Cusp

In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

More literally, 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 = x2/3;

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• 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.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Dimension: Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space....

Crescent: :For the album of the same name by Gackt, see Crescent (album). For the passenger train, see Crescent (Amtrak)....

Cone: A cone is a basic geometrical shape: see cone (solid). Several things have also been called "cones" on account of their shape:...