Space Curves
Textbook Reference
| Primary source | OpenStax Calculus Volume 3, Section 3.1: “Vector-Valued Functions and Space Curves” |
| Direct link | https://openstax.org/books/calculus-volume-3/pages/3-1-vector-valued-functions-and-space-curves |
| Textbook used in class | Stewart, Calculus, Section 13.1: “Vector Functions and Space Curves” (Examples 4, 5) |
Opening Scenario
A spring coiling upward traces a helix. A planet orbiting the sun traces an ellipse. Both can be described by vector functions $\mathbf{r}(t)$ where $t$ is time. The collection of all points traced is the space curve, and the vector function tells you where the particle is at each moment.
Quick Reference
Space curve. The set $C = \{(f(t), g(t), h(t)) : t \in I\}$ traced by $\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle$ for $t$ in interval $I$.
Helix: $\mathbf{r}(t) = \langle a\cos t, a\sin t, bt\rangle$ is a circular helix of radius $a$ that rises (or falls) at rate $b$ per radian.
Key Concepts
1. Sketching Space Curves by Eliminating Parameters
Example 1. Identify the curve $\mathbf{r}(t) = \langle \cos t, \sin t, t\rangle$, $t \geq 0$. (Stewart 13.1, Example 4.)
The $xy$-trace satisfies $x^2 + y^2 = \cos^2 t + \sin^2 t = 1$: a circle. The $z$-component increases as $t$ increases. The curve spirals upward along the cylinder $x^2 + y^2 = 1$, forming a helix.
Example 2. Find a vector function for the curve of intersection of the cylinder $x^2 + y^2 = 1$ and the plane $y + z = 2$. (Stewart 13.1, Example 5.)
Parameterize the cylinder with $x = \cos t$, $y = \sin t$. The plane gives $z = 2 - y = 2 - \sin t$.
$\mathbf{r}(t) = \langle \cos t, \sin t, 2 - \sin t\rangle$.
2. Common Space Curves
| Curve | Vector function | Description |
|---|---|---|
| Straight line | $\mathbf{r}_0 + t\mathbf{d}$ | Point plus direction times $t$ |
| Circle in a plane | $\langle a\cos t, a\sin t, c\rangle$ | Circle of radius $a$ at height $c$ |
| Helix | $\langle a\cos t, a\sin t, bt\rangle$ | Spirals up the $z$-axis |
| Ellipse | $\langle a\cos t, b\sin t, 0\rangle$ | Ellipse in the $xy$-plane |
confusing a space curve with a surface. A vector function $\mathbf{r}(t)$ with one parameter traces a one-dimensional curve, not a surface. Surfaces require two parameters. The curve is a one-dimensional set embedded in 3D space.
Common Misconceptions
a vector function $\mathbf{r}(t)$ traces a surface, not a curve.
This is the action-view-of-function error. A vector function $\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle$ has one independent parameter $t$, so its image is a one-dimensional object, a curve. Surfaces require two independent parameters $\mathbf{r}(s,t) = \langle f(s,t), g(s,t), h(s,t)\rangle$. The helix $\mathbf{r}(t) = \langle \cos t, \sin t, t\rangle$, for instance, is a curve in 3D space, not a surface, even though it lives in three dimensions.
Leveled Practice
Problem 1. Parameterize the curve of intersection of $z = x^2 + y^2$ and $z = 4$.
Show answer
$z = 4$ forces $x^2 + y^2 = 4$, a circle of radius $2$. Let $x = 2\cos t$, $y = 2\sin t$, $z = 4$. So $\mathbf{r}(t) = \langle 2\cos t, 2\sin t, 4\rangle$.