Coordinate transformation

A rule that converts coordinates describing a position or quantity in one coordinate system into coordinates in another. It preserves the underlying object while changing its representation.

Change of basis: A conversion between coordinate descriptions of the same vector using different bases of a vector space. It is the linear-algebraic form of changing coordinates while keeping the vector fixed.
Coordinate system: A scheme that assigns numbers to locations or other objects using a chosen origin, axes, or parameters. A transformation compares two such schemes for describing the same objects.
Cartesian coordinate system: A coordinate system that locates points by signed distances along perpendicular axes. Conversions between Cartesian and other coordinate systems simplify problems with different symmetries.
Passive transformation: A change in the coordinate description of an object while the object itself remains fixed. It is the coordinate-change interpretation of transforming between reference systems.
Jacobian matrix: A matrix of first partial derivatives describing the local linear behavior of a differentiable map. Its entries show how small coordinate changes in one system produce changes in another.
Coordinate chart: A map assigning coordinates to points in a region of a manifold. Charts provide the local coordinate descriptions that transformations connect.
Polar coordinate system: A two-dimensional coordinate system locating points by distance from an origin and angle from an axis. Cartesian-to-polar conversion makes circular geometry easier to express.
Active transformation: A map that moves or changes an object while leaving the coordinate system fixed. It contrasts with coordinate change, where the description changes but the object does not.
Affine transformation: A map combining a linear transformation with a translation. It captures coordinate changes that include both rotation or scaling and an origin shift.
Basis (linear algebra): A linearly independent set of vectors that spans a vector space. A basis determines the component coordinates used to represent each vector.

Linked from 18 pages