Singular point of a curve

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 where f is a polynomial function f: \R^2 \to \R. If f is expanded as If the origin (0, 0) is on the curve then a_0 = 0. If b_1 ≠ 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 b_0 ≠ 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 Then f can be written If 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 If then f = 0 has a solution of multiplicity 2 or higher and the line or is tangent to the curve. In this case, if is not 0 then the curve has a point of double contact with If the coefficient of x^2, is 0 but the coefficient of x^3 is not then the origin is a point of inflection of the curve. If the coefficients of x^2 and x^3 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. If b_0 and b_1 are both 0 in the above expansion, but at least one of c_0, c_1, c_2 is not 0 then the origin is called a double point of the curve. Again putting f can be written Double points can be classified according to the solutions of Crunode If has two real solutions for m, that is if then the origin is called a crunode.

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Conic section

A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties. The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions.

Crunode

In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ordinary double point. For a plane curve, defined as the locus of points f (x, y) = 0, where f (x, y) is a smooth function of variables x and y ranging over the real numbers, a crunode of the curve is a singularity of the function f, where both partial derivatives and vanish.

Cusp (singularity)

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 a cusp is a point where both derivatives of f and g are zero, and the directional derivative, in the direction of the tangent, changes sign (the direction of the tangent is the direction of the slope ).

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