Curvature — Question 1

Question 1 – Solution

Strategy Use $\kappa=\|\vec r'\times\vec r''\|/\|\vec r'\|^3$, viewing planar vectors as three-dimensional vectors with zero third component.

Calculation $\vec r'=\left\langle -a\sin t,a\cos t,0\right\rangle$ and $\vec r''=\left\langle -a\cos t,-a\sin t,0\right\rangle$. Thus $\|\vec r'\|=a$ and $\|\vec r'\times\vec r''\|=a^2$. Therefore $$\boxed{\kappa=\frac{a^2}{a^3}=\frac 1a},\qquad \boxed{\rho=\frac 1\kappa=a}.$$

Interpretation Small circles turn more sharply and have larger curvature; the radius of curvature $\rho$ is exactly the circle’s geometric radius.