Example 6: Space-curve plotting, parametric curves

Objective: Plot and analyze the space curve given parametrically by the following equations:

x = 2cos(3t), y = 2sin(3t), z = 3sin(t)

The DPGraph Commands: Click here to download the .dpg file on your computer for a list of commands used in the correct syntax and for later modifications. Note the use of parentheses (to clarify operations) and asterisks (for multiplication). Also note the following:


display the graph of the entire space curve over the parameter range u in the interval [0,2*pi]. This is a bounded curve since the trigonometric functions that define its coordinates are all bounded functions. A bounded curve can be fully plotted inside a finite viewing box.

The third or bottom command ensures that the graph is smooth by using a sufficiently high number of "u steps" (try re-drawing the curve using a smaller number than 100, say, 30; the resulting non-smooth curve is wrong! On the other hand, once a smooth curve is obtained with say, 100, larger step values are inefficient and should be avoided). Figures 1, (A) and (B) shows the same curve from two different view points.

Note that although in Figure 1(B) the curve appears to itersect itself 6 times, Figure 1(A) shows that it actually does so only 2 times. Can you find these points graphically? Analytically?


x^2 + y^2 = 4([cos(3t)]^2 +[sin(3t)]^2) = 4

which is the familiar equation of the cylinder in rectangular coordinates. By drawing and rendering the cylinder separately using DPGraph, and then superimposing the curve on the cylinder in MS Paint, we obtain the following figure which verifies our conclusions:

Another way of seeing the wrap-around-the-pipe feature of the curve, is by looking at it from above. This is done by adding the command:


which gives the graph on the left in Figure 3 below. We see both the circularity and the two self-intersection points in this graph. The graph on the right is obtained by adding an extra command,


that turns off the 3D perspective. In this case, we do not see the bottom of the cylinder as being farther from us, hence appearing smaller than the top; both top and bottom are treated equally when there is no 3D perspective and we clearly see the circular boundary of the cylinder that contains the curve (no longer visible) in it.