Parametrisation

Parametrisation is the representation of a geometric object by using parameters. It is implemented as a map from a parameter domain into a space, assigning to each parameter value a point of the object. The parameters describe the object in a compact, constructive way and enable calculations such as distances, areas, and integrals.

Common examples include plane curves described by x = f(t), y = g(t); space curves given by r(t) = (x(t), y(t), z(t)); and surfaces described by X(u, v) = (x(u, v), y(u, v), z(u, v)). In each case the parameter domain is typically an interval in the real line or a region in the plane, from which points of the object are traced.

Reparametrisation means changing the parameter using a smooth invertible transformation. An arclength parametrisation uses the length along the curve as the parameter, often yielding unit speed. Such parametrisations simplify certain computations and provide a natural notion of progress along the object.

Regularity and singularities are important considerations. A parametrisation is regular when its partial derivatives span the appropriate dimension (for a curve, one direction; for a surface, two). If the Jacobian drops rank, degeneracies or singular points can occur, complicating analysis and computation. Parametrisations may be local, valid in a neighbourhood, or global, covering larger regions.

Parametrisations are contrasted with implicit representations and are foundational in calculus, geometry, and computer graphics. They enable explicit construction of curves and surfaces, facilitate integration and rendering, and relate to the study of manifolds in differential geometry. In statistics, a parameterisation describes a family of distributions by a parameter vector.