Parametric Equations

Table of Contents

Parametric equations offer a different way to describe curves in the plane. Instead of writing $y$ directly as a function of $x$ (like $y = f(x)$), we describe both $x$ and $y$ in terms of a third variable, usually called a parameter and denoted by $t$.

A parametric curve in the plane is given by a pair of equations
$$
x = x(t), \quad y = y(t),
$$
together with an interval of parameter values, such as $a \le t \le b$ or $t$ in all real numbers.

Here:

$t$ is the parameter;
as $t$ varies, the point $(x(t), y(t))$ moves along a curve in the plane.

You can think of $t$ as “time” and $(x(t), y(t))$ as the position of a moving point at time $t$.

Basic examples

Straight line (parametric form)
Consider the line given by the usual (Cartesian) equation
$$
y = 2x + 1.
$$
One parametric description is
$$
x(t) = t, \quad y(t) = 2t + 1, \quad -\infty < t < \infty.
$$
As $t$ runs through all real numbers, the point $(t, 2t+1)$ runs along the entire line.

We can change the parameterization without changing the geometric line. For example,
$$
x(t) = 3t, \quad y(t) = 6t + 1
$$
traces the same line, but with a different “speed” along it.

Circle (parametric form)
The circle with center at the origin and radius $r$ has Cartesian equation
$$
x^2 + y^2 = r^2.
$$
A standard parametric representation is
$$
x(t) = r \cos t,\quad y(t) = r \sin t,\quad 0 \le t \le 2\pi.
$$
As $t$ increases from
Parabola (parameter not equal to $x$)
The parabola $y = x^2$ can be parametrized in many ways. The simplest is
$$
x(t) = t,\quad y(t) = t^2,
$$
which just renames the variable. A different parametrization is
$$
x(t) = t^2,\quad y(t) = t^4.
$$
If you eliminate $t$ (by setting $t^2 = x$ and then $t^4 = x^2$), you get $y = x^2$ again. So both pairs of equations describe the same geometric curve.

Orientation and direction

A parametric curve is not just a set of points; it also carries a direction of travel as $t$ increases. This is called the orientation.

For the circle
$$
x(t) = r \cos t,\quad y(t) = r \sin t,\quad 0 \le t \le 2\pi,
$$
the point start at $(r, 0)$ when $t = 0$, moves through the upper half of the circle as $t$ goes from $0$ to $\pi$, and returns to $(r, 0)$ at $t = 2\pi$. The motion is counterclockwise.

If we instead use
$$
x(t) = r \cos t,\quad y(t) = -r \sin t,\quad 0 \le t \le 2\pi,
$$
the same circle is traced, but now the motion goes clockwise. The geometric curve is the same, but the orientation is different.

Interval of the parameter

The interval of $t$ matters: it controls how much of the curve is drawn.

For the parabola parametrized by
$$
x(t) = t,\quad y(t) = t^2,
$$
with $-\infty < t < \infty$, you get the whole parabola.
If you restrict to $t \ge 0$, then you only get the right half of the parabola (where $x \ge 0$).
If you restrict to $0 \le t \le 1$, you only get the piece from $(0,0)$ to $(1,1)$.

Thus, when working with parametric equations, you must always keep track of both:

the formulas $x(t)$ and $y(t)$, and
the allowed range of $t$.

Eliminating the parameter

Sometimes it is useful to convert from a parametric description to a Cartesian one by eliminating the parameter $t$.

The general idea is:

Start with $x = x(t)$ and $y = y(t)$.
Solve one of these for $t$ and substitute into the other, or use an identity relating $x(t)$ and $y(t)$ directly.
Obtain a relation between $x$ and $y$ only.

Examples:

Line
Suppose
$$
x(t) = 2t + 1,\quad y(t) = -3t + 4.
$$
Solve the first equation for $t$:
$$
t = \frac{x - 1}{2}.
$$
Substitute into $y$:
$$
y = -3\left(\frac{x - 1}{2}\right) + 4
= -\frac{3}{2}x + \frac{3}{2} + 4
= -\frac{3}{2}x + \frac{11}{2}.
$$
So the corresponding Cartesian equation is
$$
y = -\frac{3}{2}x + \frac{11}{2}.
$$
Circle
For
$$
x(t) = r\cos t,\quad y(t) = r\sin t,
$$
use the identity $\cos^2 t + \sin^2 t = 1$:
$$
\left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2
= \cos^2 t + \sin^2 t = 1.
$$
Thus
$$
x^2 + y^2 = r^2.
$$
This is the familiar equation of a circle.

Eliminating the parameter gives you the shape of the curve, but it usually loses the orientation and information about how many times or in what manner the curve is traced.

Parametric description of motion

A common use of parametric equations is to describe the position of a moving object in the plane. For instance, suppose a particle moves so that
$$
x(t) = 3t,\quad y(t) = 2t + 1,\quad t \ge 0.
$$
At time $t$ the particle is at $(3t, 2t+1)$. As $t$ increases, the particle travels along a straight line with constant “horizontal speed” $3$ units per unit time and “vertical speed” $2$ units per unit time.

More complicated motions are also described parametrically. For example,
$$
x(t) = \cos t,\quad y(t) = \sin(2t),
$$
traces a curve that loops back and forth. It is not possible to describe this easily as a single formula $y = f(x)$.

Graphing parametric curves

To sketch or understand a parametric curve, a practical approach is:

Make a table of values
Choose several $t$ values in the allowed interval and compute $(x(t), y(t))$.
Plot the points
Plot each computed point on the Cartesian coordinate plane.
Indicate direction
Show the direction of motion by adding arrows along the curve in the order of increasing $t$.
Connect smoothly
Connect the points in the order of $t$ with a smooth curve that reflects the behavior of the functions $x(t)$ and $y(t)$ (for example, whether they are increasing or decreasing).

For simple parametric curves, you can predict the general shape by knowing how the basic functions like $\sin t$, $\cos t$, and polynomials behave.

Non-function curves

One advantage of parametric equations is that they can describe curves that fail the usual “vertical line test” for functions of $x$. For example, a circle cannot be written as a single function $y = f(x)$, because for many $x$-values there are two $y$-values. But the circle has a neat parametric description using sine and cosine.

Parametric equations are therefore very useful any time you want to describe:

closed shapes such as circles and ellipses,
looping or self-intersecting curves,
or motions where $x$ and $y$ depend naturally on some other quantity (often time) instead of on each other directly.