In mathematics, a plane curve is a curve in a plane, that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.
Smooth plane curve
A smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation f(x, y) = 0, where f : R2 → R is a smooth function, and the partial derivatives ∂f/∂x and ∂f/∂y are never both 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.
Algebraic plane curve
Algebraic curves were studied extensively since the 18th century.
Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation x2 + y2 = 1 has degree 2.
The non-singular plane algebraic curves of degree 2 are called conic sections, and are isomorphic to the projective completion of the circle x2 + y2 = 1 (that is the projective curve of equation x2 + y2 - z2= 0). The non-singular plane curves of degree 3 are called elliptic curves, and those of degree four are called quartic plane curves.
|Name||Implicit equation||Parametric equation||As a function||graph|
- Algebraic curve
- Differential geometry
- Algebraic geometry
- Plane curve fitting
- Projective varieties
- Two-dimensional graph
- Coolidge, J. L. (April 28, 2004), A Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0-486-49576-0.
- Yates, R. C. (1952), A handbook on curves and their properties, J.W. Edwards, ASIN B0007EKXV0.
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