parametrising

Parametrising is a mathematical process used to express a set of equations or a geometric object using a smaller set of variables, known as parameters. Instead of describing a curve or surface directly in terms of its coordinates, one introduces one or more independent parameters that can vary to generate all the points on the object. For instance, a circle in the Cartesian plane can be parametrised using an angle parameter. As the angle varies from 0 to 2π, the parametric equations trace out the entire circle.

This technique is widely applicable in various fields, including calculus, computer graphics, engineering, and physics. In calculus, parametrising curves allows for easier calculation of derivatives and integrals, simplifying complex geometric manipulations. In computer graphics, it is fundamental for defining and rendering shapes, animations, and simulations. Parametrisation provides a flexible and efficient way to represent and manipulate mathematical structures that might otherwise be difficult to handle. The choice of parameters can significantly influence the ease of computation and the interpretation of results. Often, the goal is to find a parametrisation that is smooth and covers the object exactly once.