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 , {\displaystyle f(x,y)=0,} where f is a polynomial function f : R 2 → R . {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} .} If f is expanded as
f = a 0 + b 0 x + b 1 y + c 0 x 2 + 2 c 1 x y + c 2 y 2 + ⋯ {\displaystyle f=a_{0}+b_{0}x+b_{1}y+c_{0}x^{2}+2c_{1}xy+c_{2}y^{2}+\cdots }
If the origin (0, 0) is on the curve then a0 = 0. If b1 ≠ 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 b0 ≠ 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 {\displaystyle \mathbb {R} } to the plane which defines the curve in the neighborhood of the origin. Note that at the origin
b 0 = ∂ f ∂ x , b 1 = ∂ f ∂ y , {\displaystyle b_{0}={\frac {\partial f}{\partial x}},\;b_{1}={\frac {\partial f}{\partial y}},}
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,
f ( x , y ) = ∂ f ∂ x = ∂ f ∂ y = 0. {\displaystyle f(x,y)={\frac {\partial f}{\partial x}}={\frac {\partial f}{\partial y}}=0.}
Regular points Assume the curve passes through the origin and write y = m x . {\displaystyle y=mx.} Then f can be written
f = ( b 0 + m b 1 ) x + ( c 0 + 2 m c 1 + c 2 m 2 ) x 2 + ⋯ . {\displaystyle f=(b_{0}+mb_{1})x+(c_{0}+2mc_{1}+c_{2}m^{2})x^{2}+\cdots .}
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