parameterizations

Parameterizations are representations of objects or processes using parameters. In geometry, a parameterization of a curve is a pair of functions x(t) and y(t) that map a parameter t from a domain to coordinates on the curve. A surface can be parameterized by three coordinate functions x(u,v), y(u,v), and z(u,v). The map from the parameter domain to the object is called a parameterization, and a reparameterization is a change of the parameterization via a suitable function.

Parameterizations are used to describe shapes, perform calculations, and enable sampling, integration, and rendering. A parameterization is regular (immersed) when its Jacobian has full rank, which ensures local invertibility. Special classes include conformal parameterizations, which preserve angles, and global parameterizations, which cover the whole object, possibly using several pieces.

Common examples include the unit circle parameterized by t -> (cos t, sin t), the ellipse t -> (a cos t, b sin t), and the torus t, s -> ((R + r cos s) cos t, (R + r cos s) sin t, r sin s). Parameterizations are not unique; different choices of parameters or reparameterizations describe the same object.

In other disciplines, parameterization refers to how a model is expressed in terms of unknown quantities. In statistics and machine learning, choosing a parameterization can affect numerical stability and optimization. In computer graphics, meshes are parameterized to assign texture coordinates (UV mapping). In differential geometry, a chart provides a local parameterization of a manifold, and an atlas combines charts to cover the whole space.