Sketching Space Curves in Three-Dimensional Space

On the Vector-Valued Functions page, we looked at some examples of sketching a space curve $C$ traced by a vector valued function $\vec{r}(t) = (x(t), y(t), z(t))$ that was continuous for all $t$ in the interval $I$. We will now look at some more examples of sketching space curves.

Example 1

Describe and sketch the space curve $C$ traced out by the vector valued function $\vec{r}(t) = \cos t \vec{i} + \cos t \vec{j} + \sin t \vec{k}$ for $-\infty < t < \infty$

Let's first rewrite $\vec{r}(t) = (\cos t, \cos t, \sin t)$ for convenience. First notice that $[x(t)]^2 + [z(t)]^2 = 1$. Therefore $C$ lies on the cylinder $x^2 + z^2 = 1$. Also, notice that $[y(t)]^2 + [z(t)]^2 = 1$, and so $C$ also lies on the the circular cylinder $y^2 + z^2 = 1$. Furthermore, we notice that $x(t) = y(t) = \cos t$, and so our curve is also on the plane $y = x$. The following animation represents $C$ (in red) alongside the two cylinders and the plane $C$ lies on.

Example 2

Describe and sketch the space curve $C$ traced out by the vector valued function $\vec{r}(t) = (1, t, t^2)$ for $t > 0$

We notice that for $t > 0$, $x(t), y(t), z(t) > 0$ and so $C$ lies entirely in the first octant. We also know that $\vec{r}(t)$ starts at $(1, 0, 0)$ since $\vec{r}(0) = (1, 0, 0)$.

Now further, we know that $\vec{r}(t)$ projects a parabola onto the $yz$ plane,. Also, $\vec{r}(t)$ projects a straight, half line onto the $xy$ and $xz$ planes. The following animation represents $C$ (in purple).

Example 3

Describe and sketch the space curve $C$ traced out by the vector valued function $\vec{r}(t) = (t, 2, \cos t)$ for $-\infty < t < \infty$

First notice that $C$ lies on the plane $y = 2$, and that $\vec{r}(t)$ projects the curve $x = \cos z$ onto the plane $y = 2$. The following animation represents $C$ (in magenta).