THE ROTATION INDEX OF A PLANE CURVE

Transcription

1 THE ROTATION INDEX OF A PLANE CURVE AARON W. BROWN AND LORING W. TU The rotation index of a smooth closed lane curve C is the number of comlete rotations that a tangent vector to the curve makes as it goes around the curve. According to a classical theorem of Heinz Hof [1], the rotation index of a iecewise smooth closed curve with no self-intersections is 1, deending on whether the curve is oriented clockwise or counterclockwise. Hof s theorem is a key ingredient in a roof of the Gauss Bonnet theorem for a surface ([, Proof of Th. 9.3,. 164], [3, Proof of Th. 4.4,. 187]). In this article, we generalize Hof s theorem by allowing the curve C to have self-intersections (Figure 0.1). C1 4 C1 C1 1 c.0/ 3 1 FIGURE 0.1. Local indices at self-intersection oints. We show that it is ossible to define a local index at each self-intersection such that the rotation index rot.c / of the curve C can be comuted from these local indices: rot.c / D 1 C X ; (0.1) where 1 deends on the orientation of C and the sum runs over all self-intersection oints of C. What is most remarkable about this formula is that it exresses the rotation index, a riori 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 Hof index theorem for a vector field, the Bott residue theorem, or the Lefschetz fixed-oint theorem that exress a global invariant as a sum of local invariants. Whereas the known localization theorems localize to zeros of a vector field, fixed oints of a ma, or fixed oints of a grou action, Formula (0.1) localizes to the self-intersection oints of a curve. It is the only localization formula of this tye that we are aware of. Our rotation index theorem exresses a fundamental fact about closed lane curves: the number of comlete turns and the tye of crossings of a closed lane curve are not arbitrary, but must satisfy the constraint imosed by (0.1). We first rove the formula for a iecewise smooth curve March 1, version 7. 1

2 AARON W. BROWN AND LORING W. TU in the lane and then generalize it to a iecewise smooth curve in a Riemannian manifold of dimension. 1. ORIENTATIONS ON A PLANE CURVE By a curve in the lane, we will mean either a continuous ma cw Œ0; ` R or its oriented image C WD c.œ0; `/; the context will make clear whether a curve is a ma or a oint set. The curve cw Œ0; ` R is closed if c.0/ D c.`/. By the Jordan curve theorem, a simle closed curve C in the lane searates the lane into two regions, one bounded and the other unbounded; the bounded region is called the interior of the curve. When the simle closed curve is iecewise smooth, the union M of the curve C and its interior is a manifold with boundary and ossibly with corners, and the ositive orientation on C is the orientation in Stokes s theorem, namely, if N is an outward normal vector at C and T is a tangent vector at that gives the orientation of C, then the ordered air.n ; T / gives the standard orientation ofr (Figure 1.1). T N FIGURE 1.1. Positive orientation on a curve. When the curve C has a self-intersection, such as the figure-eight (Figure 1.), the region D bounded by the curve may have more than one comonents. Because the various comonents can give rise to incomatible orientations, it may not be ossible to define the ositive orientation as the boundary orientation of D[ C. For a iecewise smooth curve that may have self-intersections, we introduce the concet of a ositive orientation with resect to a smooth extremal oint. q q FIGURE 1.. Positive orientation with resect to and q Definition 1.1. Let cw Œ0; ` R be a iecewise smooth curve. A oint C WD c.œ0; `/ is called an extremal oint of C if the curve C lies entirely in a closed half lane bounded by a line through (Figure 1.3). Lemma 1.. If is a smooth extremal oint of a iecewise smooth lane curve C and L is the line through bounding a closed half lane containing C, then the line L is tangent to C at (Figure 1.3).

3 THE ROTATION INDEX OF A PLANE CURVE 3 s q r FIGURE 1.3. and q not extremal; r and s are extremal. PROOF. Let L tan be the tangent line to C at. If L tan L, then L tan intersects L transversally at, so C also intersects L transversally at (Figure 1.4), contradicting the hyothesis that C lies on one side of L. Hence, L tan D L. C L tan L FIGURE 1.4. Transversal intersection of L tan and L. Now suose cw Œ0; ` R is a iecewise smooth, closed curve. At a smooth extremal oint D c.t 0 /, there is a well-defined unit outward normal vector N to the curve, since C lies entirely on one side of its tangent line. Let T D c 0.t 0 / be the velocity vector at D c.t 0 /. We say that the curve c is ositively oriented with resect to if the ordered air.n ; T / gives the counterclockwise orientation ofr (Figure 1.5). T N FIGURE 1.5. Positive orientation with resect to a smooth extremal oint.. THE LOCAL INDEX AT A SELF-INTERSECTION We say that a curve C 1 crosses another curve C at an isolated intersection oint C 1 \ C if C 1 asses from one side of C to the other side of C at. Figure.1 shows two examles of curves that cross each other at. Note that two curves can cross each other at and still be tangent at. Figure. shows four examles of curves that do not cross each other at. Smooth curves meeting at without crossing each other at are necessarily tangent at.

4 4 AARON W. BROWN AND LORING W. TU FIGURE.1. Branches that cross each other at. Double arrows and gray indicate the second branch. (i) (ii) (iii) (iv) FIGURE.. Branches that do not cross each other at ; local index D 0. Let cw Œ0; ` R be a smooth lane curve with finitely many self-intersection oints. In articular, all the self-intersection oints are isolated. A oint D c.t/ is an m-fold oint of c if D c.t/ for exactly m values of t. We will define the local index at a self-intersection oint in stages. First consider an isolated double (-fold) oint where the two branches of C cross each other. Near, the curve C intersects a sufficiently small circle centered at in exactly four oints. Label the initial oint and the endoint on the circle of the first branch (earlier branch) a and 1 resectively, and of the second branch (latter branch) b and resectively (Figure.1). Because the two branches cross each other at, oints 1 and are adjacent on the circle, i.e., not searated by a or b. As one walks on the circle from 1 to 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 oint is defined to be 8 C1 if the two branches cross each other at and the second branch is ˆ< clockwise from the first branch, D 1 if the two branches cross each other at and the second branch is (.1) counterclockwise from the first branch, ˆ: 0 if the two branches do not cross each other at : At an m-fold oint, the curve has m branches, labelled say 1; ; : : : ; m in chronological order. For any air 1 i < j m, branch i and branch j intersect at a double oint with local index ij D 1 or 0 as above. The local index at the m-fold oint is defined to be the sum of the

5 THE ROTATION INDEX OF A PLANE CURVE 5 m airs of local indices ij : D X ij : 1i<j m 3 D C3 r D 1 r 1 q q D 1 c.0/ FIGURE.3. A curve with a trile oint. Examle.1. The curve C in Figure.3 has three branches at the trile oint, labelled 1; ; 3 chronologically. Branches 1 and define a double oint with local index 1 D 1, since they cross each other at and branch is clockwise from branch 1. Similarly, 13 D 3 D 1. Hence, the local index at is D 1 C 13 C 3 D 1 C 1 C 1 D 3: At the double oint q, since branch is counterclockwise from branch 1, the local index is q D THE ROTATION INDEX THEOREM Let cw Œ0; ` R be a smooth unit-seed curve such that c.0/ D c.`/. Since cw Œ0; ` R is a closed curve, we can extend its domain to R without changing the image C by making c eriodic of eriod `. In articular, the domain of c can be extended to the oen interval ; ` C Œ containing Œ0; `. We will assume c 0.0/ D c 0.`/. Let.t/ N be the angle that the tangent vector c 0.t/ makes with resect to the horizontal. Because the angle is defined only u to an integer multile of, we have a well-defined smooth function NW Œ0; ` R=, called the tangent angle function. Since Œ0; ` is simly connected, by the lifting criterion of covering sace theory, N can be lifted to a smooth function W Œ0; ` R. Provided.0/ is secified, the lifted function is unique. Definition 3.1. Since c 0.`/ and c 0.0/ are the same vector,.`/ and.0/ differ by an integer multile of. The rotation angle of c is defined to be.`/.0/, and the rotation index rot.c/ of c is defined to be rot.c/ WD 1.`/.0/ : (3.1) Note that the right-hand side of (3.1) is indeendent of the choice of.0/.

6 6 AARON W. BROWN AND LORING W. TU c 0.t C 0 / q q D c.t 0 / c 0.t 0 / FIGURE 3.1. The jum angle q at q. When the curve c is iecewise smooth, at each singular oint q D c.t 0 /, we let be the outgoing tangent vector and c 0.t C 0 / D lim tt C 0 c 0.t/ c 0.t 0 / D lim tt 0 c 0.t/ the incoming tangent vector (Figure 3.1). The jum angle q at q is defined to be the angle from the incoming vector c 0.t 0 / to the outgoing vector c 0.t C 0 /. For a iecewise smooth curve c, the rotation angle rot.c/ is defined to be the sum of the changes in the angles along each smooth segment lus the sum of the jum angles, and the rotation index is again defined to be.1=/ rot.c/. Theorem 3. (Rotation index theorem). Let cw Œ0; ` R be a iecewise smooth immersion with finitely many self-intersections none of which are singularities of the ma c. Assume c.0/ D c.`/ and let S be the set of self-intersections. Suose that the initial oint c.0/ is a smooth extremal oint. Then the rotation index of c is rot.c/ D 1 C X S ; (3.) where the term 1 is ositive or negative deending, resectively, on whether c is ositively or negatively oriented with resect to its initial oint. This theorem shows that the rotation index is indeendent of the arametrization and of the initial oint, as long as the initial oint is a smooth extremal oint. We sometimes write rot.c / instead of rot.c/. Examle 3.3. For the curve C in Figure 0.1, rot.c / D 1 C 4X i D 1 C.1 C 1 1 C 1/ D 3: id1 Examle 3.4 (Curve with a trile oint). For the curve C with a trile oint in Figure.3, rot.c / D 1 C. C q C r / D 1 C.3 1 1/ D :

7 THE ROTATION INDEX OF A PLANE CURVE 7 4. THE SECANT ANGLE FUNCTION In this section, cw Œ0; ` R is a smooth unit-seed closed curve with finitely many selfintersections such that the initial oint c.0/ is not a self-intersection oint. Denote by T the closed triangular region (Figure 4.1) Let be the diagonal and P the set T D f.t 1 ; t / Œ0; ` Œ0; ` j t 1 t g: (4.1) D f.t; t/ T g: P D f.t 1 ; t / T j t 1 < t and c.t 1 / D c.t /g: (4.) The set P arametrizes the airs of times at which a self-intersection of C occurs: a double oint D c.t 1 / D c.t / of C corresonds to a unique.t 1 ; t / in P ; an m-fold oint of C, D c.t 1 / D D c.t m /; corresonds to the m oints.ti ; t j /, 1 i < j m, in P. By hyothesis, P is a finite set. Any air.t 1 ; t / T. [ P / defines a unit secant vector c.t 1 /c.t /. Let.t 1 ; t / be the angle that the secant c.t 1 /c.t / makes relative to the horizontal: if u then c.t 1 /c.t / D c.t / c.t 1 / kc.t / c.t 1 /k and N.t 1; t / D cos 1 c.t 1 /c.t / u : Since N.t 1 ; t / is well defined u to an integer multile of, it is a function from T.[P / to R=. When.t 1 ; t / aroaches a oint.t; t/ in the diagonal, the unit secant vector c.t 1 /c.t / aroaches the unit tangent vector at c.t/, and the secant angle N.t 1 ; t / aroaches the tangent angle.t/ N defined earlier. Therefore, the secant angle function N.t 1 ; t /, which a riori is defined and C 1 only on the interior of T P, can be extended to a continuous function N W T P R=. We call N W T P R= the secant angle function of the curve c, and a oint.t 1 ; t / P a ole of the secant angle function. T T 0 FIGURE 4.1. An oen neighborhood T 0 of the triangular region T. Since the domain of the smooth function cw Œ0; ` R can be enlarged to an oen interval ; ` C Œ containing Œ0; `, the domain of the secant angle function can be extended smoothly to an oen set T 0 P in R containing T P (Figure 4.1). Although N is not a real-valued function (it is R=-valued), locally on any simly connected subset of T 0 P it can be reresented by real-valued functions that differ by a constant integer multile of. Let fu i g be an oen cover of T 0 P by simly connected oen sets. For any reresentative 1 on U 1 and on U, we have d 1 D d on U 1 \ U. Hence, the forms d i iece together to give well-defined global form on T 0 P. The form is exact on any simly connected oen subset of T 0 P, but not necessarily exact on T 0 P. Being locally exact, is a closed form. We will call the secant angle form of the curve c.

8 8 AARON W. BROWN AND LORING W. TU 5. THE LOCAL INDEX AS AN INTEGRAL In this section we show that the local index at a self-intersection can be comuted as an integral of the secant angle form. Theorem 5.1 (Integral formula for the local index of a double oint). Let cw Œ0; ` R be a smooth, unit-seed, closed curve with finitely many self-intersections such that the initial oint c.0/ is not a self-intersection. Suose D c.e 1 / D c.e / is a double oint. Let T be the closed triangular region of the revious section and a counterclockwise loo in T enclosing.e 1 ; e / but no other oles of. Then the local index at is D 1 : (5.1) PROOF. By the hyothesis that the initial oint c.0/ is not a self-intersection, the oles of the secant angle function N are all in the interior of the closed triangular region T. By Stokes s theorem, since is a closed form, the integral in (5.1) is indeendent of the counterclockwise loo, as long as.e 1 ; e / is the only ole of N enclosed by. To rove the theorem, it suffices to show that for each of the three tyes of double oints in (.1), the integral.1=/ R for an aroriate loo gives the correct local index. BF F D c.f / B D c.b/ A D c.a/ D D c.d/ AF FIGURE 5.1. Local index at. ı3 AD ı ı 4 ı 1 BD Consider now the self-intersection oint D c.e 1 / D c.e / in Figure 5.1. Pick times a and b near e 1, and d and f near e such that a < e 1 < b < d < e < f and.e 1 ; e / is the only ole of the secant angle function N in the closed rectangle R D f.x; y/ T j a x b; d y f g in Figure 5.. Let A D c.a/, B D c.b/, D D c.d/, and F D c.f / be the oints on the curve at t D a; b; d; f resectively. We choose to be the the boundary of the rectangle R. The integral of over the side.a; d/.b; d/ of is the change ı 1 in the secant angle as the secant moves from AD to BD, while the initial oint of the secant goes from A to B:.a;d/.b;d/ D the change in the secant angle from AD to BD D ı 1.counterclockwise/:

9 THE ROTATION INDEX OF A PLANE CURVE 9 t f e d a e 1 b t 1 Similarly, FIGURE 5.. A small rectangular loo about..b;d/.b;f /.b;f /.a;f /.a;f /.a;d/ D ı.counterclockwise/; D ı 3.counterclockwise/ D ı 4.counterclockwise/: From Figure 5.1, we see that the sum of ı 1, ı, ı 3, ı 4 is. Hence, 1 D 1 C C.a;d/.b;d/.b;d/.b;f / D 1.ı 1 C ı C ı 3 C ı 4 / D 1./ D 1:.b;f /.a;f / C.a;f /.a;d/ The comutation of q in Figure 5.3 differs from that of in that the relative disosition of the four oints A; B; D; F are different: BD D D c.d/ B D c.b/ q A D c.a/ F D c.f / AD ı1 AF ı ı 4 ı 3 BF FIGURE 5.3. Local index at q. This time, 1 D 1.ı 1 C ı C ı 3 C ı 4 / D 1. / D 1:

10 10 AARON W. BROWN AND LORING W. TU When the two branches do not cross each other, there are four cases for the relative disosition of the four oints A; B; D; F as in Figure.. In case (i), the change in the secant angle from AD to BD to BF to AF is (Figure 5.4) F B 1 D 1.ı 1 C ı C ı 3 C ı 4 / D 0: D A AF AD BF BD FIGURE 5.4. Local index at a nontransversal double oint. The other three cases (ii), (iii), (iv) are similar and all give 0 as the local index at r. Theorem 5. (Integral formula for the local index of an m-fold oint). Let cw Œ0; ` R be a smooth, unit-seed, closed curve with finitely many self-intersections such that the initial oint c.0/ is not a self-intersection. Suose D c.e 1 / D D c.e m / is an m-fold oint. For 1 i < j m, let ij be a counterclockwise loo in T enclosing.e i ; e j / but no other oles of. Then the local index at is D 1 X : ij 1i<j m PROOF. At, the curve has m branches, labelled chronologically 1; : : : ; m. By definition, the index ij is the local index of the double oint defined by branches i and j. By Theorem 5.1, ij D 1 : ij Hence, by the definition of, D X 1i<j m ij D 1 X 1i<j m ij : 6. PROOF OF THE ROTATION INDEX THEOREM In this section we rove the rotation index theorem for a closed smooth lane curve cw Œ0; ` R with self-intersections. Note that a rotation or a translation of the lane leaves the rotation index as well as the local indices of the curve C invariant. By an aroriate rotation, one may assume that the tangent line T c.0/.c / is horizontal. Assume now that c is ositively oriented with resect to c.0/. Then the initial vector c 0.0/ oints to the right and the angle of the initial vector c 0.0/ is 0 (see Figure 6.1). Let O D.0; 0/, L D.`; `/, and Q D.0; `/ be the three vertices of the triangular region T (Figure 6.), and let T 0 be an oen neighborhood of T as in Figure 4.1.

11 THE ROTATION INDEX OF A PLANE CURVE 11 c.0/ c 0.0/ FIGURE 6.1. Horizontal initial vector at c.0/. Q D.0; `/ L D.`; `/ O D.0; 0/ FIGURE 6.. Small loos about the oles. In a simly connected neighborhood U of the line segment OL in T 0, the secant angle function N W U R= has a unique lift W U R with a given initial value.0; 0/. On U, the secant angle form is exact: D d. Hence, the rotation angle of C is.`/.0/ D.`; `/.0; 0/ D : Around each ole e of the S 1-form, we draw a small counterclockwise circle e. Let B e be the interior of e. Then T Be is a manifold with boundary T B e D OL C LQ C QO e : Since is a closed 1-form on T OL ep OL S Be, by Stokes s theorem, C C D X : (6.1) LQ QO ep e The vertex O D.0; 0/ of the triangular region T corresonds to the tangent vector c 0.0/, the vertex Q D.0; `/ to the tangent vector c 0.0/, and the vertex L D.`; `/ to the tangent vector cr 0.`/ D c 0.0/. Since the line segment OQ is simly connected, along OQ the form is exact with OQ being the change in the angle from the tangent vector c0.0/ to the tangent vector c 0.0/. Hence, D 0 D : (6.) OQ [ ep

12 1 AARON W. BROWN AND LORING W. TU Similarly, along QL the secant moves from c 0.0/ to c 0.`/ and the secant angle changes from to, so D D : Equation (6.1) becomes which gives OL QL QL OQ D D X ; OL ep e rot.c/ D 1 D 1 C 1 OL D 1 C X S X ep e (by Theorem 5.). If the curve c is negatively oriented with resect to c.0/, then the secant angle along OQ goes from to 0 and the secant angle along QL goes from 0 to. A comutation similar to the above then gives rot.c/ D 1 C X : S D 1 c.0/ FIGURE 6.3. Nonextremal initial oint. Remark 6.1 (Deendence on the initial oint). In the rotation index theorem in the lane (Theorem 3.), the initial oint must be an extremal oint. When the initial oint is not an extremal oint, for examle, as in In Figure 6.3, Equation (6.) fails and the rotation index formula (3.) also fails. Indeed, the correct formula for the rotation index in Figure 6.3 is instead rot.c / D 3 C D 3 C. 1/ D : 7. PIECEWISE SMOOTH PLANE CURVES We now extend the roof of the rotation index theorem to the iecewise smooth case. Let cw Œ0; ` R be a iecewise smooth closed curve with singularities at 1 < < < r in the oen interval 0; `Œ. Assume further the initial oint c.0/ is not a singularity and that the self-intersections of c occur away from the singularities. Define the triangular region T and the set P as in (4.1) and (4.) resectively. The secant angle is defined and continuous on T P, but it is not smooth whenever t 1 or t equals one of the i s. Thus, is defined and smooth on the comlement in T of the union of P and the vertical and horizontal lines t i D j (Figure 7.1). Since the self-intersection oints of c are not singularities of c, the oles of do not meet the lines t 1 D j or t D j and so is defined and C 1 on a unctured neighborhood of each

13 THE ROTATION INDEX OF A PLANE CURVE 13 t 1 1 FIGURE 7.1. The domain of for a iecewise smooth curve. t 1 c c FIGURE 7.. Smoothing a corner. ole. It follows that for a iecewise smooth closed curve c, the roofs above still aly so that Theorems 5.1 and 5. remain true for a iecewise smooth closed curve. When the curve c is iecewise smooth, by a rocess of smoothing corners (see [,. 161] and Figure 7.), one can find a nearby smooth curve Nc with the same rotation angle as c. Moreover, since by hyothesis the singularities of c are not self-intersection oints, Nc and c have the same self-intersection oints and the same local index at each self-intersection oint. Thus, the rotation index formula (3.) remains valid for a iecewise smooth curve. 8. GENERALIATIONS We give two generalizations of the rotation index theorem (Theorem 3.), first to an arbitrary metric in the lane, and then to an oriented Riemannian manifold. In Theorem 3., the angle relative to the horizontal is measured relative to the horizontal with resect to the usual Euclidean metric g 0 onr. In fact, we may relace g 0 by an arbitrary metric in the lane. Theorem 8.1. The formula in Theorem 3. remains valid if the angles are measured relative to the horizontal with resect to an arbitrary Riemannian metric on the lane. PROOF. Let g 1 be an arbitrary Riemannian metric onr. Then g s D.1 s/g 0 Csg 1, s Œ0; 1, is a continuous family of Riemannian metrics on R arametrized by the interval Œ0; 1. Clearly, the rotation index of the curve c with resect to the metric g s is a continuous function of s. Since it is integer-valued, it must be a constant. Similarly, the secant angle function N, which deends on g s, is a continuous function of s. Therefore, the local index D.1=/ R from Theorem 5.1 is also a continuous function of s and hence is constant as a function of s. Setting s D 1 roves that Theorem 3. is true with resect to any Riemannian metric g 1 on the lane. Recall that a coordinate chart in the differentiable structure of a smooth -manifold M is an oen set U M together with a homeomorhism W U.U / of U between U and an

14 14 AARON W. BROWN AND LORING W. TU oen subset.u / of R. With resect to the differentiable structure of M, the coordinate ma W U.U / is a diffeomorhism. We write x; y for the two comonents of. On.U; x; y/, the angle of a tangent vector at U is measured relative using the Riemannian metric. Let.U; / D.U; x; y/ be a coordinate chart on an oriented Riemannian -manifold. Suose cw Œ0; ` U is a iecewise smooth immersion that is closed (c.0/ D c.`/) and has finitely many self-intersections. Assume further that the self-intersections of c are not the singularities of c. The diffeomorhism W U.U / defines a metric on.u / R such that becomes an isometry. Using this isometry, we can transfer Theorem 8.1 to the coordinate chart U. Theorem 8. (Rotation index theorem for a coordinate chart). The rotation index formulas in Theorem 3. remains valid if c mas into a coordinate chart in a Riemannian -manifold. Remark 8.3 (Deendence on the frame). In Theorem 8., the angle is measured relative to the first coordinate of a The theorem is not true for an arbitrary frame. For examle, if U is the unctured lane R f0g with the Euclidean metric, r and are olar coordinates, and the angle is measure with resect of the then the rotation index of the circle c.t/ D.cos t; sin t/ is zero, not 1. Note that.r; / is not a coordinate system on U, since is not a continuous real-valued function on U. REFERENCES [1] H. Hof, Über die Drehung des Tangenten und Sehnen ebener Kurven, Comositio Mathematica, tome (1935), [] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Graduate Texts in Mathematics 176, Sringer, New York, [3] R. S. Millman and G. D. Parker, Elements of Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, DEPARTMENT OF MATHEMATICS, TUFTS UNIVERSITY, MEDFORD, MA address: aaron.brown@tufts.edu DEPARTMENT OF MATHEMATICS, TUFTS UNIVERSITY, MEDFORD, MA address: loring.tu@tufts.edu