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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 and are zero, and at least one of them changes of sign. Cusps are ''local singularities'' in the sense that they involve only one value of the parameter , 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 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 has degree two. A plane curve cusp may be put in one of the following forms by a diffeomorphism of the plane: ''x''2 − ''y''2''k''+1 = 0, where ''k'' ≥ 1 is an integer. == Classification in differential geometry == 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. One such family of equivalence classes is denoted by ''Ak''±, where ''k'' is a non-negative integer. This notation was introduced by V. I. Arnold. A function ''f'' is said to be of type ''Ak''± if it lies in the orbit of ''x''2 ± ''y''''k''+1, i.e. there exists a diffeomorphic change of coordinate in source and target which takes ''f'' into one of these forms. These simple forms ''x''2 ± ''y''''k''+1 are said to give normal forms for the type ''A''k''''±-singularities. Notice that the ''A''2''n''+ are the same as the ''A''2''n''− since the diffeomorphic change of coordinate (''x'',''y'') → (''x'', −''y'') in the source takes ''x''2 + ''y''2''n''+1 to ''x''2 − ''y''2''n''+1. So we can drop the ± from ''A''2''n''± notation. The cusps are then given by the zero-level-sets of the representatives of the ''A''2''n'' equivalence classes, where ''n'' ≥ 1 is an integer. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cusp (singularity)」の詳細全文を読む スポンサード リンク
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