Talk:Total curvature

Whitney–Graustein theorem

The reference to the Whitney–Graustein theorem is a bit misleading. The significant part of the Whitney–Graustein theorem is the converse, namely that two loops with the same index are regularly homotopic. Here we are referring to the comparatively trivial direction of the theorem. Tkuvho (talk) 11:33, 1 February 2011 (UTC)

Major Inconsistency

The cited papers by Milnor and Sullivan actually define the total curvature of a closed curve to be the integral of the absolute value of its curvature: $\int _{a}^{b}|k(s)|\,ds.$. In particular, for polygons, the total curvature is the sum of the absolute values of the exterior angles. This is not the same as the turning number; in particular, the total curvature of a closed curve/polygon in the plane is 2π if and only if the curve/polygon is convex. Milnor's definition is consistent with the "Generalizations" section, however. — Preceding unsigned comment added by 193.170.138.132 (talk) 13:28, 7 December 2011 (UTC)

calling the discrete version a "generalization" seems backwards

The discrete quantity here is much more specific and concrete requiring fewer abstractions / less technical machinery. Calling it a "generalization" of the continuous version seems backwards (like saying that a discrete sum is a generalization of an integral). –jacobolus (t) 17:08, 17 May 2023 (UTC)