Parametric Equations

Parametric equations are a way of representing the coordinates of a point in a plane or space using one or more independent parameters. These parameters, usually denoted as t, describe the position of the point as a function of the parameter. Parametric equations are particularly useful in situations where the relationship between the coordinates is not easily expressible as a single equation, or when the coordinates vary with respect to some external factor, like time.

In the two-dimensional Cartesian coordinate system, a set of parametric equations for a curve can be written as x = f(t) and y = g(t), where f(t) and g(t) are continuous functions of the parameter t. The curve traced by the point (x, y) as t varies over a specified interval is called the parametric curve.

Parametric equations are used extensively in various fields, including physics, engineering, and computer graphics. They are particularly well-suited for representing curves and surfaces that are difficult to describe using Cartesian coordinates alone, such as ellipses, circles, and spirals. Parametric equations can also be used to describe the motion of an object in two or three dimensions, where the parameter t represents time.

Parametric equations can be converted to Cartesian equations by eliminating the parameter t, or vice versa, by introducing a new parameter. Methods such as arc length and tangent vectors can be used to analyze and study parametric curves.