Parameterisations

Parameterisation, in mathematics and related fields, is the representation of a geometric object, physical quantity, or mathematical relation using one or more parameters. A parameterisation expresses coordinates or states as functions of a parameter t (or a pair such as (u, v)), so that varying the parameter traces the object or describes a family of objects.

Common examples illustrate the idea. A circle can be parameterised by (x, y) = (cos t, sin t), with t typically in [0, 2π). A line through a point r0 in the direction v can be written as r(t) = r0 + t v. An ellipse can be expressed as (x, y) = (a cos t, b sin t). Surfaces use two parameters, for instance a sphere can be parameterised by (x, y, z) = (sin φ cos θ, sin φ sin θ, cos φ) with φ ∈ (0, π) and θ ∈ (0, 2π). In general, a surface is described by a map r(u, v) from a region in the plane to three-space.

Regularity and charts are important concepts. A parameterisation is regular if its partial derivatives are linearly independent, ensuring a meaningful local description. In geometry and analysis, parameterisations form the basis of coordinate systems and charts on manifolds, enabling calculations such as line and surface integrals, and changes of variables in integration.

Applications span computer graphics, where Bézier and B-spline curves are parametric, to finite element methods that use parametric elements. Parameterisations also underpin model-building in physics and engineering and facilitate the study of families of objects by varying the parameters.

Limitations include potential singularities, where the parameterisation fails to be regular, or the inability to cover an entire curved object with a single global parameterisation, necessitating multiple charts. The term derives from Greek roots meaning “to measure beside.”