How do I visualize space curves?

answered 2014-06-30 13:31:28 -0600

Justin
1009 ●2 ●20 ●38

The way I see it, you have a few options. The first (and probably easiest, yet most inflexible) is to use WolframAlpha. You can type something like:

parametric plot (sin t, cos t, cos(8t))

And it will graph it for you. Unfortunately, there is no interactivity, and that's not cool at all—let alone groovy! Your second option is to use Mathematica. If you download it from the class site link, you can input your UNCA email address and get it for free. This is a reference guide on how to use the ParametricPlot3D function. All you have to do is type something like this:

ParametricPlot3D[{Sin[u], Cos[u], u}, {u, 0, 10}]

More generally, the function works like this:

ParametricPlot3D[{fx, fy, fz}, {u, umin, umax}]

Where fx is the $x$ component of your curve, fy is the $y$ component of your curve, and fz is the $z$ component of your curve. The second parameter, {u, umin, umax}, is also made up of three parts. The first part, u tells Mathematica what variable you want it to use for the calculations (like $t$ or $x$ or $u$). The second and third parameters let Mathematica determine the lower and upper values for your function, and restricts the domain.

Your third option is Mac-only. If you go into Utilities in your Application folder, you will find an app called "Grapher". It's a very nice piece of software! When you open it, select "3D Graph" then "Default". Next, in the bottom left corner of the window that appears, click the "+" then "New Equation From Template...". Choose "Cartesian Surface" and you should be good to go!

answered 2014-06-29 19:43:21 -0600

Christina
807 ●18 ●35 https://sites.google.com/...

updated 2014-06-29 19:45:08 -0600

I am finding that visualizing those curves is not easy for me either. For some of them it helped me to first visualize what was going on in the x-y plane and then consider how the z affected that. But since the problems said to "investigate", I decided that using wolfram alpha was a good idea. If you parametric plot them it gives an ok visual. At some point I want to learn how to use Mathematica for problems like these.