# Find a parametrization for the curve obtained by intersecting

August 31, 2017

Question
Find a parametrization for the curve obtained by intersecting the following sphere and cylinder:

X2 + y2 + z2 = 4 a2 , ( x.png”> a )2 + y2 = a2

Hint: x = a cos t + a, y = a sin t work in the cylinder. Plug these x and y into the sphere’s defining equation and solve for z. this curve is known as viviani’s curve. Compute the curvature and torsion of viviani’s curve and ( since we know the curve is spherical) verify that the formula

(.png”>)2 + (.png”> (.png”>)’.png”>)2 = 4a2

Holds. Finally, graph Viviani’s curve on the sphere. Compare Viviani’s curve with the curve of Exercies 1..)and make a conjecture. Can you prove your conjecture? Hints: use maple and use formulas Non-Speed Curves.Recall that you have already plotted Viviani’s curve.
Hint:Recall the trig identities:
cos(t/2)= sqrt((1+cos(t))/2) and sin(t/2)= sqrt((1-cos(t))/2)
use these in your parametrization and then use the curv and tor Maple procedures.

I send all the question to understand the idea of the question. In this question I want just first part which is to find, solve for z.