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

(\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

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

where $θ \geq 0$ and $k \neq 0$ .

Cartesian coordinates

The lituus spiral with the polar coordinates $r = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}a/√θ$ 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{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 $y$ :

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

Derivation of the equation in Cartesian coordinates

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

Geometrical properties

Curvature

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

\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:

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 $θ = θ 0$ .^[1]

Tangential angle

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

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