# Lituus

## History

The Lituus curve is studied by Roger Cotes in 1722 [Robert C Yates 1952].

The word lituus means a curved staff used by the augurs in quartering the heavens. (See: crosier), or A kind of trumpet of a somewhat curved form and shrill note.

## Description

Lituus is a spiral described by the polar equation r == 1/Sqrt[θ].

The curve is asymptotic to the positive x-axis, and the other end spiral in towards the pole. The above image is a plot from 0.1 to 20*π. As θ approachs infinity, the curve approaches the origin.

## Formula

Polar equation: r == 1/Sqrt[θ].

## Properties

It has the property that a circular sector produces the same area. That is, suppose P is a point on the curve, and X a point on the asymptote OP distance from the origin O. Suppose the area of the circular sector OPX is A. As P moves towards the center on the curve, the area remains the same.

### inverse of parabolic spiral

The inverse of Lituus with respect to the center is the parabolic spiral.

### misc

The lituus spiral is a recurring shape in art called Volute.

### See Also

Archimedes' spiral, equiangular spiral, Mathematics of Seashell Shapes.

## Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

Carving the scroll in making the violin: http://www.violins.demon.co.uk/making/carvescroll.htm.