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  2. Singular point of a curve - Wikipedia

    en.wikipedia.org/wiki/Singular_point_of_a_curve

    Hence, it is technically more correct to discuss singular points of a smooth mapping here rather than a singular point of a curve. The above definitions can be extended to cover implicit curves which are defined as the zero set ⁠ f − 1 ( 0 ) {\displaystyle f^{-1}(0)} ⁠ of a smooth function , and it is not necessary just to consider ...

  3. Singularity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Singularity_(mathematics)

    One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at other points. In fact, in this case, the x-axis is a "double tangent." For affine and projective varieties, the singularities are the points where the Jacobian matrix has a rank which is lower than at other points of the variety.

  4. Singular point of an algebraic variety - Wikipedia

    en.wikipedia.org/wiki/Singular_point_of_an...

    Points of V that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complex numbers). [1]

  5. Cusp (singularity) - Wikipedia

    en.wikipedia.org/wiki/Cusp_(singularity)

    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.

  6. Singularity theory - Wikipedia

    en.wikipedia.org/wiki/Singularity_theory

    This is another branch of singularity theory, based on earlier work of Hassler Whitney on critical points. Roughly speaking, a critical point of a smooth function is where the level set develops a singular point in the geometric sense. This theory deals with differentiable functions in general, rather than just polynomials.

  7. Algebraic curve - Wikipedia

    en.wikipedia.org/wiki/Algebraic_curve

    An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.

  8. Graph (discrete mathematics) - Wikipedia

    en.wikipedia.org/wiki/Graph_(discrete_mathematics)

    A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).

  9. Resolution of singularities - Wikipedia

    en.wikipedia.org/wiki/Resolution_of_singularities

    The Whitney umbrella x 2 = y 2 z has singular set the z axis, most of whose point are ordinary double points, but there is a more complicated pinch point singularity at the origin, so blowing up the worst singular points suggests that one should start by blowing up the origin. However blowing up the origin reproduces the same singularity on one ...