📂Calculus

Parametric Equation

Build-Up

Let’s consider the situation of expressing the position of a particle on a two-dimensional plane in a formula. The path of the particle’s movement is shown in the following figure.

It is impossible to represent the path in the above figure as a function of $x$, i.e., in the form of $y = f(x)$. This is because there are multiple $y$ values corresponding to points like $x_{0}$. (A function maps each point in its domain $x$ to a single $y$.) In such cases, it is natural to represent the values of $x$ and $y$ as functions of a new variable.

$$x = f(t) \qquad y = g(t)$$

Definition

Representing $n$ variables $x, y, z, \dots$ as functions of a new variable $t$ is called a parametric equation.

$$\begin{align*} x &= f(t)\\ y &= g(t)\\ z &= h(t)\\ &\ \ \vdots \end{align*}$$

At this time, the variable $t$ is called a parameter.

Example

The position of a particle moving in a circle with a radius of $r$ can be expressed by the following parametric equation.

$$x(t) = r \cos t \qquad y(t) = r \sin t$$