Surfaces of Revolution

A "surface of revolution" is a surface obtained by rotating a curve about an axis. More precisely, consider a curve in the "$x \geq 0$" half of the Euclidean plane, with coordinates $x$ and $z$. We then consider this plane as the $y=0$ plane in 3-dimensional Euclidean space (with $x, y, z$ as our coordinates), and rotate it about the $z$ axis. The set of points through which it rotates define a surface of revolution. The curve can be paramaterised by $\gamma (t) = (r(t), t)$, where the function $r$ denotes the "radius" of a point (distance from the origin to a point), and $t$ just denotes the value of a point's $z$-coordinate (i.e. its "height").

The lines of curvature for a surface of revolution are particularly well-behaved. They always lie in a plane which is either horizontal or vertical, passing through the origin (these vertical lines of curvature are called "meridians"). The geodesics on a surface of revolution are also well behaved: they obey Clairaut's relation, meaning their behaviour can be easily characterised.