Lituus (mathematics)

Branch for positive Template:Mvar

The lituus spiral (Template:IPAc-en) is a spiral in which the angle Template:Mvar is inversely proportional to the square of the radius Template:Mvar.

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

$(\theta ,r)=(\frac{1}{2},\pm \sqrt{2k}).$

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.

Coordinate representations

Polar coordinates

The representations of the lituus spiral in polar coordinates Template:Math is given by the equation

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

where Template:Math and Template:Math.

Cartesian coordinates

The lituus spiral with the polar coordinates Template:Math can be converted to Cartesian coordinates like any other spiral with the relationships Template:Math and Template:Math. With this conversion we get the parametric representations of the curve:

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

These equations can in turn be rearranged to an equation in Template:Mvar and Template:Mvar:

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

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: $r=\frac{a}{\sqrt{\theta }}\iff \theta =\frac{a^2}{r^2}.$
3. Substitute $\theta =\frac{a^2}{r^2}$: $\frac{y}{x}=\tan \left(\frac{a^2}{r^2}\right).$
4. Substitute $r=\sqrt{x^2+y^2}$: $\frac{y}{x}=\tan \left(\frac{a^2}{{\left(\sqrt{x^2+y^2}\right)}^2}\right)\Rightarrow \frac{y}{x}=\tan \left(\frac{a^2}{x^2+y^2}\right).$

Geometrical properties

Curvature

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

$\kappa =(8{\theta }^2-2){\left(\frac{\theta }{1+4{\theta }^2}\right)}^{\frac{3}{2}}.$

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:

$L=2\sqrt{\theta }\cdot {}_{\mathrm{2}}{\mathrm{F}}_{\mathrm{1}}(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{1}{4{\theta }^2})-2\sqrt{{\theta }_0}\cdot {}_{\mathrm{2}}{\mathrm{F}}_{\mathrm{1}}(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{1}{4{\theta }_0^2}),$

where the arc length is measured from Template:Math.

Tangential angle

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

$\varphi =\theta -\arctan 2\theta .$

External links

• Template:Springer.
• Template:Mathworld
• Interactive example using JSXGraph.
• Template:MacTutor.
• https://hsm.stackexchange.com/a/3181 on the history of the lituus curve.

Template:Geometry-stub