Tangent, Normal and Binormal Vectors — Question 10

Question 10 – Solution

Strategy Differentiate the involute and use the circle’s simple Frenet derivatives.

Circle frame $\mathbf T(t)=\left\langle-\sin t,\cos t\right\rangle$ and $\mathbf T'(t)=\left\langle-\cos t,-\sin t\right\rangle=\mathbf N(t)$.

Involute derivative Differentiating, $$\vec q'=\vec r'-\mathbf T-t\mathbf T'=\mathbf T-\mathbf T-t\mathbf N=-t\mathbf N.$$ Since $t>0$, $\|\vec q'\|=t$, so $\boxed{\mathbf T_q=-\mathbf N=\left\langle\cos t,\sin t\right\rangle}$.

Interpretation The taut string leaves the circle tangentially, while the involute’s tangent points along the outward radial direction.