Plane curve

# Plane curve

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Encyclopedia
In mathematics, a plane curve is a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

in a Euclidean plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

(cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise
Piecewise
On mathematics, a piecewise-defined function is a function whose definition changes depending on the value of the independent variable...

smooth plane curves), and algebraic plane curves.

A smooth plane curve is a curve in a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

Euclidean plane R2 and is a one-dimensional smooth manifold.
Equivalently, a smooth plane curve can be given locally by an equation where is a smooth function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

, and the partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

s and are never both 0.
In other words, a smooth plane curve is a plane curve which "locally looks like a line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...

" with respect to a smooth change of coordinates.

An algebraic plane curve is a curve in an affine
Affine
Affine may refer to:*Affine cipher, a special case of the more general substitution cipher*Affine combination, a certain kind of constrained linear combination*Affine connection, a connection on the tangent bundle of a differentiable manifold...

or projective plane
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...

given by one polynomial equation (or where ƒ is a homogeneous polynomial
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 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. Some founders of the theory are considered to be Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

and Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

, with main contributors being Niels Henrik Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

, Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

, Max Noether
Max Noether
Max Noether was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century".-Biography:...

, among others.
Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...

, as number of intersections of the curve with a generic line. For example, the circle given by the equation 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
Projection (mathematics)
Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...

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.