AP Calculus BC glossary

Orientation of a parametric curve

Also called: Direction of a parametric curve

The orientation of a parametric curve is the direction the curve is traced as the parameter increases. It belongs to the parametrisation rather than to the shape, since two parametrisations can cover the same set of points in opposite directions.

Mark it with arrows on the sketch. Plotting a few increasing values of $t$ and joining them in order shows which way the point travels, and the signs of $\frac{dx}{dt}$ and $\frac{dy}{dt}$ say the same thing instant by instant: $\frac{dx}{dt}>0$ means the point is moving right just then.

Compare $x=\cos t,\ y=\sin t$ with $x=\cos t,\ y=-\sin t$ on $0 \le t \le 2\pi$ . Both trace the unit circle and both satisfy $x^2+y^2=1$ , but the first runs anticlockwise from $(1,0)$ and the second runs clockwise. The Cartesian equation cannot tell the two apart.

The mistake

Eliminating the parameter and then answering a question about motion. The Cartesian equation keeps a curve that contains the path, usually more of it than the parametrisation actually traces, and throws away the direction, the speed and the starting point. Eliminating $t$ from $x=\cos t,\ y=\sin t$ on $0 \le t \le \pi$ gives the whole unit circle when the path is only the top half, so the domain restriction has to be carried across by hand, and anything about which way the particle moves has to come from $x(t)$ and $y(t)$ or from the signs of their derivatives.

Appears in: Unit 9: Parametric, Polar, and Vector (BC)