300px, Branch for positive
In mathematics, a lituus is a
spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:[polar equation
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...]
:
where is any non-zero constant.
Thus, the
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
is
inversely proportional
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
to the square of the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
.
This spiral, which has two branches depending on the sign of , is
asymptotic
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
to the -axis. Its
points of inflexion are at
:
The curve was named for the ancient Roman
lituus
The word ''lituus'' originally meant a curved augural staff, or a curved war-trumpet in the ancient Latin language. This Latin word continued in use through the 18th century as an alternative to the vernacular names of various musical instruments ...
by
Roger Cotes
Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadratur ...
in a collection of papers entitled ''Harmonia Mensurarum'' (1722), which was published six years after his death.
External links
*
*
Interactive example using JSXGraph*
* https://hsm.stackexchange.com/a/3181 on the history of the lituus curve.
Spirals
Plane curves
{{geometry-stub
