Lituus (mathematics)

Polar coordinates The representations of the lituus spiral in polar coordinates is given by the equation : r = \frac{a}{\sqrt{\theta}}, where and . Cartesian coordinates The lituus spiral with the polar coordinates can be converted to Cartesian coordinates like any other spiral with the relationships and . With this conversion we get the parametric representations of the curve: : \begin{align} x &= \frac{a}{\sqrt{\theta}} \cos\theta, \\ y &= \frac{a}{\sqrt{\theta}} \sin\theta. \\ \end{align} These equations can in turn be rearranged to an equation in and : : \frac{y}{x} = \tan\left( \frac{a^2}{x^2 + y^2} \right). • 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 : \kappa = \left( 8 \theta^2 - 2 \right) \left( \frac{\theta}{1 + 4 \theta^2} \right)^\frac32. 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 . Tangential angle The tangential angle of the lituus spiral can be determined using the formula : \phi = \theta - \arctan 2\theta. ==References==