In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the origin of a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
is a special
point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space.
In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of
geometric symmetry.
Cartesian coordinates
In a
Cartesian coordinate system
In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, the origin is the point where the
axes of the system intersect.
[.] The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical
coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three.
Other coordinate systems
In a
polar coordinate system
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
*the point's distance from a reference point called the ''pole'', and
*the point's direction from ...
, the origin may also be called the pole. It does not itself have well-defined polar coordinates, because the polar coordinates of a point include the angle made by the positive ''x''-axis and the ray from the origin to the point, and this ray is not well-defined for the origin itself.
In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, the origin may be chosen freely as any convenient point of reference.
The origin of the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
can be referred as the point where
real axis and
imaginary axis intersect each other. In other words, it is the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
zero.
[.]
See also
*
Coordinate frame
*
Distance from a point to a plane
*
Null vector, an analogous point of a vector space
*
Pointed space, a topological space with a distinguished point
*
Radial basis function, a function depending only on the distance from the origin
References
{{DEFAULTSORT:Origin (Mathematics)
Elementary mathematics