HW #3

1. Viviani's curve, also knows as Temple of Viviani, is the curve of intersection of the sphere $x^2+y^2+z^2=4a^2$ and the cylinder $(x-a)^2+y^2=a^2$. The curve is shown in yellow in the drawings below. [You may be interested in this brief historical note.]
   
   1. Find a parametric equation of Viviani's curve and plot it. See the textbook's Exercise 1.3.26 for a hint, but you don't need to solve that exercise.
   2. Find the vectors of the curve's Frenet triad.
   3. Compute the parametric equation of a tube of radius $r$ that surrounds the curve. Plot the tube for some $r$.
   4. Find the curve's curvature, torsion. and the Darboux vector. [The Darboux vector is defined in Exercise 1.3.12 which was a part of homework #2.]
   5. In the animation above we see the vectors of the Frenet triad (drawn in red, green, and blue) and the Darboux vector (drawn in cyan) as they travel around the curve. [The vectors have been scaled by a factor of 1.4 for aesthetic reasons.] Produce something like that animation.