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Algebraic curves in the plane may be defined as the set of points (x, y) satisfying an equation of the form (,) =, where f is a polynomial function :. If f is expanded as = + + + + + + If the origin (0, 0) is on the curve then a 0 = 0.
In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. 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 an analytic, parametric equation
The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y 2 − x 3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at ...
Point a is an ordinary point when functions p 1 (x) and p 0 (x) are analytic at x = a. Point a is a regular singular point if p 1 (x) has a pole up to order 1 at x = a and p 0 has a pole of order up to 2 at x = a. Otherwise point a is an irregular singular point.
It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves. It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
Singular point of a curve, where the curve is not given by a smooth embedding of a parameter; Singular point of an algebraic variety, a point where an algebraic variety is not locally flat; Rational singularity
A point of an algebraic variety that is not singular is said to be regular. An algebraic variety that has no singular point is said to be non-singular or smooth. The concept is generalized to smooth schemes in the modern language of scheme theory. The plane algebraic curve (a cubic curve) of equation y 2 − x 2 (x + 1) = 0 crosses itself at ...
In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on ...