Lituus (mathematics)

The representations of the lituus spiral in polar coordinates ${\displaystyle \left(r,\,\theta \right)}$ is given by the equation...

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

where ${\displaystyle \theta \geq 0}$ and ${\displaystyle k\neq 0}$.

The Lituus spiral with the polar coordinates ${\displaystyle r={\tfrac {a}{\sqrt {\theta }}}}$ can be converted to Cartesian coordinates like any other spiral with the relationships ${\displaystyle x=r\cdot \cos \left(\theta \right)}$ and ${\displaystyle y=r\cdot \sin \left(\theta \right)}$. With this conversion we get the parametric representations of the curve

${\displaystyle {\begin{aligned}x&={\frac {a}{\sqrt {\theta }}}\cdot \cos \left(\theta \right),\,\\y&={\frac {a}{\sqrt {\theta }}}\cdot \sin \left(\theta \right).\,\\\end{aligned}}}$

These equations can in turn be rearranged in such a way that an equation consisting solely of terms with 𝑥 and "𝑦" is created, which describes the lituus spiral:

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

1. Divide ${\displaystyle y}$ by ${\displaystyle x}$: ${\displaystyle {\frac {y}{x}}={\tfrac {{\frac {a}{\sqrt {\theta }}}\cdot \sin \left(\theta \right)}{{\frac {a}{\sqrt {\theta }}}\cdot \cos \left(\theta \right)}}\Rightarrow {\frac {y}{x}}=\tan \left(\theta \right)}$
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 ${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$: ${\displaystyle {\frac {y}{x}}=\tan \left({\frac {a^{2}}{\left({\sqrt {x^{2}+y^{2}}}\right)^{2}}}\right)\Rightarrow {\frac {y}{x}}=\tan \left({\frac {a^{2}}{x^{2}+y^{2}}}\right)}$

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

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

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:

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

Where the arc length is measured from ${\displaystyle \theta =\theta _{0}}$.[1]

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

${\displaystyle \phi =\theta -\arctan \left(2\cdot \theta \right)}$.[1]