Math 3298 - Hampton - Lab 3 - Due 9/25

Cornu's Spiral, and an introduction to parametric surfaces.

Exercise

1. Cornu spiral.  Use the ParametricPlot command to look at Cornu's spiral: r[t_]:= {Integrate[Cos[Pi*u^2/2], {u,0,t}], Integrate[Sin[Pi*u^2/2], {u,0,t}]}.  What is the arclength of the curve from t=0 to t=c?  What is the curvature at t = c? (This is Exploratory Exercise 1 in Chapter 12.4 in our text.  A warning: Mathematica takes a long time to compute this function.)

2. Use ParametricPlot3D to examine the parametric surfaces: (A) x = sinh(v), y = cos(u)*cosh(v), z = sin(u)*cosh(v) and (B) x = 9 v^2, y = v sinh(u), z = v cosh(u).
Classify them - what kind of quadric surfaces are they (e.g. ellipsoid, elliptic paraboloid, etc. See Chapter 11.6).