The lituus spiral (/ˈlɪtju.əs/) is a spiral in which the angle θ is inversely proportional to the square of the radius r.

Branch for positive 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.

Coordinate representations

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Polar coordinates

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The representations of the lituus spiral in polar coordinates (r, θ) is given by the equation

 

where θ ≥ 0 and k ≠ 0.

Cartesian coordinates

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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:

 
Derivation of the equation in Cartesian coordinates
  1. Divide   by  : 
  2. Solve the equation of the lituus spiral in polar coordinates:  
  3. Substitute  :  
  4. Substitute  :  

Geometrical properties

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Curvature

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

 

Arc length

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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]

Tangential angle

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

 

References

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  1. ^ a b c Weisstein, Eric W. "Lituus". MathWorld. Retrieved 2023-02-04.
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