Spherical Coordinates — Question 9

Question 9

A curve on the sphere $\rho=R$ has constant polar angle $\phi=\alpha$, while $\theta=t$ varies from $0$ to $2\pi$. Find a rectangular parameterization and the exact length of the curve.

Question 9 – Solution

Strategy Convert the fixed spherical coordinates and recognize a horizontal circle.

Parameterization $$\boxed{\vec r(t)=\left\langle R\sin\alpha\cos t,\ R\sin\alpha\sin t,\ R\cos\alpha\right\rangle}, \quad 0\le t\le 2\pi.$$

Length Differentiation gives $$\|\vec r'(t)\|=R\sin\alpha$$ for $0\le\alpha\le\pi$. Hence $$\boxed{L=2\pi R\sin\alpha}.$$

Geometry The latitude circle has Euclidean radius $R\sin\alpha$ and lies in the horizontal plane $z=R\cos\alpha$. At either pole, this radius and length are zero.