Jordan curve

closed Jordan curve
open Jordan curve
Curve that is not an open Jordan curve

Jordan curves (or simple curves ) are mathematical curves named after Camille Jordan , which are defined as a homeomorphic embedding of the circle or the interval in a topological space . (The homeomorphic embedding of is called an open Jordan curve. The embedding of is called a closed Jordan curve.) ${\ displaystyle S_ {1}}$${\ displaystyle I_ {1} = [0; 1]}$${\ displaystyle I_ {1}}$${\ displaystyle S_ {1}}$

This clearly means that the curves are continuous and free of intersection points and have a start and an end point. The term Jordan curve is also used to define planar graphs .

Examples

The unit circle with the parameterization

${\ displaystyle \ varphi (t) = (\ cos (t), \ sin (t))}$, ${\ displaystyle t \ in [0.2 \ pi]}$

is a closed Jordan curve.

The way

${\ displaystyle \ varphi (t) = (\ cos (t), \ sin (t))}$ With ${\ displaystyle t \ in [0.3 \ pi]}$

also provides the unit circle, but is not a Jordan curve in this parameterization, since z. B.

${\ displaystyle \ varphi (1) = \ varphi (2 \ pi +1)}$.

The unit square is a Jordan curve, but it is not smooth with any parameterization .

The distance

${\ displaystyle \ varphi (t) = (t, 0)}$ With ${\ displaystyle t \ in [0,1]}$

is an (open) Jordan curve.