Curvature — Question 9

Question 9 – Solution

Strategy Track how first derivatives, second derivatives, cross products, and norms scale.

Scaling $$\vec q'=c\vec r',\qquad \vec q''=c\vec r'',\qquad \|\vec q'\times\vec q''\|=c^2\|\vec r'\times\vec r''\|.$$ Also $\|\vec q'\|^3=c^3\|\vec r'\|^3$. Hence $$\boxed{\kappa_q=\frac{c^2}{c^3}\kappa_r=\frac{\kappa_r}{c}}.$$

Radius Since $\rho=1/\kappa$, $\boxed{\rho_q=c\rho_r}$. Enlargement makes every local osculating circle $c$ times larger and the curve correspondingly less sharply bent.