krzywa płaska

This is related to the fact that the plane curve has a singularity at the origin.

Further, this is the only invariant - any two plane curves with the same turning number are regular homotopic.

Let C be a plane curve (the precise technical assumptions are given below).

The catenary is the only plane curve other than a horizontal line with this property.

In the case of plane curves, where the formula becomes:

In either form, it is a plane algebraic curve of genus zero.

It is a subtle point even today that a real algebraic plane curve cannot enter a disk without leaving.

Therefore the following way to represent non plane curves may be preferred.

The circle is the plane curve enclosing the maximum area for a given arc length.

An ellipse is a closed plane curve that resembles a stretched out circle (see the figure to the right).