Parametric Curve

A particle moving along a curve $C$ in three-dimensional space has $x$, $y$, $z$ coordinates as functions of time $t$:

These are the parametric equations of motion of the particle.

Parameter

$t$ is the parameter.

Trajectory

The curve traced out by the particle. Denoted by $C$ usually. Restricting $t$ to an interval $[a, b]$ gives

Orientation

The direction in which $C$ is traced as $t$ increases is the orientation of the curve.

Examples:

• The circular helix $x = a\cos t$, $y = a\sin t$, $z = bt$, $0 \le t \le 2\pi$, for fixed real $a > 0$ and $b$.

Closed Curve

A parametric curve $C$ represented by $\boldsymbol{r}(t)$, $a \le t \le b$, is closed iff $\boldsymbol{r}(a) = \boldsymbol{r}(b)$.