Plane Curve

In mathematics, an airplane curve is a curve in a planes that can be either an Euclidean plane , an affine planes or a projective planes . The most studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves . Plane curves include the Jordan curve (curves that enclose an area of ​​the planes but do not need to be smooth) and graphs of continuous functions .

symbolic representation

A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f for a specific function . If this equation can be explicitly solved for y or x —that is, rewritten or for the specific function g or h —then it provides an alternative, explicit, form of representation. A plane curve can often be represented in Cartesian coordinates by a parametric equation of the form For specific functions and

f(x,y)=0y=g(x){\displaystyle x=h(y)}{\displaystyle (x,y)=(x(t),y(t))}x(t){\displaystyle y(t).}

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

smooth flat curve

A smooth planes curves is a curves in a real Euclidean plane 2 and is a one-dimensional smooth manifold . This means that a smooth plane curve is a plane curve that “locally looks like a line “, in the sense that near every point, it can be mapped to a line by a smooth function . Equivalently, a smooth planes curves can be given locally by an equation f ( x , y ) = 0 , where f  : 2 → r is a smooth function , and the partial derivatives fx and f / y are both never 0 at a point on the curve.

algebraic plane curve

An algebraic plane curve is a curve in an affine or projective plane given by the polynomial equation f ( x , y ) = 0 (or F ( x , y , z ) = 0 , where F is a homogeneous polynomial , in the projective case .)

Algebraic curves have been studied extensively since the eighteenth century.

Every algebraic plane curve has a degree, the degree of the equation defining it , which, in the case of an algebraically closed field, is equal to the number of intersections of the curve with a line in the general case. For example, the circle given by the equation 2 + 2 = 1 has a degree of 2. Plane algebraic curves of non-singular degree 2 are called conic sections , and all of their projective completions are isomorphic circles of projective completion 2 + 2 = 1 (the projective curve of the equation that is 2 + 2 – 2 = 0 ). Plane curves of degree 3 are called cubic plane curves and , if they are non-singular, elliptic curves . Quartic planes of degree 4 are called curves.


Many examples of plane curves are shown in the gallery of curves and listed in the list of curves . Algebraic curves of degree 1 or 2 are shown here (algebraic curves of degree less than 3 are always contained in a plane):