In my Differential Geometry course, it was given the following definition for the plane curves. But, I did some research, and I couldn't find something simillar in a book or on the internet. So, I am posting here so that you can write me your opinion.
I would like to ask what the definition above intuitively/geometrically means and if it's really useful if we do differential geometry.
Definition. We call Plane Curve a subset of the Euclidean Space $\Bbb{R}^2$ such that it's covered by a number of parametrizations and every point of the set has a neibourhood which is covered by a parametrization.
Also, I have observed that almost all the DG books when they reffer to 'curve' they actually mean the 'parametrization of a curve'. What you think is the best?
Thank you in advance.