Double point
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 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 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 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,
Assume the curve passes through the origin and write y=mx. Then f can be written
If b0+mb1 is not 0 then f=0 has a solution of multiplicity 1 at x=0 and the origin is a point of single contact with line y=mx. If b0+mb1=0 then f=0 has a solution of multiplicity 2 or higher and the line y=mx, or b0x+b1y=0, is tangent to the curve. In this case, if c0+2mc1+c2m2 is not 0 then the curve has a point of double contact with y=mx. If the coefficient of x2, c0+2mc1+c2m2, is 0 but the coefficient of x3 is not then the origin is a point of inflection of the curve. If the coefficient of x2 and x3 are both 0 then the origin is called point of undulation of the curve. This analysis can be applied to any point on the curve by translating the coordinate axes so that the origin is at the given point.
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