High Quality Content by WIKIPEDIA articles! 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 and is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation (x,y) = 0, where : R2 ? R is a smooth function, and the partial derivatives ? /?x and ? /?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. An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation (x,y) = 0 (or (x,y,z) = 0, where is a homogeneous polynomial, in the projective case.)

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