Curvature — Question 3

Question 3

For the helix $\vec r(t)=\left\langle 3\cos t,3\sin t,4t\right\rangle$, compute the curvature for all $t$. Explain geometrically why a constant answer is reasonable.

Question 3 – Solution

Strategy Compute speed and the magnitude of $\vec r'\times\vec r''$ separately.

Calculation $$\vec r'=\left\langle -3\sin t,3\cos t,4\right\rangle,\quad \vec r''=\left\langle -3\cos t,-3\sin t,0\right\rangle,$$ $$\vec r'\times\vec r''=\left\langle 12\sin t,-12\cos t,9\right\rangle.$$ Thus $\|\vec r'\|=5$ and $\|\vec r'\times\vec r''\|=15$, giving $$\boxed{\kappa=\frac{15}{5^3}=\frac 3{25}}.$$

Geometry Every point of a circular helix has the same local shape: rotation about and translation along its axis carry one point to another, so its bending is uniform.