View source for Whitney-Graustein theorem

The [[Winding number|winding number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201601.png" /> of a continuous closed curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201602.png" /> in the plane, missing the origin, counts the algebraic number of times <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201603.png" /> goes around the origin in the counterclockwise direction. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201604.png" /> is a regular closed curve in the plane, i.e., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201605.png" /> is continuously differentiable and has non-vanishing derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201606.png" />, then the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201607.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201608.png" /> is defined to be the winding number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w1201609.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016010.png" /> counts the algebraic number of turns of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016011.png" /> in the counterclockwise direction.

The Whitney–Graustein theorem states that regular homotopy classes of regular curves in the plane are completely classified by the degree, i.e., two regular curves in the plane are regularly homotopic if and only if their degrees are equal (see [[#References|[a2]]]; cf. also [[Homotopy|Homotopy]]). Furthermore, in [[#References|[a2]]] H. Whitney also shows that the degree and the self-intersection number have different parity.

The above definition of degree can be extended to a larger class of curves. If a continuous closed planar curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016012.png" /> is locally one-to-one, then its secant vectors corresponding to pairs of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016013.png" /> that are close enough, are non-zero. Therefore, the winding number of the path defined by secant vectors can be used, as before, to define the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016014.png" />. More precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016015.png" /> is the winding number of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016016.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016017.png" /> small enough, or

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016018.png" /></td> </tr></table>

Replacing regular homotopy by gentle homotopy, i.e., by a homotopy which is locally one-to-one, one gets the following analogue of the Whitney–Graustein Theorem: The gentle homotopy classes of locally one-to-one curves in the plane are completely classified by the degree, i.e., two locally one-to-one curves in the plane are gently homotopic if and only if their degrees are equal (see [[#References|[a1]]]). In this case the degree and the self-intersection number have different parity as well.

The Whitney–Graustein theorem implies that the tangent vectors of any regular closed curve in the plane point in each direction at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120160/w12016019.png" /> times. It is interesting to note that this is true even for differentiable locally one-to-one curves which are not necessarily continuously differentiable (see [[#References|[a1]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Kranjc,   "Degrees of closed curves in the plane"  ''Rocky Mtn. J. Math.'' , '''23'''  (1993)  pp. 951–978</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Whitney,   "On regular closed curves in the plane"  ''Comput. Math.'' , '''4'''  (1937)  pp. 276–284</TD></TR></table>