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

(\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.

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 $a \neq 0$ .

Cartesian coordinates

The lituus spiral with the polar coordinates $r = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠a/√θ⁠$ 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 $\theta ={\frac {a^{2}}{r^{2}}}$ : ${\frac {y}{x}}=\tan \left({\frac {a^{2}}{r^{2}}}\right).$
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^[1]

\kappa =\left(8\theta ^{2}-2\right)\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 \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 .