The mechanical properties of the curves are treated in the article [Mechanics], where various forms are illustrated. The simple catenary is shown in the figure. The cartesian equation referred to the axis and directrix is y = c cosh (x/c) or y = ½c(ex/c + e−x/c); other forms are s = c sinh (x/c) and y² = c² + s², s being the arc measured from the vertex; the intrinsic equation is s = c tan ψ. The radius of curvature and normal are each equal to c sec² ψ.
The surface formed by revolving the catenary about its directrix is named the alysseide. It is a minimal surface, i.e. the catenary solves the problem: to find a curve joining two given points, which when revolved about a line co-planar with the points traces a surface of minimum area (see [Variations, Calculus of]).
The involute of the catenary is called the tractory, tractrix or antifriction curve; it has a cusp at the vertex of the catenary, and is asymptotic to the directrix. The cartesian equation is
x = √ (c² − y²) + ½c log [ {c − √ (c² − y²)} / {c + √ (c² + y²)} ],
and the curve has the geometrical property that the length of its tangent is constant. It is named the tractory, since a weight placed on the ground and drawn along by means of a flexible string by a person travelling in a straight line, the weight not being in this line, describes the curve in question. It is named the antifriction curve, since a pivot and step having the form of the surface generated by revolving the curve about its vertical axis wear away equally (see [Mechanics]: Applied).
CATERAN (from the Gaelic ceathairne, a collective word meaning “peasantry”), the band of fighting men of a Highland clan; hence the term is applied to the Highland, and later to any, marauders or cattle-lifters.