Defining Principal Radius of Curvature for Zero Principal Curvature

Question

In differential geometry, the principal radius of curvature $\rho$ is defined as the reciprocal of the principal curvature $\kappa$. But suppose one of my principal curvatures is 0. How can the principal radius of curvature be $1/0$?

Answer 1

At such a point, we can say that the radius of curvature is infinite. It's not unusual to extend the real line with $+\infty$ and $-\infty$ to unify certain statements and computations. As long as one is careful enough to avoid indeterminate expressions like $0\cdot \infty$ or $\infty-\infty$, it works well.

Examples of usage: this answer and this one.