Coordinate system

3D Spherical

Number-line

Cylindrical Coordinates

Parabolic coordinates 3D

Coordinate system is a system that uses one or more numbers or coordinates to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

Types of Coordinate Systems

The most common coordinate systems are the two-dimensional Cartesian coordinate system and the three-dimensional Cartesian coordinate system, which are used to define the position of a point in space. However, there are many other types of coordinate systems that are used in mathematical and scientific applications, including:

• Polar coordinate system: A two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
• Cylindrical coordinate system: A three-dimensional coordinate system that extends polar coordinates by adding a height element.
• Spherical coordinate system: A three-dimensional coordinate system where each point is determined by a distance from a fixed point, an angle from a fixed direction, and an angle from a fixed plane.
• Homogeneous coordinate system: Used in projective geometry, allowing for the representation of points at infinity by finite coordinates.
• Curvilinear coordinates: Coordinates defined by a smoothly varying set of curves or surfaces, used in complex shapes and fields.

Applications

Coordinate systems are used in a wide range of fields for various applications, including:

• In physics, to describe the motion of particles, the shape and size of fields, and the relationship between different objects in space.
• In engineering, for designing and constructing structures, machines, and systems.
• In astronomy, to locate celestial objects in the sky.
• In geography and geodesy, for mapping the surface of the Earth and other celestial bodies.
• In computer graphics and computer-aided design (CAD), for modeling and rendering objects and scenes.

History

The concept of coordinate systems can be traced back to ancient civilizations, but the modern Cartesian coordinate system was developed by René Descartes in the 17th century, which laid the foundation for analytic geometry and has profoundly influenced the development of modern mathematics and physics.

See Also

• Vector space
• Linear algebra
• Geometry
• Analytic geometry

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