# Unit Normal always pointing toward concave side

1. Sep 30, 2011

### Rulesby

Physically/conceptually, I understand why the unit normal vector will always point toward the side of concavity on a curve. It's because the unit normal's direction is the derivative of the unit tangent, and the unit tangent's change in direction is always toward the concave side.

But how is this represented mathematically? I tried to prove it, but I'm stuck. I've looked, but I couldn't find any proofs about this. Does anyone know?

2. Sep 30, 2011

### AlephZero

Think about a curve defined by a parametric function, and the Taylor series expansion of the functions at any point along the curve.

The curvature and the direction of the unit normal both come from the quadratic term in the Taylor series, so the normal always points to the "same" (concave) side of the curve.