The lituus spiral () is a spiral in which the polar 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
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
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:
These equations can in turn be rearranged to an equation in and :
Divide by : Solve the equation of the lituus spiral in polar coordinates: Substitute : Substitute :
Geometrical properties
Curvature The curvature of the lituus spiral can be determined using the formula
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:
where the arc length is measured from .
Tangential angle The tangential angle (see Pitch angle of a spiral) of the lituus spiral can be determined using the formula
References
External links
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Interactive example using JSXGraph. . https://hsm.stackexchange.com/a/3181 on the history of the lituus curve.