- #1

Poirot

- 94

- 3

## Homework Statement

Calculate the circumference of the circle θ = θ

_{0}(a constant) in the spatial geometry

\begin{eqnarray*}

dS^2 = a^2(d\theta^2 + sin^2\theta cos^2\theta d\phi^2)

\end{eqnarray*}

Hence, (by finding R(z)) sketch the cross section of the surface embedded in three dimensions via

\begin{eqnarray*}

(x, y, z) = (Rsin\theta cos\phi, Rsin\theta sin\phi, Rcos\theta)

\end{eqnarray*}

## Homework Equations

## The Attempt at a Solution

Calculating the circum:

Parameterising the path set φ = τ, where τ

_{1}= 0 and τ

_{2}=2π, and θ = θ

_{0}(const.)

so

\begin{eqnarray*}

circumference = \int_{0}^{2\pi} \frac{dS}{d\tau}d\tau = \int_{0}^{2\pi} a((\frac{d\theta}{d\tau})^2 + sin^2\theta cos^2\theta (\frac{d\phi}{d\tau})^2)^{1/2} d\tau

\end{eqnarray*}

using the parametrisation we get:

\begin{eqnarray*}

circumference: \int_{0}^{2\pi} a sin\theta_0 cos\theta_0 d\tau = 2\pi a sin\theta_0 cos\theta_0 = \pi a sin2\theta_0

\end{eqnarray*}

I'm not at all sure what the next part is getting at. I've played around with rearranging z=Rcosθ and plugging this in for theta in metric but I don't think that's right.

Any help would be great appreciated, thank you!