Graphing Parametric Curves and Surfaces in Space with the MPP

Parametric Curves (or Space Curves) -

To graph curves in space, like a portion of the curve with parametric equations

x(t) = 3cos(2t), y(t) = sin(2t), z(t) = t/2 (an elliptical helix)

proceed as follows:

Step 1. Start the MPP3D program

Step 2. Press "F6" to change type, then choose option C

Step 3. Enter the dimensions of the viewing box in Section F1; if unsure, try different values in order to get good graphs. In this case, the following values give a good, uncut view of the curve:

Xmin = Ymin = Zmin = -3, Xmax = Ymax = Zmax = 3

Step 4. In Section F2, enter the function values for x, y, z given in the problem (here, as listed above)

Step 5. In Section F3, enter the parameter range: Here, tmin = -2pi, tmax = 2pi

Step 6. In Section F4, enter grid size, usually 100-300; for this problem, enter 100

Step 7. Pressing "F9" should give the graph of an elliptical helix. Press "H" for a list of things you can do; for instance, you may insert coordinate axes by pressing "A", or rotate the graph as follows: Press "R", then use arrow keys to rotate the white coordinate axes, and when a good view obtains, press "Enter" to get the graph rotated. For space curves, several rotations are necessary to gain a good visual understanding of the curve. You may also print your graph by pressing either "F" or "P".

Parametric Surfaces -

To graph a surface given parametrically like the Mobius Strip (a one-sided surface) with equations:

x = f(u,v) = (2 + u*cos(v/2))cos(v), y = g(u,v) = (2 + u*cos(v/2))sin(v), z = h(u,v) = u*sin(v/2)

Steps 1-4 and 7 are essentially the same as above for space curves, except that in Step 2 you choose option 6.

Step 5. In Section F3, enter the parameter ranges; here try

umin = -1, umax = 1, vmin = 0, vmax = 2pi

Step 6. In Section F4, enter grid size of up to 30 (here a number between 20 and 30 should be okay)