Curves in Space

A space curve is defined by three functions x = f (t), y = g(t), z = h(t) of a single variable. You can create a ``fat curve'' by specifying a radius for the curve in the Plot Properties dialog box. This radius can be constant or can be a function of t. The Sample Size is the number of computed points along the curve; the Number of Tube Points is the number of computed points in a cross section of the tube. Ranges refers to the range of computed values for the parameter t. The View Intervals include intervals for x, y, and z of the form x0xx1, y0yy1, z0zz1.

$\blacktriangleright$ To plot a space curve

1.
Enter the three defining expressions as the components of a three-element vector.

2.
With the insertion point in the vector, from the Plot 3D submenu choose Tube.

3.
To change the radius or the view, double-click the plot, click revise, and change the settings.


The ``fat curve'' is designed to show which parts of the curve are close to the observer and which are far away. Otherwise, a curve in space is difficult to visualize. In the following example, the radius is set to 1 and 0≤t≤6.28 ( $\approx$ 2π).

$\blacktriangleright$ Plot 3D + Tube

$\left[\vphantom{
\begin{array}{c}
-10\cos t-2\cos (5t)+15\sin (2t) \\
-15\cos (2t)+10\sin t-2\sin (5t) \\
10\cos (3t)
\end{array}
}\right.$$\begin{array}{c}
-10\cos t-2\cos (5t)+15\sin (2t) \\
-15\cos (2t)+10\sin t-2\sin (5t) \\
10\cos (3t)
\end{array}$$\left.\vphantom{
\begin{array}{c}
-10\cos t-2\cos (5t)+15\sin (2t) \\
-15\cos (2t)+10\sin t-2\sin (5t) \\
10\cos (3t)
\end{array}
}\right]$

dtbpF3in2.0003in0ptYou can draw a ``thin curve,'' by setting the radius to 0 in the dialog box.

By typing an expression in t for the radius and choosing the curve to be a straight line, you can get surfaces of revolution. In the following example, the radius is set to 1 - sin t, the range for t is -2πt≤2π, and the number of tube points is 30.

$\blacktriangleright$ Plot 3D + Tube

$\left[\vphantom{ t,0,0}\right.$t, 0, 0$\left.\vphantom{ t,0,0}\right]$

dtbpF3in2.0003in0pt

The spine of the surface of revolution can be any line, as illustrated by the next example plotted with Radius: 4 + sin 3t + 2 cos 5t, Axes: Frame, Style: Hidden Line, and t Range: -5≤t≤5.

$\blacktriangleright$ Plot 3D + Tube

$\left(\vphantom{ 2t,-3t,t}\right.$2t, - 3t, t$\left.\vphantom{ 2t,-3t,t}\right)$

dtbpF3in2.0003in0pt