Plane curves

Plane curves
Name	Plane curves
Field	Mathematics
Related	Analytic geometry, Algebraic geometry, Differential geometry

Plane curves are one-dimensional subsets of a two-dimensional affine or projective Euclidean space studied across Analytic geometry, Algebraic geometry, and Differential geometry. They arise in constructions by figures in the history of Greek mathematics, in models used by Isaac Newton and Gottfried Wilhelm Leibniz, and in modern research influenced by institutions such as the Clay Mathematics Institute and the European Mathematical Society. Plane curves connect with computational projects at companies like Google and research groups at universities such as University of Cambridge, Harvard University, and University of Göttingen.

Definition and basic properties

A plane curve may be defined as the image of a continuous map from an interval or circle into the plane of René Descartes's coordinate system, or as the zero set of a function studied by Joseph-Louis Lagrange and Carl Friedrich Gauss. Basic properties include degree, genus (in the sense used by Bernhard Riemann), and topological type examined in contexts associated with Élie Cartan and Henri Poincaré. Important invariants were developed in correspondence between mathematicians at the Royal Society and academies like the Académie des Sciences.

Classification and types

Classification schemes trace to work by Newton on ovals and asymptotic behavior and by Philip Furtwangler and later Federico Enriques on algebraic types. Common types include rational curves studied by Felix Klein, elliptic curves central to Andrew Wiles's proof and the Taniyama–Shimura conjecture, and transcendental curves such as those investigated by Sofya Kovalevskaya. Real plane curves are classified up to isotopy in work influenced by Vladimir Arnold and Oleg Viro, while projective classification connects to ideas of Alexander Grothendieck and David Mumford.

Parametric and implicit representations

Parametric representations were systematized by Johannes Kepler and formalized by Augustin-Louis Cauchy and allow expressions like rational parametrizations used by Niels Henrik Abel and Jacobi. Implicit equations f(x,y)=0, central to Isaac Newton's classification and to Galois theory as developed by Évariste Galois, provide algebraic descriptions; computational elimination techniques are linked to work at Institut des Hautes Études Scientifiques and projects in symbolic computation at Wolfram Research. Transition between forms uses algorithms influenced by researchers at Massachusetts Institute of Technology and Stanford University.

Singularities and local behavior

The study of singular points—nodes, cusps, tacnodes—was advanced by Bernhard Riemann and Felix Klein and later by Oscar Zariski and Heisuke Hironaka on resolution of singularities. Puiseux series expansions echo work by Victor Puiseux and are used in local classification; techniques connect with papers from Princeton University and seminars at the Institute for Advanced Study. Milnor numbers and other local invariants relate to research by John Milnor and applications in analyses influenced by René Thom.

Differential geometry of plane curves

Curvature and torsion notions were developed by Joseph-Louis Lagrange and Gaspard Monge; curvature functions and evolutes relate to problems studied by Jean-Victor Poncelet and Siméon Denis Poisson. Frenet–Serret type formulas for planar Frenet frames appear in texts tied to Élie Cartan and applications in robotics researched at Carnegie Mellon University and ETH Zurich. The study of geodesics on curves and variational problems links to methods by Leonhard Euler and modern calculus of variations groups working with the American Mathematical Society.

Algebraic plane curves

Algebraic plane curves defined by homogeneous polynomials are central to the classical work of Bernhard Riemann, Alexander Grothendieck, and David Mumford. Topics include Bézout's theorem originating in correspondences involving the Paris Academy of Sciences, intersection multiplicities formalized by Jean-Pierre Serre, and moduli spaces developed in the tradition of Grothendieck and Pierre Deligne. Modern computational algebraic approaches connect to research at ETH Zurich and consortia like the European Research Council.

Notable examples and families

Famous families include conics studied by Apollonius of Perga and René Descartes, cubic curves such as the elliptic curves pivotal in results by Andrew Wiles and used in standards by National Institute of Standards and Technology, and quartic plane curves appearing in the works of Max Noether and David Hilbert. Classical special curves include the cycloid from Galileo Galilei and Christiaan Huygens, the catenary analyzed by Johann Bernoulli, the lemniscate of Bernoulli family and Carl Friedrich Gauss, and the folium of Pierre de Fermat. Contemporary study of fractal-like plane curves involves researchers affiliated with Santa Fe Institute and computational explorations at Microsoft Research.

Category:Mathematics