Lituus (mathematics)

The lituus spiral is a spiral in which the angle $r$ is inversely proportional to the square of the radius $θ$.

This spiral, which has two branches depending on the sign of $r$, is asymptotic to the $r$ axis. Its points of inflexion are at


 * $$(\theta, r) = \left(\tfrac12, \pm\sqrt{2k}\right).$$

The curve was named for the ancient Roman lituus by Roger Cotes in a collection of papers entitled Harmonia Mensurarum (1722), which was published six years after his death.

Polar coordinates
The representations of the lituus spiral in polar coordinates $(r, θ)$ is given by the equation


 * $$r = \frac{a}{\sqrt{\theta}},$$

where $θ ≥ 0$ and $k ≠ 0$.

Cartesian coordinates
The lituus spiral with the polar coordinates $r = a⁄√θ$ can be converted to Cartesian coordinates like any other spiral with the relationships $x = r cos θ$ and $y = r sin θ$. With this conversion we get the parametric representations of the curve:


 * $$\begin{align}

x &= \frac{a}{\sqrt{\theta}} \cos\theta, \\ y &= \frac{a}{\sqrt{\theta}} \sin\theta. \\ \end{align}$$

These equations can in turn be rearranged to an equation in $x$ and $x$:


 * $$\frac{y}{x} = \tan\left( \frac{a^2}{x^2 + y^2} \right).$$


 * 1) Divide $$y$$ by $$x$$:$$\frac{y}{x} = \frac{\frac{a}{\sqrt{\theta}} \sin\theta}{\frac{a}{\sqrt{\theta}} \cos\theta} \Rightarrow \frac{y}{x} = \tan\theta.$$
 * 2) Solve the equation of the lituus spiral in polar coordinates: $$r = \frac{a}{\sqrt{\theta}} \Leftrightarrow \theta = \frac{a^2}{r^2}.$$
 * 3) Substitute $$\theta = \frac{a^2}{r^2}$$: $$\frac{y}{x} = \tan\left( \frac{a^2}{r^2} \right).$$
 * 4) Substitute $$r = \sqrt{x^2 + y^2}$$: $$\frac{y}{x} = \tan\left( \frac{a^2}{\left( \sqrt{x^2 + y^2} \right)^2} \right) \Rightarrow \frac{y}{x} = \tan\left( \frac{a^2}{x^2 + y^2} \right).$$

Curvature
The curvature of the lituus spiral can be determined using the formula


 * $$\kappa = \left( 8 \theta^2 - 2 \right) \left( \frac{\theta}{1 + 4 \theta^2} \right)^\frac23.$$

Arc length
In general, the arc length of the lituus spiral cannot be expressed as a closed-form expression, but the arc length of the lituus spiral can be represented as a formula using the Gaussian hypergeometric function:


 * $$L = 2 \sqrt{\theta} \cdot \operatorname{_2 F_1}\left( -\frac{1}{2}, -\frac{1}{4}; \frac{3}{4}; -\frac{1}{4 \theta^2} \right) - 2 \sqrt{\theta_0} \cdot \operatorname{_2 F_1}\left( -\frac{1}{2}, -\frac{1}{4}; \frac{3}{4}; -\frac{1}{4 \theta_0^2} \right),$$

where the arc length is measured from $θ = θ_{0}$.

Tangential angle
The tangential angle of the lituus spiral can be determined using the formula


 * $$\phi = \theta - \arctan 2\theta.$$