Curvature — Question 5

Question 5

Let $\vec r(s)$ be a twice-differentiable unit-speed curve. Prove that $\kappa(s)=\|\vec r''(s)\|$ and that the acceleration $\vec r''(s)$ is perpendicular to the tangent.

Question 5 – Solution

Strategy Differentiate the unit-speed identity $\vec r'\cdot\vec r'=1$.

Orthogonality Because $\|\vec r'\|=1$, $$\vec r'\cdot\vec r'=1\quad\Longrightarrow\quad 2\vec r'\cdot\vec r''=0.$$ Thus $\vec r''\perp\vec r'=\mathbf T$.

Curvature For an arc-length parameter, $\mathbf T=d\vec r/ds$. By definition, $$\boxed{\kappa=\left\|\frac{d\mathbf T}{ds}\right\| =\left\|\frac{d^2\vec r}{ds^2}\right\|}.$$

Interpretation Unit speed removes tangential acceleration. Any remaining second derivative measures only the rate at which the direction turns.