Plane curve

Smooth plane curve

A smooth plane curve is a curve in a real Euclidean plane ⁠${\displaystyle {\mathbb{R}}^2}$⁠ 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 ${\displaystyle f(x,y)=0,}$ where ⁠${\displaystyle f:{\mathbb{R}}^2\to \mathbb{R}}$⁠ is a smooth function, and the partial derivatives ⁠${\displaystyle \mathrm{\partial }f/\mathrm{\partial }x}$⁠ and ⁠${\displaystyle \mathrm{\partial }f/\mathrm{\partial }y}$⁠ are never both 0 at a point of the curve.

Algebraic plane curve

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

Algebraic curves have been 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 ${\displaystyle x^2+y^2=1}$ has degree 2.

The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle ${\displaystyle x^2+y^2=1}$ (that is the projective curve of equation ${\displaystyle x^2+y^2-z^2=0}$). The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree 4 are called quartic plane curves.

Examples

Numerous examples of plane curves are shown in Gallery of curves and listed at List of curves. The algebraic curves of degree 1 or 2 are shown here (an algebraic curve of degree less than 3 is always contained in a plane):