Plane Curves: Global Properties
DOI link for Plane Curves: Global Properties
Plane Curves: Global Properties book
Global properties concern attributes about the curve taken as a whole. The tangential indicatrix of a curve lies entirely on the unit circle in the plane but with possibly a complicated parametrization. The winding number of a regular curve around a point offers a strategy to prove the Jordan Curve Theorem in the case when the curve is regular. The concept of a vertex is obviously a local property of the curve, but if one were to experiment with a variety of closed curves, one would soon guess that there must be a restriction on the number of vertices. Isoperimetric inequality is an example of a global theorem since it relates quantities that take into account the entire curve at once.