Lituus (mathematics) explained

thumb|300px|Branch for positive The lituus spiral is a spiral in which the angle is inversely proportional to the square of the radius .

This spiral, which has two branches depending on the sign of, is asymptotic to the axis. Its points of inflexion are at

(\theta, r) = \left(\tfrac 12 , \pm\sqrt{ 2 k}\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.

Coordinate representations

Polar coordinates

The representations of the lituus spiral in polar coordinates is given by the equation

r = (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 & = (a)/(\sqrt{\theta) } \cos\theta, \\ y &= \frac \sin\theta. \\\end

These equations can in turn be rearranged to an equation in and :

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

1. Divide

y by x :(y)/(x) = ((a)/(\sqrt{\theta))/(\sin\theta}{ (a)/(\sqrt{\theta)) } \cos\theta} \Rightarrow \frac = \tan\theta.

1. Solve the equation of the lituus spiral in polar coordinates:

r = (a)/(\sqrt{\theta) } \Leftrightarrow \theta = \frac.

1. Substitute

\theta = (a ^{2})/(r ^{2}) : (y)/(x) = \tan\left( (a ^{2})/(r ^{2}) \right).

1. Substitute

r = \sqrt{x ^{2} + y ^{2}} : (y)/(x) = \tan\left( (a ^{2})/(\left( \sqrt{x ^{2} + y ^{2}) \right) ^{2}} \right) ⇒ (y)/(x) = \tan\left( (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( (\theta)/(1+4 \theta ^{2}) \right) ^{(23 .)/()}

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} ⋅ \operatorname{ _{2} F _{1}\left(} - (1)/(2) , - (1)/(4) ; (3)/(4) ; - (1)/(4 \theta ^{2}) \right) - 2 \sqrt{\theta _{0}} ⋅ \operatorname{ _{2} F _{1}\left(} - (1)/(2) , - (1)/(4) ; (3)/(4) ; - (1)/(4 \theta ^{2} _{0}) \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.

External links

• .
  • Interactive example using JSXGraph.
• .
• https://hsm.stackexchange.com/a/3181 on the history of the lituus curve.