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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. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions.

## Symbolic representation

A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form ${\displaystyle f(x,y)=0}$ for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as ${\displaystyle y=g(x)}$ or ${\displaystyle x=h(y)}$ for specific function g or h – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form ${\displaystyle (x,y)=(x(t),y(t))}$ for specific functions ${\displaystyle x(t)}$ and ${\displaystyle y(t).}$Cartesian coordinates by an implicit equation of the form ${\displaystyle f(x,y)=0}$ for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as ${\displaystyle y=g(x)}$ or ${\displaystyle x=h(y)}$ for specific function g or h – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form ${\displaystyle (x,y)=(x(t),y(t))}$ for specific functions ${\displaystyle x(t)}$ and ${\displaystyle y(t).}$

Plane curves can sometimes also be represented in alternative coordinate systems, such as polar coordinates that express the location of each point in terms of an angle and a distance from the origin.

## Smooth plane curve

A smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function. 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/∂

Plane curves can sometimes also be represented in alternative coordinate systems, such as polar coordinates that express the location of each point in terms of an angle and a distance from the origin.

A smooth plane curve is a curve in a real Euclidean plane R2 and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function. 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 at a point of the curve.