In mathematics, a plane curve is a curve in a Euclidian plane (cf. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object space curve). In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. In Mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of the Independent

A smooth plane curve is a curve in a real Euclidian plane R² is a one-dimensional smooth manifold. In Mathematics, the real numbers may be described informally in several different ways A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Equivalently, a smooth plane curve can be given locally by an equation f(x,y) = 0, where f is a smooth function of two variables, and the partial derivatives f_x and f_y are not simultaneously equal to 0. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In other words, a smooth plane curve is a plane curve which "locally looks like a line" with respect to a smooth change of coordinates.

An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation f(x,y) = 0 (or f(x,y,z) = 0, where f is a homogeneous polynomial, in the projective case. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics )

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 Riemann et. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements al. , with some main contributors being Niels Henrik Abel, Antoni Poincaré, Max Noether, et. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation Max Noether ( 24 September 1844 - 13 December 1921) was a German Mathematician who worked on Algebraic geometry and al. Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field, as number of intersections of the curve with a generic line. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients For example, a circle x² + y² = 1 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 projection of the circle x² + y² = 1. In Mathematics, a singular point of an Algebraic variety V is a point P that is 'special' (so singular in the geometric sense that V In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a projection is any one of several different types of functions mappings operations or transformations for example the following In 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. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician elliptic curves, Weierstrass P-function). In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O In Mathematics, Weierstrass's elliptic functions are Elliptic functions that take a particularly simple form (cf Jacobi's elliptic functions) they are named

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 curve
• Algebraic geometry
• Projective varieties

References

1. Coolidge, J. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety L.

A Treatise on Algebraic Plane Curves. New York: Dover, 1959.

1. Yates, R. C.

A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, 1947.

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