Hint: You can find the derivative of just by entering and the second derivative by entering (that is two apostrophe's, NOT quotation marks). Also, you can test your curvature function by using it to find the curvature of , which should be (for all t).
Many curves in 3 dimensions actually lie on surfaces of some sort. For example, the curve:
lies on the surface of the unit sphere about the origin. If you graph r and the unit sphere on the same axes, you can see this:
This defines a unit sphere we can overlay this onto. (You can use this definition later in the exercises if you wish.)
(If you use the Mesh->False directive in your Show command, you can make it easier to see the curve. Unfortunately, there is a bug in the RealTime3D package that ignores any color directives. This means you can either rotate your pictures with the mouse or control the colors, not both. I'll be glad when they fix this one...)
You can show algebraically that this does lie on the sphere by showing that its components satisfy the equation for a sphere:
This shows that as it should be for a sphere. (It is usually easier to set the Cartesian equation equal to 0 to check it, especially on more complicated surfaces.) For most of the surfaces we will be looking at, you should apply the //FullSimplify at the end of the line (it does things like take care of trig identities, etc.).
You will probably need to set PlotPoints to 100 or 200 to get smooth looking curves here. Also, you will probably need to experiment around with your domain for t a bit. If the domain is too large, you will get a mess (too many line segments, which can give a false impression of the curve). If it is too small, you may not be able to tell what surface the curve lies on.