Induced Metric on Surface of Revolution/Corollary

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

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

Let the smooth local parametrization of $C$ be:


 * $\map \gamma t = \tuple {\map a t, \map b t}$

Let $\gamma$ be a unit-speed curve.

Then the induced metric on $S_C$ is:


 * $g = d t^2 + \map a t^2 d \theta^2$

Proof
By definition of the unit-speed curve:


 * $\size {\map {\gamma'} t}_g = 1$

In our case we are working with the Euclidean space.

Hence:


 * $\sqrt {\map {a'^2} t + \map {b'^2} t} = 1$

or:


 * $ \map {a'^2} t + \map {b'^2} t = 1$

Substitution of this into the induced metric of $S_C$ yields the desired result.