Plane Curves and Space Curves

Plane Curves and Space Curves

We are about to look at the definitions of some types of curves, but before we do, we must first look at the definition of a smooth curve.

 Definition: If $C$ is a curve given by the vector-valued function $\vec{r}(t)$ on an interval $I$, then $C$ is said to be Smooth if: a) $\vec{r'}(t)$ is continuous on $I$. b) $\vec{r'}(t) \neq \vec{0}$ except possibly at the endpoints of $I$.

Geometrically, a curve $C$ is smooth if it does not any sharp points, kinks, or cusps. The following image represents a smooth curve in contrast with a curve that is not smooth. Definition: Let $\vec{r}(t)$ be a vector-valued function. Then for the interval $I$, if $\vec{r}(t)$ is continuous on $I$ and the curve $C$ traced by $\vec{r}(t)$ can lie on a single plane, then $\vec{r}(t)$ is called a Plane Curve.

We have already dealt with tons of plane curves. For example, the curve $f(x) = x^3$ is a plane curve because the graph of $f$ lies on the $xy$-plane. All lines in $\mathbb{R}^3$ are plane curves as well. Another plane curve could be defined by the vector equation $\vec{r}(t) = (1, t, t^2)$ which represents the graph of the parabola $y = z^2$ onto the plane $x = 1$ as depicted below: If a curve is not a plane curve, then it will be what is called a space curve.

 Definition: Let $\vec{r}(t)$ be a vector-valued function. Then for the interval $I$, if $\vec{r}(t)$ is continuous on $I$ then the curve $C$ traced by the parametric equations $x = x(t)$, $y = y(t)$, and $z = z(t)$ is called a Space Curve.

We will primarily be looking at space curves in $\mathbb{R}^3$.