Q:

Arc length of curve of intersection between cylinder and sphere

Given the sphere $x^2+y^2+z^2 = \frac{1}{8}$ and the cylinder $8x^2+10z^2=1$, find the arc length of the curve of intersection between the two.

I tried parametrizing the cylinder (the task specifies this as a hint). My attempt:
$$x(t) = \frac{1}{\sqrt{8}}  \sin(t)$$
$$z(t) = \frac{1}{\sqrt{10}} \cos(t)$$
Plugging this into $x^2+y^2+z^2 = \frac{1}{8}$, I solve for $y$ to get 
$$y = \sqrt{\frac{\cos(2t)+1}{4\sqrt{5}}}$$
I then tried integrating $|x(t), y(t), z(t)|$ from $0$ to $2\pi$
with no luck. I suspect my parametrization is wrong as my expression for $y$ looks rather ugly. Any ideas?

A:

The intersection of the cylinder with the sphere produces two curves, each of which is a great circle.

Since the radius of the sphere is $\dfrac{1}{\sqrt{8}}$, the length of each great circle is $\dfrac{2 \pi}{\sqrt{8}} = \dfrac{\pi}{\sqrt{2}}$.

Using SymPy to verify:
>>> from sympy import *
>>> t = Symbol('t')
>>> x = sin(t) / sqrt(8)
>>> y = cos(t) / sqrt(40)
>>> z = cos(t) / sqrt(10)
>>> x_dot = diff(x,t)
>>> y_dot = diff(y,t)
>>> z_dot = diff(z,t)
>>> v = Matrix([x_dot, y_dot, z_dot])
>>> v.T * v
Matrix([[sin(t)**2/8 + cos(t)**2/8]])
>>> simplify(sin(t)**2/8 + cos(t)**2/8)
1/8

Note that the Euclidean norm of the velocity vector is $\dfrac{1}{\sqrt{8}}$ and, thus, independent of $t$.

