Smooth Local Parametrization of Surface of Revolution

Theorem

Let $S_C$ and $C$ be the surface of revolution and its generating curve.

Let the smooth local parametrization of $C$ be:

$\map \gamma t = \tuple {\map x t, \map y t}$

Then the smooth local parametrization of $S_C$ can be written as:

$\map X {t, \theta} = \tuple {\map y t \cos \theta, \map y t \sin \theta, \map x t}$

where $\tuple {t, \theta}$ belongs to a sufficiently small open set in the plane, and the revolution is done around the $x$-axis.

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics