Cusp(2)

cuspidal point

A singular point of a curve, the two branches of which have a common semi-tangent there. In the case of a plane curve one distinguishes cusps of the first and the second kind. In the former case the curve lies on one side of the tangent cone (Fig.a); in the second, on different sides (Fig.b).

Figure: c027420a

Figure: c027420b

Comments

In the above the word "branch" is used in a naive and non-technical sense as follows. View a curve as the image of a finite or infinite interval in Euclidean space . Let be a single-valued analytic function defined on some interval. If (or ) defines a subset of , one speaks of a branch of . Here is taken for convenience. There is a second more technical (and more precise) notion of a branch in algebraic and analytic geometry which defines the branches at a point as the points above on the normalization of the curve (cf. Normal scheme). Using this concept a cusp is a singular point of a curve which has only one branch at this point.

A curve with a cusp of the first kind (Fig.a) is, e.g., , and one with a cusp of the second kind (Fig.b) — e.g. .

The word "cusp" is also used in the theory of modular forms (see Fuchsian group; Modular form).

References

[a1]	R.J. Walker, "Algebraic curves" , Springer (1978)