Tangent, Normal and Binormal Vectors — Question 4

Question 4 – Solution

Strategy Apply the chain rule and track the sign introduced by the decreasing parameter.

Tangent Since $\vec q'=\vec r'(\phi)\phi'$, normalization gives $\mathbf T_q=-\mathbf T_r$.

Normal Differentiating $\mathbf T_q=-\mathbf T_r(\phi)$ with respect to $u$ introduces another negative factor $\phi'$, so its direction agrees with $\mathbf N_r$. Hence $\mathbf N_q=\mathbf N_r$.

Binormal $\mathbf B_q=\mathbf T_q\times\mathbf N_q=-\mathbf B_r$. Thus $$\boxed{(\mathbf T,\mathbf N,\mathbf B)\mapsto(-\mathbf T,\mathbf N,-\mathbf B)}.$$ The bending side is geometric, while forward and right-handed directions reverse.