Cartesian coordinate system
lastmod 2024-03-07created 2023-01-095 edits
The Cartesian coordinate system is one of the coordinate systems that represent a Euclidean space (or, more generally, an inner product space) of arbitrary dimension. It is named after René Descartes, the French philosopher and mathematician who invented it after being inspired by a fly moving around on the ceiling. The two-dimensional Cartesian coordinate system is also called the coordinate plane (座標平面), and the three-dimensional Cartesian coordinate system the coordinate space (座標空間). Orthogonal coordinate system (直交座標系) is a general term for all coordinate systems whose unit vectors parallel to the coordinate axes are always perpendicular to each other, including the Cartesian coordinate system as well as polar, cylindrical, and spherical coordinate systems. Unlike other coordinate systems such as the polar coordinate system, the Cartesian coordinate system easily generalizes to any dimension. The Cartesian coordinate system is useful when the object it represents has translational symmetry, but it does not easily represent other kinds of symmetry such as rotational symmetry. In general, given a basis and an origin in a Euclidean space, a Cartesian coordinate system can be defined from them.
In the most common cases, the coordinate plane and the coordinate space, Cartesian coordinates are usually written with the Latin letters x, y, z. In four dimensions, w or (when dealing with spacetime in physics) t is used. For arbitrary dimensions, they are written with subscripts in the form xn.
The coordinate plane in particular is significant in that a unified rule of information applies, allowing information about sets, functions, polynomials, and matrices to be represented in a single space.
Moreover, the information in the Cartesian coordinate system can be extended not only to higher-dimensional Cartesian coordinate systems but also to information in other coordinate systems, which makes it even more significant, and it is still in use today.
Coordinate plane
Example of a coordinate plane The coordinate plane used today is defined by the mutually orthogonal x-axis (horizontal) and y-axis (vertical) that form the xy-plane. The point where the x-axis and y-axis meet is called the origin.
Korean: 사분면
The quadrants (四分面) of the coordinate plane are the four parts of the coordinate plane divided by the x-axis and y-axis. They are divided into the first through fourth quadrants.
First quadrant (+,+), second quadrant (-,+), third quadrant (-,-), fourth quadrant (+,-)
See Quadrant (plane geometry) for details.
Conversion between Korean: 극좌표 and Korean: 데카르트 좌표
See Polar coordinate system#Conversion between polar and Cartesian coordinates.