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The Rotation Index of a Plane Curve

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The Rotation Index of a Plane Curve THE ROTATION INDEX OF A PLANE CURVE AARON W. BROWN AND LORING W. TU The rotation index of a smooth closed plane curve C is the number of complete rotations that a tangent vector to the curve makes as it goes around the curve. According to a classical theorem of Heinz Hopf [1], the rotation index of a piecewise smooth closed curve with no self-intersections is ˙1, depending on whether the curve is oriented clockwise or counterclockwise. Hopf’s theorem is a key ingredient in a proof of the Gauss–Bonnet theorem for a surface ([2, Proof of Th. 9.3, p. 164], [3, Proof of Th. 4.4, p. 187]). In this article, we generalize Hopf’s theorem by allowing the curve C to have self-intersections (Figure 0.1). C1 C1 b b p2 p1 p4 b C1 p3 b 1 b c.0/ FIGURE 0.1. Local indices at self-intersection points. We show that it is possible to define a local index p at each self-intersection p such that the rotation index rot.C/ of the curve C can be computed from these local indices: rot.C/ D ˙1 C p; (0.1) Xp where ˙1 depends on the orientation of C and the sum runs over all self-intersection points of C . What is most remarkable about this formula is that it expresses the rotation index, a priori a global invariant of the curve, as a sum of local contributions at the self-intersections. Thus, our formula fits into the general framework of localization theorems such as the Hopf index theorem for a vector field, the Bott residue theorem, or the Lefschetz fixed-point theorem that express a global invariant as a sum of local invariants. Whereas the known localization theorems localize to zeros of a vector field, fixed points of a map, or fixed points of a group action, Formula (0.1) localizes to the self-intersection points of a curve. It is the only localization formula of this type that we are aware of. Our rotation index theorem expresses a fundamental fact about closed plane curves: the number of complete turns and the type of crossings of a closed plane curve are not arbitrary, but must satisfy the constraint imposed by (0.1). We first prove the formula for a piecewise smooth curve March 21, version 7. 1 2 AARON W. BROWN AND LORING W. TU in the plane and then generalize it to a piecewise smooth curve in a Riemannian manifold of dimension 2. 1. ORIENTATIONS ON A PLANE CURVE By a curve in the plane, we will mean either a continuous map cW Œ0;` ! R2 or its oriented image C WD c.Œ0;`/; the context will make clear whether a curve is a map or a point set. The curve cW Œ0;` ! R2 is closed if c.0/ D c.`/. By the Jordan curve theorem, a simple closed curve C in the plane separates the plane into two regions, one bounded and the other unbounded; the bounded region is called the interior of the curve. When the simple closed curve is piecewise smooth, the union M of the curve C and its interior is a manifold with boundary and possibly with corners, and the positive orientation on C is the orientation in Stokes’s theorem, namely, if Np is an outward normal vector at p 2 C and Tp is a tangent vector at p that gives the orientation 2 of C , then the ordered pair .Np; Tp/ gives the standard orientation of R (Figure 1.1). Tp b Np FIGURE 1.1. Positive orientation on a curve. When the curve C has a self-intersection, such as the figure-eight (Figure 1.2), the region D bounded by the curve may have more than one components. Because the various components can give rise to incompatible orientations, it may not be possible to define the positive orientation as the boundary orientation of D [C . For a piecewise smooth curve that may have self-intersections, we introduce the concept of a positive orientation with respect to a smooth extremal point. b b b b p q p q FIGURE 1.2. Positive orientation with respect to p and q Definition 1.1. Let cW Œ0;` ! R2 be a piecewise smooth curve. A point p 2 C WD c.Œ0;`/ is called an extremal point of C if the curve C lies entirely in a closed half plane bounded by a line through p (Figure 1.3). Lemma 1.2. If p is a smooth extremal point of a piecewise smooth plane curve C and L is the line through p bounding a closed half plane containing C , then the line L is tangent to C at p (Figure 1.3). THEROTATIONINDEXOFAPLANECURVE 3 b s b b r b q p FIGURE 1.3. p and q not extremal; r and s are extremal. PROOF. Let Ltan be the tangent line to C at p. If Ltan ¤ L, then Ltan intersects L transversally at p, so C also intersects L transversally at p (Figure 1.4), contradicting the hypothesis that C lies on one side of L. Hence, Ltan D L. C Ltan L b p FIGURE 1.4. Transversal intersection of Ltan and L. Now suppose cW Œ0;` ! R2 is a piecewise smooth, closed curve. At a smooth extremal point p D c.t0/, there is a well-defined unit outward normal vector Np to the curve, since C lies 0 entirely on one side of its tangent line. Let Tp D c .t0/ be the velocity vector at p D c.t0/. We say that the curve c is positively oriented with respect to p if the ordered pair .Np; Tp/ gives the counterclockwise orientation of R2 (Figure 1.5). Tp Np b FIGURE 1.5. Positive orientation with respect to a smooth extremal point. 2. THE LOCAL INDEX AT A SELF-INTERSECTION We say that a curve C1 crosses another curve C2 at an isolated intersection point p 2 C1 \ C2 if C1 passes from one side of C2 to the other side of C2 at p. Figure 2.1 shows two examples of curves that cross each other at p. Note that two curves can cross each other at p and still be tangent at p. Figure 2.2 shows four examples of curves that do not cross each other at p. Smooth curves meeting at p without crossing each other at p are necessarily tangent at p. 4 AARON W. BROWN AND LORING W. TU FIGURE 2.1. Branches that cross each other at p. Double arrows and gray indi- cate the second branch. p p p p b b b b (i) (ii) (iii) (iv) FIGURE 2.2. Branches that do not cross each other at p; local index p D 0. Let cW Œ0;` ! R2 be a smooth plane curve with finitely many self-intersection points. In particular, all the self-intersection points are isolated. A point p D c.t/ is an m-fold point of c if p D c.t/ for exactly m values of t. We will define the local index at a self-intersection point in stages. First consider an isolated double (2-fold) point p where the two branches of C cross each other. Near p, the curve C intersects a sufficiently small circle centered at p in exactly four points. Label the initial point and the endpoint on the circle of the first branch (earlier branch) a and 1 respectively, and of the second branch (latter branch) b and 2 respectively (Figure 2.1). Because the two branches cross each other at p, points 1 and 2 are adjacent on the circle, i.e., not separated by a or b. As one walks on the circle from 1 to 2 without hitting a and b, the direction is either clockwise or counterclockwise. In this way, one can say whether the second branch is clockwise or counterclockwise from the first branch. The local index at a double point p is defined to be C1 if the two branches cross each other at p and the second branch is 8 clockwise from the first branch, p D ˆ1 if the two branches cross each other at p and the second branch is (2.1) ˆ < counterclockwise from the first branch, ˆ 0 if the two branches do not cross each other at p: ˆ :ˆ At an m-fold point p, the curve has m branches, labelled say 1;2;:::;m in chronological order. For any pair 1 i < j m, branch i and branch j intersect at a double point with local index ij D ˙1 or 0 as above. The local index p at the m-fold point p is defined to be the sum of the THEROTATIONINDEXOFAPLANECURVE 5 m 2 pairs of local indices ij : p D ij : 1i<jXm p DC3 3 p 2 b 1 r D1 b r b q q D1 b c.0/ FIGURE 2.3. A curve with a triple point. Example 2.1. The curve C in Figure 2.3 has three branches at the triple point p, labelled 1; 2; 3 chronologically. Branches 1 and 2 define a double point with local index 12 D 1, since they cross each other at p and branch 2 is clockwise from branch 1. Similarly, 13 D 23 D 1. Hence, the local index at p is p D 12 C 13 C 23 D 1 C 1 C 1 D 3: At the double point q, since branch 2 is counterclockwise from branch 1, the local index is q D1.
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