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In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. ==Algebraic curves in the plane== Algebraic curves in the plane may be defined as the set of points (''x'', ''y'') satisfying an equation of the form ''f''(''x'', ''y'')=0, where ''f'' is a polynomial function ''f'':R2→R. If ''f'' is expanded as : If the origin (0, 0) is on the curve then ''a''0=0. If ''b''1≠0 then the implicit function theorem guarantees there is a smooth function ''h'' so that the curve has the form ''y''=''h''(''x'') near the origin. Similarly, if ''b''0≠0 then there is a smooth function ''k'' so that the curve has the form ''x''=''k''(''y'') near the origin. In either case, there is a smooth map from R to the plane which defines the curve in the neighborhood of the origin. Note that at the origin : so the curve is non-singular or ''regular'' at the origin if at least one of the partial derivatives of ''f'' is non-zero. The singular points are those points on the curve where both partial derivatives vanish, : 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「singular point of a curve」の詳細全文を読む スポンサード リンク
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