The Scribble Curve 

or 

Why are your examples all so cheezy? 

Maple worksheet: scribble_curve.mw 

You may notice that there are a limited number of good examples of curves whose arc length, TNB-frames, and curvature can be computed (nicely) by hand. We typically stick to planar curves, circles, helixes and the like. The purpose of this worksheet is to demonstrate how out of hand things can get when we start with the "wrong" relatively simple curve. 

>	restart; with(plots): with(VectorCalculus):

 

Here's a nice curve which I like to call "The Scribble". 

I would like to point out that the formula defining this curve is not all that complicated: where . 

>	r := t -> <5*cos(sqrt(t))+cos(t)*cos(t),5*sin(sqrt(t))+sin(t)*cos(4*t),sin(t)>: 'r(t)'=r(t); ScribblePlot := spacecurve(r(t),t=0..2*Pi,scaling=constrained,numpoints=10000,thickness=3,color=blue,axes=boxed,viewpoint=circleleft): display(ScribblePlot);

 

Here I compute the unit tangent function for the scribble... 

>	T := t -> simplify(subs(u=t,1/Norm(diff(r(u),u))*diff(r(u),u))): 'T(t)' = T(t);

 

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I'm glad I didn't have to do that by hand! 

Now let's make Maple draw some tangent arrows. Specifically, I'll have Maple draw 20 unit tangents along the scribble curve. 

WARNING: The following command make take a minute to execute. 

>	SomeTangents := seq(arrow(r(2*j*Pi/20),T(2*j*Pi/20),shape=arrow,color=red),j=1..20): display({ScribblePlot,SomeTangents},orientation=[37,173],viewpoint=circleleft);

 

Finally, let's look at the graph of the curvature of the scribble curve. Seeing how it winds all over the place we should expect the curvature should go up and down. 

WARNING: The following command make take a minute to execute. 

>	kappa := Curvature(r(t),t): plot(kappa,t=0..2*Pi);

 

Display kappa  using "[> simplify(kappa);" (remove the comment marker "#") ... I dare you. 

I could never compute such a thing by hand! 

>	simplify(kappa);

 

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