parametrizations

Parametrizations are representations of geometric objects as the image of a map from a parameter domain into Euclidean space. A parameterization consists of a parameter space, often an open subset U of R^m, and a parameterization function gamma: U -> R^n. The image gamma(U) is the subset described by the parameterization. The parameter t (or vector t in R^m) serves as a flexible coordinate to traverse the object.

For curves (m = 1), a parameterization gives coordinate functions x(t), y(t), possibly z(t) in higher dimensions. For surfaces and higher-dimensional objects, gamma maps from a higher-dimensional parameter domain, such as U subset of R^2, into R^3 (or higher). A parameterization is regular or an immersion if its differential has full rank at points in U, ensuring the mapped image locally looks like the parameter domain.

Reparameterization refers to changing the parameter, t -> phi(s) with phi differentiable and often invertible, yielding another parameterization of the same geometric set. Different parameterizations may travel at different speeds but describe the same shape.

Common examples include the circle gamma(t) = (cos t, sin t) for t in [0, 2π), and the sphere gamma(θ, φ) = (sin φ cos θ, sin φ sin θ, cos φ) with appropriate ranges for θ and φ. In algebraic geometry, rational parameterizations express curves or surfaces via rational functions in parameters, and a curve is rational if it admits a birational parameterization from a simple parameter space.

In differential geometry, parameterizations underpin coordinate charts and atlases on manifolds, enabling local Euclidean descriptions and calculations of geometric quantities such as arc length and surface area.