# 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.