<< RealTime3D`

<< Graphics`

Calculus III - Lab 2a: 
3-Dimensional Graphs - Curves

Warning: Save early and save often when you are working on this lab.  Some of the computations are fairly memory intensive and you really don't want to suddenly have your computer lock up on you and lose an hour's work you never got around to saving.  You have been warned.

Curves

Curvature

For a curve r [t] , find κ[t] , the curvature function at time t (define it as a function).

Hint:  You can find the derivative of r[t] just by entering r'[t] and the second derivative by entering r''[t] (that is two apostrophe's, NOT quotation marks).  Also, you can test your curvature function by using it to find the curvature of r[t_] := {Cos[t],  Sin[t],  t}, which should be (1/2) (for all t).

Graph r[t_] := {(t^4), (t^3), (t^2) Sin[4  t]} and κ[t]  over the same domain for t (-1 to 1 should be okay).  Explain how these two graphs relate to each other.  What important fact about the curve does the curvature graph make clear (it isn't at all obvious on the actual graph or r)?

Curves on a surface

Many curves in 3 dimensions actually lie on surfaces of some sort.  For example, the curve:

r[t_] := {Cos[12  t] (Sqrt[1 - (t^4)]), Sin[12  t] (Sqrt[1 - (t^4)]), 
        (t^2)}

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:

curve = 
      ParametricPlot3D[Evaluate[r[t]], {t, -1, 1}, 
        PlotPoints -> 200]

This defines a unit sphere we can overlay this onto.  (You can use this definition later in the exercises if you wish.)

unitSphere = 
      ParametricPlot3D[{Cos[u] Sin[v], Sin[u] Sin[v], Cos[v]}, {u, 0, 
          2  π}, {v, 0, π}, Boxed -> False, 
        Axes -> False]

Show[unitSphere, curve,  Mesh -> False]

(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:

(((Cos[12  t] (Sqrt[1 - (t^4)])))^2) + (((Sin[12  t] (Sqrt[1 - (t^4)])))^2) + ((((t^2)))^2) - 1 // FullSimplify

This shows that (x^2) + (y^2) + (z^2) - 1 = 0 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.).

For each of the following, decide what surface it lies on.  Show this graphically and algebraically (as above).  For simplicity sake, you can assume that each surface will be as simple as possible (i.e., the coefficients are all 1).

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.

1.

r[t_] := {Sin[(t^(1/3))] Cos[3 t], Sin[3  t] Sin[(t^(1/3))], 
        Cos[(t^(1/3))]}

2.

r[t_] := {((( Cos[Log[6  (t^2) + 1]]))^2),  
        Cos[Sin[5  t]] Cos[Log[6  (t^2) + 1]],  
        Cos[Log[6  (t^2) + 1]] Sin[Sin[5  t]]}

3.

r[t_] := {(Sqrt[1 - (((Sin[(t^(1/2))]))^2)]), 
        Cos[(t^(1/2))] Sin[1 - (t^2)], 
        Cos[(t^(1/2))] Cos[1 - (t^2)]}

Extra credit:  For all the examples above, it is relatively easy to check to see that the curve really lies on the surface by using the implicit form of the equation for the surface (i.e., an equation in X, Y, and Z only).  However, if all you have are the parametric equations for a surface, explain how you can generate many curves on that surface easily without having to find the implicit form to check.  Give a few examples and graph them (as above)...