Notes on the helix curve

Parametric equations of a right-handed helix with radius \(R\), pitch \(P\), and axis along the positive \(z\)-direction are

(1)\[\begin{equation} x = R \cos t, \quad y = R \sin t, \quad z = \left(\frac{P}{2 \pi}\right) t. \end{equation}\]

The point \(r = \left( x, y, z\right)\) corresponding to \(t = 0\) is \(\left( R, 0, 0\right)\).

Arclength parametrized form of the above equations is

(2)\[\begin{equation} x(s) = R \cos \left( \frac{s}{c}\right), \quad y(s) = R \sin \left( \frac{s}{c}\right), \quad z(s) = \frac{bs}{c}, \end{equation}\]

where \(R > 0\) and \(c = \sqrt{R^2+b^2}\). The pitch \(P = 2\pi b\).

The equations for the tangent \(\bm{T}\), normal \(\bm{N}\), and binormal \(\bm{B}\) of the Frenet frame are

(3)\[\begin{align} \bm{T}(s) & = \left[ -\frac{R}{c} \sin \left(\frac{s}{c}\right), \frac{R}{c} \cos \left(\frac{s}{c}\right), \frac{b}{c} \right] \\ \bm{N}(s) & = \left[ -\cos \left(\frac{s}{c}\right), -\sin \left(\frac{s}{c}\right), 0 \right] \\ \bm{B}(s) & = \left[ \frac{b}{c} \sin \left(\frac{s}{c}\right), -\frac{b}{c} \cos \left(\frac{s}{c}\right), \frac{R}{c} \right]. \end{align}\]

The curvature \(\kappa(s)\) and torsion \(\tau(s)\) are

(4)\[\begin{align} \kappa(s) &= \lVert \bm{T}'(s)\rVert = \frac{R}{c^2} = \frac{R}{R^2+b^2} \\ \tau(s) &= \frac{b}{c^2} = \frac{b}{R^2+b^2}. \end{align}\]

At \(s = 0\), \(\bs{\beta}(s) = \left[x(s), y(s), z(s)\right] = \left[R, 0, 0\right]\) and

(5)\[\begin{align} \bm{T}(0) &= \left[0, \frac{R}{c}, \frac{b}{c} \right]\\ \bm{N}(0) &= \left[-1, 0, 0 \right]\\ \bm{B}(0) &= \left[0, -\frac{b}{c}, \frac{R}{c} \right]\\ \end{align}\]

For more details see Barrett O’Neill, Elementary Differential Geometry 2nd ed. p. 60.

If \(l\) is the curvature along \(s\) and \(m\) the twist along \(s\),

(6)\[\begin{gather} R = \frac{l}{l^2+m^2} \quad \mathrm{and} \quad b = \frac{m}{l^2+m^2} \\ l = \frac{R}{R^2+b^2} \quad \mathrm{and} \quad m = \frac{b}{R^2+b^2}. \end{gather}\]

Helix angle \(\tan \psi = P/2 \pi R \implies P = 2\pi R \tan \psi\).

Number of turns \(N = L/\sqrt{4\pi^2R^2+P^2}\), where \(L\) is the contour length of the helix.

Extent of the helix along its axis \(h = NP = PL/\sqrt{4\pi^2R^2+P^2}\).

Distance along the axis \(\delta = w/\cos \psi = w \sqrt{P^2+4\pi^2 R^2}/2\pi R\).

Condition for non-intersection. \(P \geq \delta \implies P \geq 2\pi R w /\sqrt{4\pi^2 R^2 - w^2}\).

Relations between radius/pitch and curvature/twist.

(7)\[\begin{gather} R = \frac{l}{l^2+m^2} \quad \mathrm{and} \quad P = \frac{2\pi m}{l^2+m^2} \\ l = \frac{4\pi^2 R}{4\pi^2 R^2 + P^2} \quad \mathrm{and} \quad m = \frac{2\pi P}{4 \pi^2 R^2 + P^2}. \end{gather}\]

Perpendicularity. If two helices on a cylinder are at right angles, \(\tan \psi_1 \tan \psi_2 = -1 \implies P_2 = -4\pi^2 R_1 R_2/P_1 = -4\pi^2 R^2/P_1\).

Helix axis from quaternion integration.

(8)\[\begin{equation} \hat{\bm{a}} = \left(-\frac{\upomega_1}{\omag}, 0, \frac{\upomega_3}{\omag} \right), \end{equation}\]

where \(\bs{\upomega} = \left( l, 0, -m \right)\).

Translation surface. Let \(H_1\) be a helix with parameters \(l_1\) and \(m_1\) and a director frame \(\left( \bm{d}_1, \bm{d}_2, \bm{d}_3\right)\). Let \(H_2\) another helix with handedness opposite to that of \(H_1\) with parameters \(l_2\) and \(m_2\). Then the curvature \(n\) along \(\bm{d}_1\) at any point on the midline of the translation surface formed by translating \(H_2\) along \(H_1\) is \(n = m^2/l\).