# Help understanding John Lee's definition of curvature

On page 3 of his book Riemannian Manifolds, John Lee states the following

If you want to continue your study of plane geometry beyond figures constructed from lines and circles, sooner or later you will have to come to terms with other curves in the plane. An arbitrary curve cannot be completely described by one or two numbers such as length or radius; instead the basic invariant is curvature, which is defined using calculus and is a function of position on the curve.

Formally, the curvature of a plane curve $\gamma$ is defined to be $\kappa(t) := |\ddot{\gamma}(t)|$, the length of the acceleration vector, when $\gamma$ is given a unit speed parameterization.

Two Questions:

(1) Regarding the first paragraph, why isn't length sufficient to study curves?

(2) What is a "unit speed parameterization"? Does he mean velocity or speed (ie $v$ or $\|v\|$)?

## 1 Answer

1- If length were sufficient a circle of length (perimeter) $2\pi$ would be the same as a square whose side is $\frac{2\pi}{4}$.

2- a unit speed parametrization is defined as follows: if $x(t)$ is an arbitrary parametrization take $s$ to be defined by $\frac{ds}{dt}=\vert\vert x'(t)\vert\vert$. You check that $\vert\vert \frac{dx}{ds} \vert\vert=1$. $x(s)$ is the unit speed parametrization of the curve