Parametric Equations

Introduction to Parametric Equations

Parametric equations are a set of equations that express the coordinates of the points that make up a geometric object such as a curve or surface, in terms of one or more independent parameters.
This can be helpful when it’s easier or more practical to define the x and y coordinates separately, for the same value of the parameter (commonly denoted as t).
By varying t, we get different points on the curve, which, when joined together, give the required curve.

A function of x can be expressed in parametric form if both x and y are expressed in terms of another parameter, usually denoted as t.
Any point (x, y) on the graph of a function f can be expressed as (x, f(x)).
If x can be expressed as g(t), then the point (x, f(x)) becomes (g(t), f(g(t))), making f(g(t)) the parametric form of f(x).

To eliminate the parameter, express one variable in terms of the other, assuming that it can be done using algebraic manipulation.
This forms a Cartesian equation, which can be plotted without reference to t.
Although the Cartesian equation gives you the whole locus, it loses the information about the direction of the parameter’s movement.

To differentiate y with respect to x in parametric form, differentiate both x and y with respect to t separately, then divide dy/dt by dx/dt to get dy/dx.
Chain rule is essentially used here.

Parametric equations are prevalently used in physics, engineering, and computer graphics to represent trajectories of moving objects or to model the shape and movement of surfaces in 3D space.
For these applications, it’s more convenient to express the object’s coordinates x and y in terms of time t or another suitable parameter.

Parametric equations can be used to find the equation of the tangent or normal to a curve at a particular point.
The coordinates of the point and the gradient of the curve at that point (found by differentiating) are used to find the equation of the tangent or normal.