# Lituus (mathematics)

The lituus spiral (/ˈlɪtju.əs/) is a spiral in which the angle θ is inversely proportional to the square of the radius r.

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

${\displaystyle (\theta ,r)=\left({\tfrac {1}{2}},\pm {\sqrt {2k}}\right).}$

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.

## Coordinate representations

### Polar coordinates

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

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

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:

{\displaystyle {\begin{aligned}x&={\frac {a}{\sqrt {\theta }}}\cos \theta ,\\y&={\frac {a}{\sqrt {\theta }}}\sin \theta .\\\end{aligned}}}

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

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

## Geometrical properties

### Curvature

The curvature of the lituus spiral can be determined using the formula[1]

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

### 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:

${\displaystyle L=2{\sqrt {\theta }}\cdot \operatorname {_{2}F_{1}} \left(-{\frac {1}{2}},-{\frac {1}{4}};{\frac {3}{4}};-{\frac {1}{4\theta ^{2}}}\right)-2{\sqrt {\theta _{0}}}\cdot \operatorname {_{2}F_{1}} \left(-{\frac {1}{2}},-{\frac {1}{4}};{\frac {3}{4}};-{\frac {1}{4\theta _{0}^{2}}}\right),}$

where the arc length is measured from θ = θ0.[1]

### Tangential angle

The tangential angle of the lituus spiral can be determined using the formula[1]

${\displaystyle \phi =\theta -\arctan 2\theta .}$

## References

1. ^ a b c Weisstein, Eric W. "Lituus". MathWorld. Retrieved 2023-02-04.