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 $\mathbb{R}^3$.