# Lituus (mathematics)

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In mathematics, a lituus is a spiral with polar equation

${\displaystyle r^{2}\theta =k}$

where k is any non-zero constant. Thus, the angle θ is inversely proportional to the square of the radius r.

This spiral, which has two branches depending on the sign of ${\displaystyle r}$, is asymptotic to the ${\displaystyle x}$ axis. Its points of inflexion are at ${\displaystyle (\theta ,r)=({\tfrac {1}{2}},{\sqrt {2k}})}$ and ${\displaystyle ({\tfrac {1}{2}},-{\sqrt {2k}})}$.

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.

• "Lituus", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• Interactive example using JSXGraph
• O'Connor, John J.; Robertson, Edmund F., "Lituus", MacTutor History of Mathematics archive , University of St Andrews .
• https://hsm.stackexchange.com/a/3181 on the history of the lituus curve.