## Curvature

# Simple Definition of Curvature

Before we start formally defining curvature, we must at least have a basic idea of what it should be. So curvature is **how curvy something is. **What I mean is, if something doesn't curve at all, like y = x, the curvature is 0. If something is extremely curvy, then it's curvature is really high. Now, I know that's very hand-wavy, so let's get to some real math.

# Definitions

So, we need to define a few things.

**unit tangent vector**is defined as: $$T(t) = \dfrac{r'(t)}{|r'(t)|}$$ Because $|T(t)|$ is constant (it's always 1), by a theorem we proved before (which isn't on this site, sorry), $T(t)$ is perpendicular to $T'(t)$. So, now we have our unit normal vector. The

**unit normal vector**is defined as: $$N(t) = \dfrac{T'(t)}{|T'(t)|}$$ This is

**always**perpendicular to the normal vector.