Cusp

A cusp is a sharp point or apex, such as occurs in two dimensions at the end of a crescent, or in three dimensions at the tip of a cone or horn. There are a number of technical terms derived from it.

In mathematics, a cusp might mean

• a singular point of a curve such as that seen near (0,0) for

y = x^2/3;

• a conical singular point seen in hyperbolic geometry;
• a missing point on a modular curve that has to be supplied by compactification.

The term cusp form in modular form theory refers to the third of these meanings, implying a form that vanishes at cusps. There is an intricate theory of the connection of cuspidal phenomena with parabolic subgroups. Note, however, that hyperbolic geometry, Riemann surfaces and modular forms are related; in this case, a cusp can be visualized as a cone-shaped protrusion, and be given a precise geometrical definition through a cusp neighborhood.

In a separate field, cusp is a term in astrology for the boundary day(s) between zodiac signs.