Parametric Curves and Motion

A parameter records more than a shape

A parametric curve is $\mathbf r(t)=(x(t),y(t))$ . It records location, direction of traversal and possibly repeated visits. Eliminating $t$ can reveal the curve's shape, but often loses its orientation or the part actually traced.

Where $x'(t)\ne0$ , the chain rule gives $dy/dx=y'(t)/x'(t)$ . Differentiating once more with respect to $x$ requires another division by $x'$ :

\frac{d^2y}{dx^2}=\frac{d}{dt}\left(\frac{y'}{x'}\right)\frac1{x'}.

A vertical tangent may occur where $x'=0$ and $y'\ne0$ . If both vanish, these tests are inconclusive and the local curve needs further analysis.

Visual guide

VISUAL GUIDEA parameter gives direction as well as shape

Parametric ellipse

The ellipse x = 2 cos t, y = sin t is traversed counterclockwise as t increases. The arrow at (2, 0) points upward because the velocity there is (0, 1). Speed is not constant around this ellipse.

Worked example: tangent and curvature

For $x=t^2$ , $y=t^3$ , with $t>0$ , the slope is $3t^2/(2t)=3t/2$ . At $t=2$ the point is $(4,8)$ and the tangent is $y-8=3(x-4)$ . The second derivative is $(3/2)/(2t)=3/(4t)$ , positive on this branch.

At $t=0$ , both first derivatives vanish. Dividing by them would be invalid. The curve has a cusp there when both positive and negative parameters are included.

Worked example: distance traveled

The speed is $\|\mathbf r'(t)\|=\sqrt{x'^2+y'^2}$ , so the traveled distance is $\int_a^b\sqrt{x'^2+y'^2}dt$ . For $x=2\cos t$ , $y=2\sin t$ , speed is $2$ . From $0$ to $2\pi$ the distance is $4\pi$ , one circumference, while net displacement is zero.

From $0$ to $4\pi$ the distance doubles because the circle is traversed twice. The geometric curve still has circumference $4\pi$ . Specify whether the task asks for the length of a curve traced once or total distance traveled.

Practice

Find the slope for $x=1+2t$ , $y=t^2$ at $t=3$ .
Find the distance for $x=3t$ , $y=4t$ , $0\le t\le2$ .
At which parameters on the unit circle are tangents vertical?

Show worked solutions

$dy/dx=2t/2=t$ , giving slope $3$ .
Speed is $5$ , so distance is $10$ .
$x'=-\sin t=0$ and $y'=\cos t\ne0$ : $t=k\pi$ for integer $k$ .