EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE

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1 Journal of Singularities Volume 0 204, Proc. of 2th International Workshop on Singularities, São Carlos, 202 DOI: /jsing.204.0f EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE T. FUKUNAGA AND M. TAKAHASHI Dedicated to Professor Shyuichi Izumiya on the occasion of his 60th birthday Abstract. The evolute of a regular curve in the Euclidean plane is given by not only the caustics of the regular curve, envelope of normal lines of the regular curve, but also the locus of singular points of parallel curves. In general, the evolute of a regular curve has singularities, since such points correspond to vertices of the regular curve and there are at least four vertices for simple closed curves. If we repeat an evolute, we cannot define the evolute at a singular point. In this paper, we define an evolute of a front and give properties of such an evolute by using a moving frame along a front and the curvature of the Legendre immersion. As applications, repeated evolutes are useful to recognize the shape of curves.. Introduction The evolute of a regular plane curve is a classical object cf. [5, 8, 9]. It is useful for recognizing the vertex of a regular plane curve as a singularity generically, a 3/2 cusp singularity of the evolute. The caustics evolutes are related to general relativity theory, see for instance [6, 0]. The properties of evolutes are discussed by using distance squared functions and the theories of Lagrangian and Legendrian singularities cf. [, 2, 3, 3, 4, 7, 20]. Moreover, the singular points of parallel curves of a regular curve sweep out the evolute. By using this property, we define an evolute of a front in 2. In order to consider properties of an evolute of a front, we introduce a moving frame along a front a Legendre immersion cf. [7]. In [7], we give existence and uniqueness for a Legendre curve in the unit tangent bundle like for regular plane curves. It is quite useful to analyze a Legendre curve or, a frontal in the unit tangent bundle. In 3, we give another representation for the evolute of a front by using the moving frame and the curvature of the Legendre immersion Theorem 3.3. By the representation, we give properties of the evolutes of fronts, for example, the evolute of a front is also a front. It follows that we can consider the repeated evolutes, namely, the evolute of an evolute of a front, see Theorem 4. in 4. Moreover, we extend the notion of the vertex for a front or, a Legendre immersion and give a kind of four vertex theorem for a front, see Proposition 3.. Furthermore, the evolute of a front is also given by the envelope of normal lines of the front. A singular point of the evolute of the evolute of a regular curve measure to the contact of an involute of a circle. We give the n-th evolute of a front in 5. In 6, we give examples of the evolutes of fronts. In the appendix, we give the condition of contact between regular curves. All maps and manifolds considered here are differentiable of class C. 200 Mathematics Subject Classification. 58K05, 53A04, 57R45. Key words and phrases. evolute, parallel curve, front, Legendre immersion. The authors would like to thank the referee for helpful comments to improve the original manuscript. The second author was supported by a Grant-in-Aid for Young Scientists B No

2 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE Definitions and basic concepts Let I be an interval or R. Suppose that γ : I R 2 is a regular plane curve, that is, γt 0 for any t I. If s is the arc-length parameter of γ, we denote ts by the unit tangent vector ts = γ s = dγ/dss and ns by the unit normal vector ns = Jts of γs, where J is the anticlockwise rotation by π/2. Then we have the Frenet formula as follows: t s n s = 0 κs κs 0 ts ns where κs = t s ns is the curvature of γ and is the inner product on R 2. Even if t is not the arc-length parameter, we have the unit tangent vector tt = γt/ γt, the unit normal vector nt = Jtt and the Frenet formula ṫt ṅt = 0 γt κt γt κt 0, tt nt where γt = dγ/dtt, γt = γt γt and κt = det γt, γt/ γt 3 = ṫt nt/ γt. Note that κt is independent of the choice of a parametrization. In this paper, we consider evolutes of plane curves. The evolute Evγ : I R 2 of a regular plane curve γ is given by Evγt = γt + κt nt, away from the points where κt = 0 cf. [5, 8, 9]. If γ is not a regular curve, then we cannot define the evolute as above, since the curvature may diverge at a singular point. However, we define an evolute of a front in the Euclidean plane, see Definition 2.0 and Theorem 3.3. It is a generalization of the evolute of regular plane curves. We say that γ : I R 2 is a front or, a wave front in the Euclidean plane, if there exists a smooth map ν : I S such that the pair γ, ν : I R 2 S is a Legendre immersion, namely, γt, νt 0, 0 and γt, νt θ = 0 for each t I. Here θ is the canonical contact structure on T R 2 = R 2 S, and S is the unit circle. We remark that the second condition is equivalent to γt νt = 0 for each t I cf. [, 2, 3]. Throughout the paper, we assume that the pair γ, ν is co-orientable, the singular points of γ are finite and γ has no inflection points. The first and second conditions can be removed, see Remarks 3.4 and 3.5. However, we add these conditions for the sake of simplicity. We give examples of fronts. See [, 4, ] for other examples. Example 2.. One of the typical examples of a front is a regular plane curve. Let γ : I R 2 be a regular plane curve. In this case, we may take ν : I S by νt = nt. Then it is easy to check that γ, ν is a Legendre immersion. Example 2.2. Let γ : R R 2 be a 3/2 cusp A 2 -singularity given by γt = t 2, t 3. In this case, 0 is a singular point of γ. If we take ν : R S by νt = / 9t t, 2, then we can show that γ, ν is a Legendre immersion. Hence the 3/2 cusp is an example of a front. The 3/2 cusp is the generic singularity of fronts and also evolutes in the Euclidean plane. Example 2.3. Let γ : R R 2 be a 4/3 cusp E 6 -singularity given by γt = t 3, t 4. In this case, 0 is also a singular point of γ. If we take ν : R S by νt = / 6t t, 3, then we can show that γ, ν is a Legendre immersion. Hence the 4/3 cusp is also an example of a front, see Example 6.3. Example 2.4. Let γ : R R 2 be a 5/2 cusp A 4 -singularity given by γt = t 2, t 5. In this case, 0 is also a singular point of γ. However, the 5/2 cusp is not a front. By the condition,

3 94 T. FUKUNAGA AND M. TAKAHASHI γt νt = 0, we take ν : R S by νt = ±/ 25t t 3, 2. Then γ, ν is not an immersion at t = 0 and hence γ is not a front but γ is a frontal, see [7]. Remark 2.5. By the definition of the Legendre immersion, if γ, ν is a Legendre immersion, then γ, ν is also. We have the following Lemma cf. [4,, 2]. Lemma 2.6. Let γ : I R 2 be a front and t 0 I. If γ i t 0 = 0 for each i k and γ k t 0 0, then γ at t 0 is diffeomorphic to the curve t k, t k+ + ot k+ at t = 0. Moreover, if k = 2 respectively, k = 3, the curve at t 0 is diffeomorphic to a 3/2 respectively, 4/3 cusp. Let γ, ν : I R 2 S be a Legendre immersion. We define a parallel curve γ λ : I R 2 of γ by γ λ t = γt + λνt for each λ R. Then we have following results. Proposition 2.7. For a Legendre immersion γ, ν : I R 2 S, the parallel curve γ λ : I R 2 is a front for each λ R. Proof. We take ν λ : I S by ν λ t = νt. Since γ λ t = γt + λνt, it holds that γ λ t = γt + λ νt. If γ λ t 0 = 0 at a point t 0 I, then we have γt 0 + λ νt 0 = 0. If ν λ t 0 = νt 0 = 0, then γt 0 = 0. It contradicts the fact that γ, ν is an immersion. Hence γ λ, ν λ is an immersion. By νt νt =, we have νt νt = 0. Then γ λ t ν λ t = γt + λ νt νt = γt νt + λ νt νt = 0 holds. It follows that γ λ, ν λ is a Legendre immersion and hence γ λ is a front. We denote the curvature of the parallel curve γ λ t by κ λ t, when γ λ is a regular curve. Proposition 2.8. Let γ, ν be a Legendre immersion. If γ is a regular curve and λ /κt, then a parallel curve γ λ is also regular and Evγ λ t is consistent with Evγt. Proof. Since γ λ t = γt + λnt, it holds that γ λ t = γt λκttt. By the assumption λ /κt, γ λ is a regular curve. By a direct calculation, we have Hence we have κ λ t = κt λκt, n λt = λκt λκt nt. Evγ λ t = γ λ t + κ λ t n λt = γt + λnt + = γt + nt = Evγt κt λκt κt λκt λκt nt Remark 2.9. Let γ, ν be a Legendre immersion. If t 0 is a singular point of the front γ, then lim t t0 κt =. By the equality κ λ t = κt/ λκt, we have lim t t0 κ λ t 0, see also Remark 3.2. We now define an evolute of a front in the Euclidean plane. Definition 2.0. Let γ, ν : I R 2 S be a Legendre immersion. We define an evolute Evγ : I R 2 of the front γ as follows: { γt + κt nt if t is a regular point, Evγt = γ λ t + κ λ t n λt if t t 0 δ, t 0 + δ, t 0 is a singular point of γ,

4 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE 95 where δ is a sufficiently small positive real number, λ R is satisfied the condition λ /κt and κt 0. Remark 2.. By the assumption of the finiteness of singularities of a front, there exists λ R with the condition λ /κt. Moreover, by Proposition 2.8, we can glue on the regular interval of γ and γ λ. Then the evolute of a front is well-defined. Furthermore, by definition, the evolute of a front Evγ is a C map. In order to consider properties of the evolute of a front, we need a moving frame along a front or, a Legendre immersion cf. [7]. Let γ, ν : I R 2 S be a Legendre immersion. If γ is a regular curve around a point t 0, then we have the Frenet formula of γ in 2. On the other hand, if γ is singular at a point t 0, then we don t define such a frame. However, ν is always defined even if t is a singular point of γ. Therefore, we have the Frenet formula of a front as follows. We put µt = Jνt. We call the pair {νt, µt} is a moving frame along a front γt in R 2 and we have the Frenet formula of a front which is given by νt µt = 0 lt lt 0 νt µt where lt = νt µt. Moreover, if γt = αtνt + βtµt for some smooth functions αt, βt, then αt = 0 follows from the condition γt νt = 0. Hence, there exists a smooth function βt such that 2 γt = βtµt. Since γ, ν is an immersion, we have lt, βt 0, 0 for each t I. The pair l, β is an important invariant of Legendre curves or, frontals in the unit tangent bundle like as the curvature of a regular plane curve, for more detail, see [7]. We call the pair l, β the curvature of the Legendre curve. Since we assume that γ, ν is a Legendre immersion, so that we call l, β the curvature of the Legendre immersion. For the related properties, see [5, 6]., 3. Properties of the evolutes of fronts In this section, we consider properties of the evolutes of fronts. Let γ, ν : I R 2 S be a Legendre immersion with the curvature of the Legendre immersion l, β. First we give a relationship between the curvature of the Legendre immersion lt, βt and the curvature κt if γ is a regular curve. Lemma 3.. If γ is a regular curve, then lt = βt κt. 2 If γ λ is a regular curve, then lt = βt + λlt κ λ t. Proof. By a direct calculation, γt = βtµt, γt = βtµt βtltνt and det γt, γt det βtµt, βtµt βtltνt κt = γt 3 = βt 3 = βt2 lt βt 3 = lt βt. Therefore we have lt = βt κt. 2 We can also prove by the same calculations of. Remark 3.2. Since lt, βt 0, 0, if t 0 is a singular point of γ, then γ λ is a regular curve. By Lemma 3. 2, lt 0 = λlt 0 κ λ t 0. It follows from λlt 0 0 that κ λ t 0 0. We give another representation of the evolute of a front by using the moving frame of a front {νt, µt} and the curvature of the Legendre immersion lt, βt.

5 96 T. FUKUNAGA AND M. TAKAHASHI Theorem 3.3. Under the above notations, the evolute of a front Evγt is represented by 3 and Evγ is a front. curvature Proof. Evγt = γt βt lt νt, More precisely, Evγt, Jνt is a Legendre immersion with the lt, d dt βt. lt First suppose that γ is a regular curve. Since γt = βtµt, we have βt 0 and tt = βt βt µt, By Lemma 3., κt = lt/ βt and lt 0. Then Evγt = γt + βt nt = γt + κt lt βt nt = βt νt. βt βt νt = γt βt lt νt. Second suppose that t 0 is a singular point of γ and γ λ is a regular curve with λ /κt. Since γ λ t = βt + λltµt, we have βt + λlt 0 and t λ = βt + λlt βt + λlt µt, n βt + λlt λ = βt + λlt νt. By Lemma 3. 2, κ λ t = lt/ βt + λlt and lt 0. Then Evγ λ t = γ λ t + κ λ t n βt + λlt βt + λlt λt = γt + λνt + νt lt βt + λlt = γt βt lt νt. If we take νt = Jνt = µt, then Evγt, νt is a Legendre immersion. νt = ltjµt 0 and by the form of Evγt = βtlt βt lt lt 2 νt = d βt 4 Jµt, dt lt In fact, we have Evγt νt = 0. It follows that Evγt, Jνt is a Legendre immersion with the curvature lt, d/dtβt/lt and hence Evγ is a front. This completes the proof of the Theorem. Remark 3.4. By the representation 3, we may define the evolute of a front even if γ have non-isolated singularities, under the condition lt 0. By Lemma 3. and Remark 3.4, for a Legendre immersion γ, ν with the curvature of the Legendre immersion l, β, we say that t 0 is an inflection point of the front γ or, the Legendre immersion γ, ν if lt 0 = 0. Since βt 0 0 and Proposition 3., lt 0 = 0 is equivalent to the condition κt 0 = 0. Remark 3.5. Let γ, ν be a Legendre immersion, then γ, ν is also Remark 2.5. However, Evt does not change. It follows that we can define an evolute of a non co-orientable front, by taking double covering of γ.

6 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE 97 Remark 3.6. By Definition 2.0, the evolute of a front is independent on the parametrization of γ, ν. The curvature of the Legendre immersion l, β is depended on the parametrization of γ, ν, see [7]. If s = st is a parameter changing on I to I, then lt = lstṡt and βt = βstṡt. It also follows from the representation 3 that the evolute of a front is independent on the parametrization of γ, ν. If t 0 is a singular point of γ, then βt 0 = 0. As a corollary of Theorem 3.3, we have the following. Corollary 3.7. If t 0 is a singular point of γ, then Evγt 0 = γt 0. Proposition 3.8. Let γ, ν : I R 2 S be a Legendre immersion without inflection points. Suppose that t 0 is a singular point of γ. Then t 0 is a regular point of Evγt if and only if γt 0 0. Proof. By the assumption, βt 0 = 0. Let t 0 be a regular point of Evγt. Since 4 and lt 0 0, we have βt 0 0. By the differentiate of γt = βtµt, we have γt = βtµt βtltνt It follows that γt 0 = 0 and γt 0 = βt 0 µt 0 0. The converse is also holded by reversing the arguments. Note that by Lemma 2.6 and Proposition 3.8, the conditions follows that γ is diffeomorphic to the 3/2 cusp at t 0. Hence, we can recognize the 3/2 cusp of original curve by the regularity of the evolute of a front, see Examples 6.2 and 6.3. The most degenerate case of the evolute of a front, we have classified as follows: Proposition 3.9. If Evγt 0, then γ is a part of a circle or a point. Proof. By the condition Evγt 0, there exists a constant c R such that βt/lt c, if and only if βt = clt. If c = 0, then γt = βtµt = 0. It follows that γ is a point. Suppose that c 0. By the existence and the uniqueness of a front in [7], we take νt = cos ltdt, sin ltdt, µt = sin ltdt, cos ltdt. By γt = βtµt, we have γt = c lt sin ltdt dt + a, c lt cos = c cos ltdt + a, c sin ltdt + b ltdt dt + b for some constants a, b R. Therefore, γ is a part of a circle. As a well-known result, a singular point of Evγ of a regular plane curve γ is corresponding to a vertex of γ, namely, κt = 0 cf. [5, 8, 8, 9]. We extend the notion of vertex. For a Legendre immersion γ, ν with the curvature of the Legendre immersion l, β, t 0 is a vertex of the front γ or a Legendre immersion γ, ν if d/dtβ/lt 0 = 0, namely, d/dtevt 0 = 0. Note that if t 0 is a regular point of γ, the definition of the vertex coincides with usual vertex for regular curves. Therefore, this is a generalization of the notion of the vertex of regular plane curves.

7 98 T. FUKUNAGA AND M. TAKAHASHI Remark 3.0. Let γ, ν be a Legendre immersion. If t 0 is a singular point of γ which degenerate more than 3/2 cusp, then t 0 is a vertex of a front γ. In fact, d β t 0 = βt 0 lt 0 βt 0 lt 0 dt l lt 0 2 = 0, since βt 0 = βt 0 = 0 by Proposition 3.8. In this paper, a Legendre immersion γ, ν : [a, b] R 2 S is a closed Legendre immersion if γ n a, ν n a = γ n b, ν n b for all n N {0} where γ n a, ν n a, γ n b and ν n b means one-sided differential. If γ, ν : [a, b] R 2 S is a closed Legendre immersion, then both a and b are regular points or both a and b are singular points of γ. When a and b are singular points of γ, we treat these singular points as one singular point. Proposition 3.. Let γ, ν : [a, b] R 2 S be a closed Legendre immersion without inflection points. If γ has at least two singular points which degenerate more than 3/2 cusp, then γ has at least four vertices. 2 If γ has at least four singular points, then γ has at least four vertices. Proof. Suppose that γ has at least two singular points which degenerate more than 3/2 cusp. By Remark 3.0, these singularities are vertices of γ, therefore it is sufficient to show that there is at least one vertex between two adjacent singular points. Since γ has no inflection points, the sign of the curvature of γ on regular points is constant. Therefore, either lim t t0 κt = for all t 0 Σγ or lim t t0 κt = for all t 0 Σγ, where Σγ is the set of singular points of γ. This concludes there exist t t, t 2 such that κt = 0 for singular points t and t 2 of γ. Suppose that a and b are singular points which degenerate more than 3/2 cusp. Since we treat a and b as the one singular point, there exists at least one singular point t a, b which degenerate more than 3/2 cusp by the assumption. In this case, there exist at least two vertices v a, t and v 2 t, b. Moreover, a and t are also vertices. Therefore, there exist at least four vertices. Next, suppose that a and b are regular points or 3/2 cusps. Then there exist at least two singular points t and t 2 we assume t < t 2 in a, b which degenerate more than 3/2 cusp. In this case, there exists at least one vertex v t, t 2. Moreover, since γ, ν is closed, there exists a point v 2 [a, t t 2, b] such that κv 2 = 0. Therefore, γ has at least four vertices. 2 Suppose that γ has at least four singular points. Since γ has no inflection points, the sign of the curvature of γ on regular points is constant. Therefore, either lim t t0 κt = for all t 0 Σγ or lim t t0 κt = for all t 0 Σγ. This concludes there exist t t, t 2 such that κt = 0, that is, there is at least one vertex between two adjacent singular points. Suppose that a and b are singular points of γ. Since we treat a and b as the one singular point, there exist at least three singular points t, t 2 and t 3 of γ in a, b, which we assume to be ordered so that a < t < t 2 < t 3 < b. Since there is at least one vertex between two adjacent singular points, there exist at least four vertices v a, t, v 2 t, t 2, v 3 t 2, t 3 and v 4 t 3, b. Next, suppose that a and b are regular points of γ. Let t, t 2, t 3 and t 4 be singular points of γ we assume a < t < t 2 < t 3 < t 4 < b. Since there is at least one vertex between two adjacent singular points, there exist at least three vertices v t, t 2, v 2 t 2, t 3, v 3 t 3, t 4. Moreover, since γ, ν is closed, there exists a point v 4 [a, t t 4, b] such that κv 4 = 0. Therefore, γ has at least four vertices. Finally, in this section, we consider the evolute of a front as a wave front of a Legendre immersion by using a family of functions.

8 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE 99 We define a family of functions by F t, x, y = γt x, y µt. F : I R 2 R Proposition 3.2. Let γ, ν : I R 2 S be a Legendre immersion with the curvature of the Legendre immersion l, β. F t, x, y = 0 if and only if there exists a real number λ such that x, y = γt λνt. 2 F t, x, y = F/ tt, x, y = 0 if and only if lt 0 and x, y = γt βt/ltνt. Proof. γt x, y µt = 0 if and only if there exists λ R such that γt x, y = λνt. 2 F/ tt, x, y = γt µt+γt x, y µt = βt λlt. If lt = 0, then βt = 0. This is a contradiction for lt, βt 0, 0. It follows that λ = βt/lt. The converse is also holded. One can show that F is a Morse family, in the sense of Legendrian singularity theory cf. [, 4, 20], namely, F, F/ t : I R 2 R R is a submersion at t, x, y DF, where DF = {t, x, y F t, x, y = F/ tt, x, y = 0}. It follows that the evolute of a front Evγ is a wave front of a Legendre immersion and is given by the envelope of normal lines of the front. 4. Evolutes of the evolutes of fronts By Theorem 3.3, the evolute of a front is also a front without inflection points. We consider a repeated evolute of an evolute of a front and give properties of a singular point of it. Let γ, ν be a Legendre immersion with the curvature of the Legendre immersion l, β and without inflection points. Theorem 4.. The evolute of an evolute of a front is given by 5 EvEvγt = Evγt βtlt βt lt lt 3 µt. Proof. At this proof, we denote γt = Evγt. By the proof of Theorem 3.3, γt, νt = Evγt, µt is a Legendre immersion. Since µt = J νt = νt and the derivative of the evolute of the front 4, we have βt = βtlt βt lt lt 2, where γt = βt µt. Moreover lt = lt by the Frenet formula of a front. It follows that EvEvγt = Ev γt = γt βt lt νt = Evγt βtlt βt lt lt 3 µt. We can also prove Theorem 4. by a direct calculation of the definition of the evolute of a front Definition 2.0. We need to divide into four cases, that is, γ is a regular or a singular, and Evγ is a regular or a singular. All cases coincide with 5. We also call EvEvγ the second evolute of a front. Now we consider a geometric meaning of a singular point of EvEvγt.

9 00 T. FUKUNAGA AND M. TAKAHASHI Lemma 4.2. Suppose that γ and Evγ are both regular curves. If EvEvγt 0, then γ is an involute of a circle. Proof. We may assume that t is the arc-length parameter of γ. It follows that βt = and hence lt = κt by Lemma 3.. Moreover, we have βt 2 = and βt = 0. Since tt = βtµt and nt = βtνt, we have µt = βttt and νt = βtnt. Then and hold. It follows that Evγt = γt βt lt EvEvγt = Evγt + Evγt = κt κt 2 nt, βt νt = γt βtnt = γt + κt κt nt βt κt κt 3 βttt = Evγt + κt κt 3 tt EvEvγt = κtκt 3 κt2 κt 4 tt. By the assumptions, κt 0, κt 0 and κtκt 3 κt 2 0, it follows that 2 d κt κt = 2. dt κt κt Solving this differential equation, there exist constants C, C 2 R with C 2 0 such that κt = C 2 2t + C. A curve having the curvature / 2ct for a constant c R \ {0} is an involute of a circle with radius c. By the existence and the uniqueness theorems of regular plane curves, see for example [8, 9], γ is an involute of a circle cf. [9, P.38]. Let γ : I R 2 be a regular curve and t 0 I. The involute of a regular curve is defined by Invγ, t 0 : I R 2 ; t Invγ, t 0 t = γt γu du tt. t 0 Note that EvInvγ, t 0 t = γt, for more detail see [5, 8, 9]. Theorem 4.3. Suppose that γ and Evγ are regular curves. If t 0 is a singular point of EvEvγ, then γ is at least 4-th order contact to an involute of a circle at the point t = t 0 up to congruent. Proof. We may assume that t is the arc-length parameter of γ. By the same arguments in the proof of Lemma 4.2, we have κt 0 0, κt 0 0 and κt 0 κt 0 3 κt 0 2 = 0. We set κt 0 = a and κt 0 = b. Then we define a curve γt whose curvature is given by a κt = a b 2t + 2t0 + a, respectively, κt = a a b b 2t 2t0 a b if ab > 0 respectively, ab < 0. Then κt 0 = κt 0 = a and κt 0 = κt 0 = b. Since κt 0 κt 0 3 κt 0 2 = 0 and κt κt 3 κt 2 0, we have κt 0 = κt 0. By the Theorem A. in the appendix, γ and γ are at least 4-th order contact at the point t = t 0 up to congruent. It follows that γ and an involute of a circle are at least 4-th order contact at the point t = t 0 up to congruent. This completes the proof of Theorem.

10 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE 0 Remark 4.4. Suppose that γ is a regular curve. If t 0 is a singular point of Evγt and EvEvγt, then κt 0 = κt 0 = 0 by the same calculations of the proof of Lemma 4.2. It follows that γ and the osculating circle are at least 4-th order contact at the point t = t 0. Proposition 4.5. Let γ, ν : I R 2 S be a Legendre immersion without inflection points. Suppose that t 0 is a singular point of both γ and Evγ. Then t 0 is a regular point of EvEvγ if and only if... γ t 0 0. Proof. Let t 0 be a regular point of EvEvγ. By Proposition 3.8, βt 0 = βt 0 = 0 and lt 0 0. Since d dt EvEvγt = βtlt 2 βtlt lt 3 βtlt lt + 3βt lt 2 lt 4 µt, it holds that d/dtevevγt 0 = βt 0 lt 0 2 µt 0 0 if and only if βt 0 0. By the differentiate of γt = βtµt, we have... γ t = βt βtlt 2 µt 2 βtlt + βt ltνt It follows that... γ t 0 = βt 0 µt 0 0. The converse is also shown by reversing the arguments. Note that by Lemma 2.6 and Proposition 4.5, the conditions follows that γ is diffeomorphic to the 4/3 cusp at t The n-th evolutes of fronts Let γ, ν : I R 2 S be a Legendre immersion with the curvature l, β and without inflection points. We give the form of the n-th evolute of a front, where n is a natural number. We denote Ev 0 γt = γt and Ev γt = Evγt for convenience. We define Ev n γt = EvEv n γt, β 0 t = βt, and β n t = d βn t dt lt inductively. Theorem 5.. Ev n γ, J n ν : I R 2 S is a Legendre immersion with the curvature l, β n, where the n-th evolute of the front is given by where J n is n-times operations of J. Proof. and Let n = and n = 2, then Ev n γt = Ev n γt β n t J n νt, lt Ev γt = Ev 0 γt β 0t lt J 0 νt = γt βt lt νt Ev 2 γt = Ev γt β t lt J νt = Evγt d dt = Evγt βtlt βt lt lt 3 µt. βt lt lt Jνt These are nothing but the evolute of a front 3 and the second evolute of a front 5.

11 02 T. FUKUNAGA AND M. TAKAHASHI Next suppose that j k is holded, namely, Ev j γt = Ev j γt β j t J j νt lt for j k. We consider EvEv k γt. Suppose that Ev k γt, J k νt is a Legendre immersion with the curvature lt, β k t. By Theorem 3.3, we have k + -th evolute of the front Ev k+ γt = Ev k γt β kt lt J k νt. Since d dt Evk+ γt = d dt Evk γt d dt βk t J k νt β kt lt lt J k νt = β k tj k+ t + β k+ tj k+2 νt β k tj k+ νt = β k+ tj k+2 νt, d dt J k+ νt = J k+ νt = J k+ ltµt = ltj k+ Jνt = ltj k+2 νt, it holds that Ev k+ γ, J k+ ν is a Legendre immersion with the curvature lt, β k+ t. By the induction, this completes the proof of Theorem. As a generalization of Propositions 3.8 and 4.5, we have the following result: Proposition 5.2. Let γ, ν : I R 2 S be a Legendre immersion with the curvature of the Legendre immersion l, β and without inflection points. Suppose that t 0 is a singular point of γ. Then the following are equivalent: t 0 is a singular point of Ev i γt for i =,..., n. 2 d i β/dt i t 0 = 0 for i =,..., n. 3 d i γ/dt i t 0 = 0 for i = 2,..., n +. Proof. 6 First, we show that β i t is given by the form β i t and lower terms of β i t, namely, β i t = βi t lt i + Lβt,..., β i t for some smooth function L which contain lt and derivatives of lt. Since β t = d βt = βt dt lt lt + βt d, dt lt the case of i = is holded. Suppose that i = k is holded, namely, there exists a smooth function L such that β k t = βk t lt k + Lβt,..., βk t. Then β k+ t = d dt βk t = βk+ t lt lt k+ + Lβt,..., β k t, for some smooth function L. By the induction, we conclude the assertion. Second, assume that t 0 is a singular point of Ev i γt for i =,..., n. By Theorem 5., d/dtev i γt 0 = 0 if and only if β i t 0 = 0. Since 6 and βt 0 = 0, it holds that β i t 0 = 0 for i =,..., n if and only if β i t 0 = 0 for i =,..., n. It follows that implies 2. By the reversing arguments, the converse follows from 2.

12 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE 03 Finally, since γt = βtµt, we can also show that 2 is equivalent to 3 by the induction. 6. Examples We give examples to understand the phenomena for evolutes of fronts. Example 6.. Let γt = a cos t, b sin t be an ellipse with a, b > 0 and a b. Since νt = a2 sin 2 t + b 2 cos 2 t b cos t, a sin t, µt = a sin t, b cos t, a2 sin 2 t + b 2 cos 2 t we have lt = ab a 2 sin 2 t + b 2 cos 2 t, βt = a 2 sin 2 t + b 2 cos 2 t. The evolute, the second evolute and the third evolute of the ellipse γ are given by Evγt = EvEvγt = a 2 b 2 cos 3 t, a2 b 2 sin 3 t, a b a 2 b 2 ab 2 cos t b 2 cos 4 t + 3a 2 sin 4 t + b 2 sin 2 2t, a2 b 2 a 2 sin t a 2 sin 4 t + 3b 2 cos 4 t + a 2 sin 2 2t, b and Ev 3 γt = a 2 b 2 8a 3 b 2 cos3 t 45a 4 0a 2 b 2 3b a 4 + 4a 2 b 2 + b 4 cos 2t + 5a 2 b 2 2 cos 4t, a 2 b 2 8a 2 b 3 sin3 t 3a 4 + 0a 2 b 2 45b 4 + 2a 4 + 4a 2 b 2 5b 4 cos 2t 5a 2 b 2 2 cos 4t. The ellipse γ and its evolute red curve are showed in Figures left and 2 center. Moreover, the second evolute yellow curve, see Figure center, and the third evolute green curve, see Figures right and 2 right. The evolute is useful to recognize the difference of the sharp of curves. In Figure 2, the left is a circle and the center is an ellipse and its evolute. We can observe the evolute of the ellipse, however, it is very small red curve. If we consider the repeated evolute, we can easy to observe it. The right in Figure 2 is the second and the third evolute of the ellipse.

13 04 T. FUKUNAGA AND M. TAKAHASHI Figure. The ellipse and evolutes Figure 2..0 Example 6.2. Let γt = 3 cos t cos 3t, 3 sin t sin 3t = 6 cos t 4 cos 3 t, 4 sin 3 t be the nephroid, see Figure 3 left. Since νt = sin 2t, cos 2t and µt = cos 2t, sin 2t, we have lt = 2, βt = 6 sin t. The evolute and the second evolute of the nephroid are as follows, see Figure 3 center and right: Evγt = 2 cos 3 t, 3 sin t 2 sin 2 t, 3 EvEvγt = 2 cos t cos3 t, sin 3 t. We can observe that γt/4 = EvEvγt.

14 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE Figure 3. The nephroid and evolutes. Example 6.3. Let γt = t 3, t 4 be the 4/3 cusp, Figure 4 left. Since νt = / 6t t, 3 and µt = / 6t , 4t, we have lt = 2/6t 2 + 9, βt = t 2 6t The evolute and the second evolute of the 4/3 cusp are as follows, see Figure 4 center and right: Evγt = 2t 3 63 t5, 94 t2 + 5t 4, EvEvγt = 27 8 t 23t3 32t 5, 9 4 t2 23t t Figure 4. The 4/3 cusp and evolutes. 2 Appendix A. Contact between regular curves In this appendix, we discuss contact between regular curves. Let γ : I R 2 ; t γt and γ : Ĩ R2 ; u γu be regular plane curves, respectively. We say that γ and γ have k-th order contact at t = t 0, u = u 0 if dγ γt 0 = γu 0, dt t 0 = d γ du u d k γ 0,, dt k t 0 = dk γ du k u d k+ γ 0, dt k+ t 0 dk+ γ du k+ u 0. Moreover, we say that γ and γ have at least k-th order contact at t = t 0, u = u 0 if dγ γt 0 = γu 0, dt t 0 = d γ du u d k γ 0,, dt k t 0 = dk γ du k u 0. Let γ, γ 2 : I R 2 be regular plane curves. We say that γ and γ 2 are congruent if there exists a congruence C such that γ 2 t = Cγ t = Aγ t + b for all t I, where the congruence is given by a rotation A and a translation b on R 2. Let γ : I R 2 ; t γt and γ : Ĩ R2 ; u γu be regular plane curves. We take the arc-length parameter for γt and γu, respectively. In general, we may assume that γt and

15 06 T. FUKUNAGA AND M. TAKAHASHI γu have at least first order contact at any point t = t 0, u = u 0 up to congruent. We denote the curvatures κt of γt and κu of γu, respectively. Theorem A.. Let γ : I R 2 and γ : Ĩ R2 be regular plane curves. If γt and γu have at least k + 2-th order contact at t = t 0, u = u 0 then dκ κt 0 = κu 0, dt t 0 = d κ du u d k κ 7 0,, dt k t 0 = dk κ du k u 0. Conversely, if t and u are the arc-length parameter of γ and γ respectively, and the condition 7 holds, then γ and γ have at least k + 2-th order contact at t = t 0, u = u 0 up to congruent. Proof. We may assume that t and u are the arc-length parameter of γ and γ respectively. Suppose that γ and γ have at least third order contact. Since the Frenet formula, we have dγ/dtt = tt, d 2 γ/dt 2 t = κtnt and d γ/duu = tu, d 2 γ/du2 u = κuñu. It follows that tt 0 = tu 0, nt 0 = ñu 0 and κt 0 = κu 0. Hence, the case of k = holds. Suppose that γ and γ have at least k + 2-th order contact and dκ κt 0 = κu 0, dt t 0 = d κ du u d k κ 0,, dt k t 0 = dk κ du k u 0 hold. Since d 3 γ/dt 3 t = dκ/dttnt κt 2 tt, the form of d k+ γ/dt k+ t is given by d k κ tnt + f κt,, dk 2 κ t tt + g κt,, dk 2 κ t nt, dtk dtk 2 dtk 2 for some smooth functions f and g. Then d k+2 γ dt k+2 t = dk κ dt k tnt + F κt,, dk κ t tt + G κt,, dk κ t nt dtk dtk for some smooth functions F and G. By the same calculations, we have d k+2 γ dk κ u = duk+2 du k uñu + F κu,, dk κ u tu + G κu,, dk κ u ñu. duk duk It follows that d k κ/dt k t 0 = d k κ/du k u 0. By the induction, we have the first assertion. By the reversing arguments, we can prove the converse assertion up to congruent. References [] V. I. Arnol d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps vol. I. Birkhäuser 986. [2] V. I. Arnol d, Singularities of Caustics and Wave Fronts. Mathematics and Its Applications 62 Kluwer Academic Publishers 990. [3] V. I. Arnol d, Topological properties of Legendre projections in contact geometry of wave fronts.. St. Petersburg Math. J , [4] J. W. Bruce and T. J. Gaffney, Simple singularities of mappings C, 0 C 2, 0. J. London Math. Soc , DOI: 0.2/jlms/s [5] J. W. Bruce and P. J. Giblin, Curves and singularities. A geometrical introduction to singularity theory. Second edition. Cambridge University Press, Cambridge, 992. DOI: 0.07/CBO [6] J. Ehlers and E. T. Newman, The theory of caustics and wave front singularities with physical applications, J. Math. Physics , DOI: 0.063/ [7] T. Fukunaga and M. Takahashi, Existence and uniqueness for Legendre curves. J. Geom , DOI: 0.007/s [8] C. G. Gibson, Elementary geometry of differentiable curves. An undergraduate introduction. Cambridge University Press, Cambridge, 200. DOI: 0.07/CBO [9] A. Gray, E. Abbena and S. Salamon, Modern differential geometry of curves and surfaces with Mathematica. Third edition. Studies in Advanced Mathematics. Chapman and Hall/CRC, Boca Raton, FL, 2006.

16 EVOLUTES OF FRONTS IN THE EUCLIDEAN PLANE 07 [0] W. Hasse, M. Kriele and V. Perlick, Caustics of wavefronts in general relativity, Class. Quantum Grav , DOI: 0.088/ /3/5/027 [] G. Ishikawa, Zariski s moduli problem for plane branches and the classification of Legendre curve singularities. Real and complex singularities, World Sci. Publ. Hackensack, NJ, 2007, [2] G. Ishikawa and S. Janeczko, Symplectic bifurcations of plane curves and isotropic liftings. Q. J. Math , DOI: 0.093/qjmath/ [3] S. Izumiya, D. Pei, T. Sano and E. Torii, Evolutes of hyperbolic plane curves. Acta Math. Sin. Engl. Ser , DOI: 0.007/s y [4] S. Izumiya and M. Takahashi, Caustics and wave front propagations: applications to differential geometry. Geometry and topology of caustics CAUSTICS 06, Banach Center Publ., , DOI: /bc [5] K. Saji, M. Umehara and K. Yamada, The geometry of fronts. Ann. of Math , DOI: /annals [6] K. Saji, M. Umehara and K. Yamada, The duality between singular points and inflection points on wave fronts. Osaka J. Math , [7] A. Saloom and F. Tari, Curves in the Minkowski plane and their contact with pseudo-circles. Geom. Dedicata , DOI: 0.007/s [8] M. Umehara, A computation of the basic invariant J + for closed 2-vertex curves, J. Knot Theory Ramifications 6 997, DOI: 0.42/S [9] M. Umehara, A unified approach to the four vertex theorems. I. Differential and symplectic topology of knots and curves, Amer. Math. Soc. Transl. Ser. 2, , [20] V. M. Zakalyukin, Reconstructions of fronts and caustics depending on a parameter and versality of mappings. J. Soviet Math , DOI: 0.007/BF Tomonori Fukunaga, Kyushu Sangyo University, Fukuoka , Japan. address: tfuku@ip.kyusan-u.ac.jp Masatomo Takahashi, Muroran Institute of Technology, Muroran , Japan. address: masatomo@mmm.muroran-it.ac.jp