc length and curvature.

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