Torus as Surface of Revolution

Theorem
Let $\struct {\R^3, d}$ be the Euclidean space.

Let $S_C \subseteq \R^3$ be the surface of revolution.

Let $C$ be a circle defined by $x^2 + \paren {y - 2}^2 = 1$ in the open upper half-plane.

Let the smooth local parametrization of $C$ be:


 * $\map \gamma t = \tuple {2 + \cos t, \sin t}$

Then the induced metric on $S_C$ is:


 * $g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$

Proof
We have that:


 * $\map {\gamma'} t = \tuple {- \sin t, \cos t}$

Furthermore:


 * $\paren {- \sin t}^2 + \paren {\cos t}^2 = 1$

Hence, $\map \gamma t$ is a unit-speed curve.

By the corollary of the induced metric on the surface of revolution:


 * $g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$