Lituus (mathematics)

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.

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:

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

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

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

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]

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

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