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Plane curve

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	Topics Plane curve Topic Home In mathematics, a plane curve is a curveCurve Overview In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and con... in a Euclidian planePlane (mathematics) In mathematics, a plane is a fundamental two-dimensional object.... (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewisePiecewise In mathematics, a function f of a real number variable x is defined piecewise, if it is continuous on all but a fini... smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a realReal number In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers.... Euclidian plane is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation , where is a smooth functionSmooth function Summary In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders:... of two variables, and the partial derivatives and are not simultaneously equal to 0. In other words, a smooth plane curve is a plane curve which "locally looks like a lineLine The word *line* derives from the Latin lingui, meaning flax plant from which linen is produced; at one time, a stretch... " with respect to a smooth change of coordinates. An algebraic plane curve is a curve in an affineAffine Affine may refer to:Affine geometry, a geometry not involving any notions of origin, length or angle... or projective planeProjective plane In mathematics, a projective plane* has two possible definitions, one of them coming from linear algebra, and another coming... given by one polynomial equation (or , where is a homogeneous polynomial, in the projective case.) Algebraic curves were studied extensively in the 18th to 20th centuries, leading to a very rich and deep theory. The founders of the theory are Issac Newton, Bernhard RiemannBernhard Riemann Georg Friedrich Bernhard Riemann was a German mathematician who made important contributions to analysis and differential ... et.al., with some main contributors being Niels Henrik AbelNiels Henrik Abel Summary Niels Henrik Abel , Norwegian mathematician, was born in Nedstrand, near Finny where his father acted as rector.... , Antoni Poincaré, Max NoetherMax Noether Max Noether was a German mathematician.... , et.al. Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed fieldAlgebraically closed field In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with co... , as number of intersections of the curve with a generic line. For example, a circle has degree 2. An important classical result states that every non-singular plane curve of degree 2 in a projective plane is isomorphic to the projectionProjection (mathematics) In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, f... of the circle . However, the theory of plane curves of degree 3 is already very deep, and connected with the Weierstrass's theory of bi-periodic complex analytic functions (cf. elliptic curves, Weierstrass P-function). There are many questions in the theory of plane algebraic curves for which the answer is not known as of the beginning of the 21st century. See also Smooth manifolds Differential geometry Algebraic curveAlgebraic curve In algebraic geometry, an algebraic curve is an algebraic variety of dimension one.... Algebraic geometryAlgebraic geometry Overview Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative... Projective varieties