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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 ⁠ ⁠ of a smooth function, and it is not necessary just to consider algebraic varieties. The definitions can be ...

  3. Quadric - Wikipedia

    en.wikipedia.org/wiki/Quadric

    In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections.

  4. 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 ...

  5. Quadric (algebraic geometry) - Wikipedia

    en.wikipedia.org/wiki/Quadric_(algebraic_geometry)

    A singular quadric surface, the cone over a smooth conic curve. If q can be written (after some linear change of coordinates) as a polynomial in a proper subset of the variables, then X is the projective cone over a lower-dimensional quadric. It is reasonable to focus attention on the case where X is not a cone.

  6. Cusp (singularity) - Wikipedia

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

    The definitions for plane curves and implicitly-defined curves have been generalized by René Thom and Vladimir Arnold to curves defined by differentiable functions: a curve has a cusp at a point if there is a diffeomorphism of a neighborhood of the point in the ambient space, which maps the curve onto one of the above-defined cusps.

  7. 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.

  8. Singularity - Wikipedia

    en.wikipedia.org/wiki/Singularity

    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

  9. 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.