Alysoid definition, graph & formula

Calculus Curves >

The alysoid (from the Greek Alusion, meaning “little chain”) is a transcendental curve first studied by Italian mathematician Ernesto Cesàro in 1886.
alysoid graph

The curve can be formed by a flexible chain of an infinite number of links, hanging by its own weight [1], but there are other ways to form the curve. More precisely, it is a curve where the center of curvature describes a parabola rolling over — and perpendicular to — a straight line.

The curve is sometimes called a catenary, which is actually a special case of the alysoid [2]. In other words, the alysoid could more accurately be described as a generalization of the catenary.

Equations for the Alysoid

Intrinsic equations:

  1. aRc = s2 + b2; b ≠ 0.
  2. s = b tan(kφ), k = b/a.

The form: s = tan kφ, where s is the arc length, is called the intrinsic Whewell equation.

Cartesian parameterization:
alysoid cartesian equation

Relationship to some other curves

  • The alysoid is similar to, but not the same as a catenary. The alysoid is a catenary if a = b (k = 1).
  • When b is zero, the curve becomes a particular case of pseudo-spiral of Pirondini, called antiloga.


[1] Schwartzman, S. (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms. Mathematical Association of America.
[2] Yates, R. Curves and their properties. National Council of Teachers of Mathematics, Washington D.S. Classics in Mathematics Education, Volume 4 (1906) from:

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