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 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.

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: These equations can in turn be rearranged to an equation in x and y: The curvature of the lituus spiral can be determined using the formula[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: where the arc length is measured from θ = θ0.

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