Math 344: Calculus III |
A space curve is the curve traced out by the terminal
points of the vectors in the image of a vector function
$$
{\bf r}(t) = \langle x(t), y(t), z(t) \rangle.
$$
We look at some examples below. Equivalently, we may regard
a space curve as the graph of a set of parametric functions in
$\mathbb{R}^3$; the component functions of the vector function.
Example 1. The toroidal spiral has parametric equations
$$ \begin{cases}
x(t) = \big(4 + \sin(20 t)\big)\cos t,\\[1 ex]
y(t) = \big(4 + \sin(20 t)\big)\sin t,\\[1 ex]
z(t) = \cos(20 t).
\end{cases}$$
Example 2. The trefoil knot has parametric equations
$$ \begin{cases}
x(t) = \big(2 + \cos(\tfrac{3}{2} t)\big)\cos t,\\[1 ex]
y(t) = \big(2 + \cos(\tfrac{3}{2} t)\big)\sin t,\\[1 ex]
z(t) = \sin(\tfrac{3}{2} t)
\end{cases}$$
Example 3. The twisted cubic has parametric equations
$$ \begin{cases}
x(t) = t, \\
y(t) = t^2 \\
z(t) = t^3
\end{cases}$$
Back to main page
Your use of Wichita State University content and this material is subject to our Creative Common License.