In mathematics, a rotation of axes in two dimensions is a mapping from an ''xy''-

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured ...

to an ''x′y′''-Cartesian coordinate system in which the

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

is kept fixed and the ''x′'' and ''y′'' axes are obtained by rotating the ''x'' and ''y'' axes counterclockwise through an angle

\theta

. A point ''P'' has coordinates (''x'', ''y'') with respect to the original system and coordinates (''x′'', ''y′'') with respect to the new system. In the new coordinate system, the point ''P'' will appear to have been rotated in the opposite direction, that is, clockwise through the angle

\theta

. A rotation of axes in more than two dimensions is defined similarly. A rotation of axes is a

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

and a

rigid transformation In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformatio ...

Motivation

Coordinate systems are essential for studying the equations of

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s using the methods of analytic geometry. To use the method of coordinate geometry, the axes are placed at a convenient position with respect to the curve under consideration. For example, to study the equations of ellipses and