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
:
where is any non-zero constant.
Thus, the
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
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 const ...
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
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...
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.