Vector Calculus | Part IA, 2005

Given a curve γ(s)\gamma(s) in R3\mathbb{R}^{3}, parameterised such that γ(s)=1\left\|\gamma^{\prime}(s)\right\|=1 and with γ(s)0\gamma^{\prime \prime}(s) \neq 0, define the tangent t(s)\mathbf{t}(s), the principal normal p(s)\mathbf{p}(s), the curvature κ(s)\kappa(s) and the binormal b(s)\mathbf{b}(s).

The torsion τ(s)\tau(s) is defined by

τ=bp\tau=-\mathbf{b}^{\prime} \cdot \mathbf{p}

Sketch a circular helix showing t,p,b\mathbf{t}, \mathbf{p}, \mathbf{b} and b\mathbf{b}^{\prime} at a chosen point. What is the sign of the torsion for your helix? Sketch a second helix with torsion of the opposite sign.

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