UPSC 2024 Maths Optional Paper 1 Q5e-ii — Step-by-Step Solution

5 marks · Section B

Serret-Frenet formulae · Vector Analysis · asked 2× in 13 yrs · Read the full method →

Question

Show that the principal normals at two consecutive points of a curve do not intersect unless torsion $\tau$ is zero.

Technique

Frenet–Serret derivatives + scalar triple product as the coplanarity criterion for two lines.

Solution

Setup. Let $\bar r(s)$ be a $C^3$ curve parameterised by arc length, with Frenet frame $(\hat t,\hat n,\hat b)$ . The principal normal line at $P=\bar r(s)$ is $L_P=\{P+\lambda\hat n(s):\lambda\in\mathbb R\}$ ; at the consecutive point $Q=\bar r(s+ds)$ it is $L_Q$ .

Step 1 — First-order expansions.

Q-P\approx\hat t\,ds.

By Frenet–Serret, $\hat n'=-\kappa\hat t+\tau\hat b$ , so

\hat n(s+ds)\approx\hat n(s)+(-\kappa\hat t+\tau\hat b)\,ds.

Step 2 — Coplanarity criterion.

Two lines through $P,Q$ along $\hat n(s),\,\hat n(s+ds)$ intersect iff the scalar triple product $[Q-P,\;\hat n(s),\;\hat n(s+ds)]=0$ .

[\hat t\,ds,\;\hat n,\;\hat n+(-\kappa\hat t+\tau\hat b)\,ds]=[\hat t,\hat n,\hat n]\,ds+ds^2[-\kappa[\hat t,\hat n,\hat t]+\tau[\hat t,\hat n,\hat b]].

The first term vanishes (two equal vectors). The second:

[\hat t,\hat n,\hat t]=0,\qquad[\hat t,\hat n,\hat b]=\hat t\cdot(\hat n\times\hat b)=\hat t\cdot\hat t=1.

To leading non-vanishing order the triple product equals $\tau\,ds^2$ .

Conclusion. The two principal normal lines intersect iff $\tau\,ds^2=0$ , i.e., iff $\tau=0$ at that point.

Answer

\boxed{\;\text{Principal normals at consecutive points intersect only if }\tau=0.\;}