Plane curve

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 : R2R 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

    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 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 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.

    Examples

    Name Implicit equation Parametric equation As a function graph
    Straight line
    Circle
    Parabola
    Ellipse
    Hyperbola

    See also

    References

    • 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 .

    External links


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