Parametric Equations
Throw a ball across a field and photograph it every tenth of a second. Where is it? You can't
answer with a single function
y = f(x), because the ball's position isn't really controlled by
how far across it has gone — it is controlled by time. At each moment
t the ball has an across-position and an up-position, and
each of them follows its own rule:
x = 15\,t, \qquad y = 20\,t - 5\,t^2.
Feed in a time t and out drop two numbers — an ordered
pair (x, y) — which you plot as a single dot on the
coordinate plane.
Let t run and the dot sweeps out the whole flight path. That third,
behind-the-scenes variable is called the parameter, and a pair of rules like
this is a set of parametric equations.
A curve can be described by giving x and y
each as a function of a third variable, the parameter (usually
t):
x = f(t), \qquad y = g(t).
- As t increases, the point (x(t),\, y(t))
moves along the curve, so the curve gains a direction of travel, its
orientation.
- Because x is free to double back, parametric form can draw
circles, loops and spirals — curves a single y = f(x) can never
manage.
A graph y = f(x) is a slave to its input: for every
x there is at most one y (that is the
vertical-line test).
So it can never draw a full circle — a vertical line through it hits top and
bottom, two y's for one x.
Parametric form has no such rule. Both x and
y answer to t, not to each other, so the
point is free to swing left again, cross its own path, or loop the loop. That is exactly how a
computer draws the arc of a thrown ball, a spiralling firework, the figure-eight of a stunt
plane — and the cycloid, the elegant arch traced by a single chalk-mark on a rolling
bicycle wheel (x = t - \sin t,\ y = 1 - \cos t). One parameter, and
the whole moving world opens up.
See it move: t drives the point
Here is the workhorse example, x = a\cos t and
y = b\sin t. Drag the t slider and watch a single
dot walk around the curve, dragging the traced arc (bright) behind it out of the full path
(faint). The dashed spoke from the origin is the radius; the little arrow shows the
direction of travel. Notice the point starts at t = 0
on the right and sweeps anticlockwise. Switch between the circle
(a = b) and the ellipse (a \ne b).
Read a few samples straight off the moving label: at t = 0 you are
at (a, 0); at t = \tfrac{\pi}{2} at the
top (0, b); at t = \pi on the left; and
by t = 2\pi you are home again, one full lap done.
Worked example 1: a circle, and eliminating the parameter
Take the parametrisation with a = b = 1, over one full turn:
x = \cos t, \qquad y = \sin t, \qquad 0 \le t < 2\pi.
What single Cartesian curve is this? To find out we eliminate the parameter —
get rid of t and find a direct relation between
x and y. Sometimes you solve one equation
for t and substitute; here it is even cleaner to lean on a known
identity. The Pythagorean identity says \cos^2 t + \sin^2 t = 1 for
every t, so substitute x and
y straight in:
x^2 + y^2 = \cos^2 t + \sin^2 t = 1.
So the point always lies on x^2 + y^2 = 1 — the
unit circle, centre the origin, radius 1. And
because t ran over the full range
0 \le t < 2\pi, the point traces the whole circle, once,
anticlockwise.
Worked example 2: a parabola, by substitution
Not every pair needs a trig identity — often the plain "solve and substitute" route is
easiest. Consider:
x = t + 1, \qquad y = t^2.
The first equation is easy to solve for the parameter: t = x - 1.
Now substitute that into the second wherever a t appears:
y = t^2 = (x - 1)^2.
The Cartesian equation is y = (x - 1)^2 — an ordinary upward
parabola, shifted one step right. This one does pass the vertical-line test, so a
graph
y = f(x) could have drawn it too. But the parametric version tells
you something the graph can't: as t climbs from negative to
positive, the point sweeps in from the upper-left, dips through the vertex at
t = 0 (the point (1, 0)), and climbs out
to the upper-right. Same curve, but now with a direction.
A small toolkit of standard parametrisations
A handful of parametric forms come up again and again — worth recognising on sight:
- Line through (x_0, y_0) in direction
(a, b):
x = x_0 + a\,t,\ \ y = y_0 + b\,t. At
t = 0 you sit at the start point; each extra unit of
t steps you along by (a, b).
- Circle of radius r, centre origin:
x = r\cos t,\ \ y = r\sin t. (Eliminate:
x^2 + y^2 = r^2.)
- Ellipse with half-widths a and
b: x = a\cos t,\ \ y = b\sin t.
(Eliminate: \tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1.)
- Parabola — the easiest of all, just let
t be x:
x = t,\ \ y = t^2.
In fact every ordinary graph y = f(x) is secretly
parametric — set x = t and y = f(t).
Parametric form is the bigger, roomier idea; the plain graph is just the special case where the
parameter happens to be the horizontal position.
Turning parametric equations into a single Cartesian equation is handy, but the Cartesian
equation is forgetful — it can throw away two things the parametric form knew:
- The direction (orientation). Both
x = \cos t,\ y = \sin t (anticlockwise) and
x = \cos t,\ y = -\sin t (clockwise) eliminate to the very same
x^2 + y^2 = 1. The equation cannot tell you which way round you
were going.
- The extent. If t only runs over
part of its range — say 0 \le t \le \tfrac{\pi}{2} — the
point traces just the top-right quarter-circle, an arc, not the whole shape.
But x^2 + y^2 = 1 describes the entire circle. The bare equation
has silently drawn in the missing three-quarters.
The fix is a habit: always state the range of the parameter, and note the
direction of travel, alongside the eliminated equation. The Cartesian curve tells you
where; only the parameter tells you which part and which way.
See it explained