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In mathematics, a plane curve is a curve in a Euclidian plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.

A smooth plane curve is a curve in a real Euclidian plane R² is an one-dimensional smooth manifold. 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 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.)

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.al., with some main contributors being Niels Henrik Abel, Antoni Poincaré, Max Noether, et.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. 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. 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.

• Smooth manifolds
• Differential geometry
• Algebraic curve
• Algebraic geometry
• Projective varieties

1. Coolidge, J. 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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