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

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

*a*R_{c}=*s*^{2}+*b*^{2};*b*≠ 0.*s*=*b*tan(kφ),*k*=*b*/*a*.

The form: *s *= tan *k*Ï†, where s is the arc length, is called the *intrinsic Whewell 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.

## References

[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: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.853.7639&rep=rep1&type=pdf