\(MATH\ 257\ Calculus\ III				Week\ of\ May\ 19, \ 2003\)
Lab 8 :  Curvature
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This lab is quick and easy.  Do it and then go down to the 7th floor of Old Chem and get some food!

Doing Curvature on Mathematica

As we've seen in class, computing the curvature of a space curve can, and usually is, a royal pain.  On Mathematica it's much easier, as long as the program doesn't barf on you.

For example, let's find a general formula for the curvature of the twisted cubic.  First we gotta define all the functions we'll need...

To do all this we'll definitely need to define a function that returns the magnitude of a general vector:

mag[
        v_] := (Sqrt[v[[1]]^2 + v[[2]]^2 + v[[3]]^2])

Then we're ready to define the unit tangent vector:

T[t_] := r'[t]/mag[r'[t]]

Then we can go right ahead and define the curvature function:

kappa[t_] := mag[T'[t]]/mag[r'[t]]

All right, then!  Let's see what the general formula for the curvature of the twisted cubic looks like!  We need to define r[t] to be the twisted cubic vector function, and then just ask what kappa[t] is!

r[t_] = {t, (t^2), (t^3)}
kappa[t]

Doesn't that look wonderful?  OK, we probably should simplify it:

Simplify[%]

And there it is; our very own function for the curvature of the twisted cubic at any point we want.  Why do we care?  Well, we could graph this sucker

Plot[kappa[t],  {t,  -3, 3}]

So it looks like the twisted cubic has the greatest curvature when t=0.  Neat!

Question 1: Find a general formula for the curvature of the helix vector function < 3Sin[t], 3Cos[t], t/2 >.  (You do NOT need to redefine the functions T[t] and kappa[t].  Just redefine r[t] and you should be all set to go.  Ain't Mathematica kewl?)  Make sure you simplify.

Question 2: Find a general formula for the curvature of the vector function < Cos[t]-Cos[3t]Sin[t], 2Sin[t]-Sin[3t], 0 >.  Yes, this curve lives in the x-y plane, but we can still find the curvature.  Do the following:  plot this space curve, find the formula for it's curvature, and make a plot of the curvature formula.  (You should make  0 ≤ t ≤ 2π.)

Yep, that's it.  You're done.  Send it to me via the Digital Dropbox and then go to the Kade Center (7th Floor of Old Chem, in the center of the floor.  It's the atrium area)  and get some food and cheer on Tom Warburg, who won the Calculus Contest!)