Lituus (mathematics)
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

 

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 of the lituus spiral can be determined using the formula

References

External links
 . Interactive example using JSXGraph. . https://hsm.stackexchange.com/a/3181 on the history of the lituus curve. Spirals
Plane curves