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:

• Unlike standard graphics calculators or the textbook, DPGraph uses the letter "u" instead of "t" for the parameter.

• The order in which the coordinates appear inside the "rectangular(...)" command is interpreted by the DPGraph as the standard order: x, then y, then z.

• Given that the term "sin(u)" has the longest period of 2*pi in the three components, the first two of the following commands:

GRAPH3D.MINIMUMU := 0
GRAPH3D.MAXIMUMU := 2*PI
GRAPH3D.STEPSU := 100

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?

Analysis:

• To Determine the orientation of the curve as shown in Figure 1(B), adjust the GRAPH3D.MAXIMUMU command to a number close to zero, and redraw the graph; note which part of the curve is drawn and use this observation to determine the correct direction on the curve. Note also from the given equations for the curve, that when u = t = 0, we obtain the point (2,0,0) on the curve, and this same point is obtained when u = t = 2*pi.

• The curve lies on and wraps around a right circular cylinder (or a "pipe") of radius 2. This is easily seen by computing:

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:

GRAPH3D.VIEW := TOP

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,

GRAPH3D.PERSPECTIVE := FALSE

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.

• To find the length of the curve, it is necessary to use the arc-length formula and integrate from 0 to 2*pi. This may be done using the MPP software, or alternately, on the TI-83 (and similar) graphics calculator as follows:

• Enter the appropriate functions into the arc-length formula, and simplify as much as possible to conclude that the length of the curve is obtained by integrating the square root of 4 + [cos(t)]^2 from 0 to pi, and then multiplying the answer by 6.

• Enter the square root of 4 + [cos(x)]^2 as function "Y1" into the TI-83, noting that the calculator uses x as the standard variable, not t.

• Set the calculator "window" large enough that the graph of Y1 is fully visible (if not sure, graph Y1 first).

• Press [2nd] [trace] to access the [calc] menu, then press [7] for the integral (or scroll down to it, then press [enter]). The calculator draws the graph and prompts for a "lower limit?" Enter 0 or move the cursor to 0, then press [enter]. When the calculator prompts for the "upper limit?" enter 3.14 for pi (it seems that TI-83 accepts only two decimal places; for more accurate results, you need to move the cursor manually, or use the MPP software). Pressing [enter] again should give an answer of about 39.94 (after rounding to two decimal places) for the length of our curve.