📂Geometry

Rotational Surfaces in Differential Geometry

Definition¹

Let $z$ be the variable on the given axis, and $r>0$ be the distance from the $z-$ axis. Then, one can consider the curve $\alpha$ on the $rz-$ plane as shown in the figure below.

As shown in the figure below, the surface obtained by rotating the curve $\alpha$ about the $z-$ axis is called a surface of revolution.

The surface of revolution is expressed as follows.

$$\mathbf{x}(t, \theta) = \left( r(t)\cos \theta, r(t)\sin \theta, z(t) \right)$$

The $t-$ parametric curve of the surface of revolution is called a meridian, and the $\theta-$ parametric curve is called a circle of latitude or parallel.

Explanation

Even if it is called a solid, it might not significantly impede communication, but strictly speaking, it is not a solid of revolution because it is hollow inside.

All meridians are geodesics. Unlike this, certain conditions are required for circles of latitude to become geodesics.

Theorem

If the curve $\alpha (t) = \left( r(t), z(t) \right)$ is regular and one-to-one, then the surface of revolution $\mathbf{x}(t, \theta) = \left( r(t)\cos \theta, r(t)\sin \theta, z(t) \right)$ formed by $\alpha$ is a simple surface. The condition for $\theta$ is $-\pi \lt \theta \lt \pi$.