Spiral equations

Polar spirals

These spirals are usually described by a function in polar coordinates \(r = f(\theta)\).

Involute of a circle

Polar equation. \(r = b \sqrt{1+\theta^2}\), where \(b >= 0\) is the radius of the circle.

Arc length. \(s = \frac{b}{2} \theta^2 \bigl . \bigr \rvert_{\theta_0}^{\theta}\), from \(\theta_0\) to \(\theta\), where \(\theta > \theta_0\).

Arc length derivative. \(\frac{ds}{d\theta} = b \theta\)

Curvature. \(\kappa = \frac{1}{b \theta}\). The curvature goes to \(\infty\) as \(\theta \rightarrow 0\), i.e. at the point where the curve begins on the circle.

Tangent.

(1)\[\begin{align} t_x &= \frac{\theta \cos \theta - (1+\theta^2) \sin \theta} {\sqrt{\theta^2 + (1+\theta^2)^2}}\\ t_y &= \frac{\theta \sin \theta + (1+\theta^2) \cos \theta} {\sqrt{\theta^2 + (1+\theta^2)^2}} \end{align}\]

Generalized Archimedean spiral

Polar equation. \(r = b \theta^n\), where \(b >= 0\) and \(n\) is non-zero (\(n=0\) is a circle.)

Arc length. \(s = \frac{b}{n} \theta^n \lvert n\rvert \; _2F_1(-\frac{1}{2}, \frac{n}{2}; 1+\frac{n}{2}; -\frac{\theta^2}{n^2}) \bigl . \bigr \rvert_{\theta_0}^{\theta}\), from \(\theta_0\) to \(\theta\), where \(\theta > \theta_0\).

Arc length derivative. \(\frac{ds}{d\theta} = b \sqrt{\theta^{2n} + n^2\theta^{2n-2}}\)

Curvature. \(\kappa = \frac{\theta^{1-n}}{b} \frac{\theta^2+n^2+n}{\left(\theta^2+n^2\right)^{3/2}}\).

Tangent.

(2)\[\begin{split}\begin{aligned} t_x &= \frac{n \cos \theta - \theta \sin \theta} {\sqrt{n^2 + \theta^2}}\\ t_y &= \frac{n \sin \theta + \theta \cos \theta} {\sqrt{n^2 + \theta^2}} \end{aligned}\end{split}\]

For more details see Diedrichs (2019).

Archimedes spiral

Polar equation. \(r = b \theta\), where \(b >= 0\) and \(n\) is non-zero (\(n=0\) is a circle.)

Arc length. From \(\theta_0\) to \(\theta\), where \(\theta > \theta_0\) \( s = \Bigl . \frac{b}{2} \left\{ \theta \sqrt{1+\theta^2} + \log \left( \theta + \sqrt{1+\theta^2} \right) \right \} \Bigr \rvert_{\theta_0}^{\theta} \).

Arc length derivative. \(\frac{ds}{d\theta} = b \sqrt{1 + \theta^2}\)

Curvature. \(\kappa = \frac{\theta^2 + 2} {b\left(\theta^2 + 1\right)^{3/2}}\).

Tangent. Set \(n = 1\) in equation (2).

Fermat spiral

Polar equation. \(r = b \theta^{1/2}\), where \(b >= 0\) and \(n\) is non-zero (\(n=0\) is a circle.)

Arc length. \(s = \left . b \sqrt{\theta} \; _2F_1\left(-\frac{1}{2}, \frac{1}{4}; \frac{5}{4}; -4\theta^2\right) \right \rvert_{\theta_0}^{\theta}\), from \(\theta_0\) to \(\theta\), where \(\theta > \theta_0\).

Arc length derivative. \(\frac{ds}{d\theta} = \frac{b}{2} \sqrt{4 \theta + \frac{1}{\theta}}\)

Curvature. \(\kappa = 2 r \frac{4 r^4 + 3 b^4} {\left(4r^4 + b^4\right)^{3/2}}\).

Tangent. Set \(n = \frac{1}{2}\) in equation (2).

Hyperbolic spiral

Polar equation. \(r = b \theta^{-1}\), where \(b >= 0\) and \(n\) is non-zero (\(n=0\) is a circle.)

Arc length. \(s = \left . b \left\{-\frac{\sqrt{1+\theta^2}}{\theta} + \log \left(\theta + \sqrt{1+\theta^2} \right) \right\} \right \rvert_{\theta}^{\theta_0}\), from \(\theta_0\) to \(\theta\), where \(\theta < \theta_0\).

Arc length derivative. \(\frac{ds}{d\theta} = b \frac{\sqrt{1+\theta^2}}{\theta^2}\).

Curvature. \(\kappa = \frac{\theta^4}{b \left(1+\theta^2\right)^{3/2}}\).

Tangent. Set \(n = -1\) in equation (2).

Lituus

Polar equation. \(r = b \theta^{-1/2}\), where \(b >= 0\) and \(n\) is non-zero (\(n=0\) is a circle.)

Arc length. \(s = \left . 2 \sqrt{\theta}\; _2F_1\left( -\frac{1}{2}, -\frac{1}{4}; \frac{3}{4}; -\frac{1}{4\theta^2} \right) \right \rvert_{\theta}^{\theta_0}\), from \(\theta_0\) to \(\theta\), where \(\theta < \theta_0\).

Arc length derivative. \(\frac{ds}{d\theta} = \frac{1}{2\theta} \sqrt{\frac{1}{\theta}+4\theta}\).

Curvature. \(\kappa = \left(8 \theta^2 - 2\right) \left(\frac{\theta}{1+4\theta^2}\right)^{3/2}\). There is a point of inflexion at \(\theta = \frac{1}{2}\).

Tangent. Set \(n = -\frac{1}{2}\) in equation (2).

Cartesian spirals

These spirals are unsually given in terms of a Césaro equation \(\kappa = f(s)\).

Polynomial spiral

Césaro equation. \(\kappa = P(s)\), where \(P(s)\) is a polynomial in \(s\).

Tangent. The Whewell equation gives a relation between the tangential angle and the arc length: \(\frac{d \psi}{ds} = \kappa(s)\), integrating which we obtain \(\psi = \int_{s_0}^s \kappa(s)\). The tangent given by

(3)\[\begin{gather} t_x = \frac{dx}{ds} = \cos \psi \\ t_y = \frac{dy}{ds} = \sin \psi \end{gather}\]

Cartesian coordinates. The equations for the cartesian coordinates are

(4)\[\begin{align} x &= \int_{s_0}^s \cos \psi\;\mathrm{d}s \\ y &= \int_{s_0}^s \sin \psi\;\mathrm{d}s. \end{align}\]

For more details see Dillen (1990.

Cornu spiral

Césaro equation. \(\kappa = a s\).

Tangent.

(5)\[\begin{align} t_x &= \cos \left(\frac{1}{2} a s^2 \right) \\ t_y &= \sin \left(\frac{1}{2} a s^2 \right) \end{align}\]

Cartesian coordinates.

(6)\[\begin{align} x &= m\; C( s/m ) \\ y &= m\; S( s/m ) \end{align}\]

where \(m = \sqrt{\pi/a}\) and

(7)\[\begin{align} S(z) &= \int_0^z \sin \left(\frac{\pi t^2}{2}\right)\;\mathrm{d}t\\ C(z) &= \int_0^z \cos \left(\frac{\pi t^2}{2}\right)\;\mathrm{d}t \end{align}\]

are the Fresnel integrals. The SciPy implementation is scipy.special.fresnel.

Nielsen spiral

Césaro equation. \(\kappa = a \exp \left( b s \right)\)., where \(b\) may be positive or negative.

Tangent.

(8)\[\begin{align} t_x &= \cos \psi \\ t_y &= \sin \psi, \end{align}\]

where the tangential angle \(\psi = \frac{a}{b} e^{bs} + \frac{\pi}{2}\). The constant \(\frac{\pi}{2}\) comes from integrating the Whewell equation with a convenient starting point \(s_0\).

Cartesian coordinates.

(9)\[\begin{align} x &= -\frac{1}{b}\left(\mathrm{Si} - \mathrm{Si_0} \right) \\ y &= \frac{1}{b}\left(\mathrm{Ci} - \mathrm{Ci_0} \right) \end{align}\]

where \(\mathrm{Si_0}, \mathrm{Ci_0} = \mathtt{sici(a/b)}\) and \(\mathrm{Si}, \mathrm{Ci} = \mathtt{ sici((a/b)*exp(b*s)) }\). \(\mathtt{sici}\) are sici integrals as implemented in the SciPy function scipy.special.sici.