Parametric Curves

A graph $y = f(x)$ can't loop back or pass the vertical line test twice. Parametric equations describe a curve by giving both coordinates as functions of a parameter $t$:

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

Think of $t$ as time and $(x(t), y(t))$ as the position of a moving point. As $t$ increases, the point traces the curve in a direction called its orientation.

Examples

Curve	Parametric equations
line through $(x_0, y_0)$ in direction $(a, b)$	$x = x_0 + at$, $\ y = y_0 + bt$
circle of radius $r$	$x = r\cos t$, $\ y = r\sin t$, $\ 0 \le t \le 2\pi$
ellipse	$x = a\cos t$, $\ y = b\sin t$
graph of $y = f(x)$	$x = t$, $\ y = f(t)$

Eliminating the parameter

To see which curve it is, remove $t$:

• Solve for $t$ in one equation and substitute. $x = t + 1$, $y = t^2$ gives $t = x - 1$, so $y = (x - 1)^2$.
• Use an identity. $x = 3\cos t$, $y = 3\sin t$: $x^2 + y^2 = 9(\cos^2 t + \sin^2 t) = 9$, a circle.

Watch the domain: $x = e^t$, $y = e^{2t}$ gives $y = x^2$, but only the part with $x > 0$, because $e^t$ is always positive.

One curve, many parametrizations

$x = \cos t$, $y = \sin t$ and $x = \cos 2t$, $y = \sin 2t$ trace the same circle, the second one twice as fast. $x = \cos t$, $y = -\sin t$ traces it clockwise. Parametric equations carry more information than the curve alone: when and which way.

Projectiles

A ball launched at speed $v_0$ and angle $\alpha$ (ignoring air resistance, $g = 9.8$ m/s²):

$$x = (v_0\cos\alpha)\,t, \qquad y = (v_0\sin\alpha)\,t - 4.9t^2$$

Horizontal motion is steady; vertical motion is free fall (06.01).