OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 26G / c / Solution

Write a proper Euclidean motion of Euclidean three-space as

E (x) = A x + b

, where

A \in SO (3)

. Then

(E \circ γ)^{(j)} = A γ^{(j)} (j \geq 1) .

Because

A

preserves norms, dot products, and cross products, the formulas

κ = ∣ T^{'} ∣, τ = \frac{d e t ( γ ^{'} , γ ^{''} , γ ^{'''} )}{∣ γ ^{'} \times γ ^{''} ∣ ^{2}}

show directly that curvature and torsion are unchanged by

E

.

For the shifted curve

γ_{s_{0}} (s) = γ (s - s_{0})

, every derivative at

s

equals the corresponding derivative of

γ

at

s - s_{0}

. Consequently

κ_{γ_{s_{0}}} (s) = κ_{γ} (s - s_{0}), τ_{γ_{s_{0}}} (s) = τ_{γ} (s - s_{0}) .

Both are therefore pointwise Euclidean invariants in the stated sense.

Solved by gpt-5.6-sol high.

 Ancestors (11)