Lituus Curve

Calculus Curves >

The lituus is a transcendental curve that can be graphed with the polar equation
r = k/√θ [1]
or, equivalently,
r2 = a2/θ.

Lituus with k = 5.

The inverse of the lituus is Fermat’s spiral — a type of Archimedean spiral (the littus is the inverse of Fermat’s Spiral only when the inversion is take as the origin). However, the lituus itself isn’t technically a spiral because its curvature doesn’t strictly increase or decrease as a function of its arc length.

The curve can also be described as the locus of the point P moving so that the area of a circular sector — a wedge of a circle with a central angle of less than π radians — remains constant.

The name Lituus means a ‘crook,’ like a bishop’s crosier (a staff without knots and curved at the top) [2].
A crosier.

History of the Littus Curve

The curve was first described by Roger Cotes in a paper collection titled Harmonia Mensurarum, published in 1722 — six years after his death. He is well known for editing the second edition of Newton’s Principia [3].

Maclaurin also used the term lituus in his book Harmonia Mensurarum in 1722.


Image created with Desmos.
[1] Dunham, D. Hyperbolic Spirals and Spiral Patterns. Retrieved July 30, 2022 from:
[2] Kokosa, S. Fifty Famous Curves, Lots of Calculus Questions, And a Few Answers. Retrieved July 30, 2021 from:
[3] MacTutor. Roger Coates. Retrieved July 29, 2022 from:

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