Editing Winding number (section)

==Turning number==
[[File:Winding Number Around Point.svg|thumb|200px|This curve has [[total curvature]] 6{{pi}}, ''turning number'' 3, though it only has ''winding number'' 2 about {{mvar|p}}.]]
One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated at the beginning of this article has a winding number of 3, because the small loop ''is'' counted.

This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential [[Gauss map]].

This is called the '''turning number''', '''rotation number''',<ref>{{cite book |first=Harold |last=Abelson |title=Turtle Geometry: The Computer as a Medium for Exploring Mathematics |publisher=MIT Press |year=1981 |page=24}}</ref> '''rotation index'''<ref>{{cite book |first=Manfredo P. |last=Do Carmo |title=Differential Geometry of Curves and Surfaces |year=1976 |publisher=Prentice-Hall |isbn=0-13-212589-7 |chapter=5. Global Differential Geometry |page=393}}</ref> or '''index of the curve''', and can be computed as the [[total curvature]] divided by 2{{pi}}.

=== Polygons ===
{{See|Density (polytope)#Polygons}}
In [[polygon]]s, the '''turning number''' is referred to as the [[polygon density]]. For convex polygons, and more generally [[simple polygon]]s (not self-intersecting), the density is 1, by the [[Jordan curve theorem]]. By contrast, for a regular [[star polygon]] {''p''/''q''}, the density is ''q''.

=== Space curves ===
Turning number cannot be defined for space curves as [[Degree of a continuous mapping|degree]] requires matching dimensions. However, for [[locally convex]], closed [[space curve]]s, one can define '''tangent turning sign''' as <math>(-1)^d</math>, where <math>d</math> is the turning number of the [[stereographic projection]] of its [[tangent indicatrix]]. Its two values correspond to the two [[regular homotopy#non-degenerate homotopy|non-degenerate homotopy]] classes of [[locally convex]] curves.<ref>{{Cite journal|last=Feldman|first=E. A.|date=1968|title=Deformations of closed space curves|journal=Journal of Differential Geometry|language=en|volume=2|issue=1|pages=67–75|doi=10.4310/jdg/1214501138 |s2cid=116999463 |doi-access=free}}</ref><ref>{{Cite journal|last1=Minarčík|first1=Jiří|last2=Beneš|first2=Michal|date=2022|title=Nondegenerate homotopy and geometric flows|journal=Homology, Homotopy and Applications|language=en|volume=24|issue=2|pages=255–264|doi=10.4310/HHA.2022.v24.n2.a12 |s2cid=252274622 |url=https://qmro.qmul.ac.uk/xmlui/handle/123456789/79678 |arxiv=1807.01540}}</ref>