Cusp (singularity)
In mathematics a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point on the curve must start to move backward. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.
For a plane curve defined by a differentiable parametric equation
a cusp is a point where both derivatives of f and g are zero, and at least one of them changes sign. Cusps are local singularities in the sense that they involve only one value of the parameter t, contrarily to self-intersection points that involve several values.
For a curve defined by an implicit equation
cusps are points where the terms of lowest degree of the Taylor expansion of F are a power of a linear polynomial; however not all singular points that have this property are cusps. In some contexts, and in the remainder of this article, one restricts the definition of a cusp to the case where the non-zero part of lowest degree of the Taylor expansion of F has degree two.
A plane curve cusp may be put in one of the following forms by a diffeomorphism of the plane: x2 − y2k+1 = 0, where k ≥ 1 is an integer.
Consider a smooth real-valued function of two variables, say f(x, y) where x and y are real numbers. So f is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target. This action splits the whole function space up into equivalence classes, i.e. orbits of the group action.
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