Angle Change of Plane Curve

Abstract

In classical analysis, a curve’s length can be defined as the supremum of the length of a polygonal line with turning points in the curve. To know the change of angles when one travles from one endpoint to another along the curve, a similar method can be taken. The analog is also treated in comlex analysis, but a more natural way to deal with such a problem exists, that is, define the change of angles to be the limit of polygonal line with turning points in the curve. Angle change between vectors and the sum of angle chage of a polygonal line are both well defined, then the way to find the angle change of a curve is showed here. A conjecture is posed. An abstract angle change function is also constructed. Further work is to solve the conjecture, to find the sufficient and necessary condition for a plane curve to be summable respect to total sum of angle change, and to study angle variation and the angle change function.