Evolutes and Involutes of Frontals in the Euclidean Plane

Abstract We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. T he definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals.


Introduction
The notions of evolutes and involutes (also known as evolvents) were studied by C. Huygens in his work [13] and studied in classical analysis, differential geometry and singularity theory of planar curves (cf. [3,4,6,10,11,12,16]). The evolute of a regular curve in the Euclidean plane is given by not only the locus of all its centres of the curvature (the caustics of the regular curve), but also the envelope of normal lines of the regular curve, namely, the locus of singular loci of parallel curves (the wave front of the regular curve). On the other hand, the involute of a regular curve is the trajectory described by the end of stretched string unwinding from a point of the curve. Alternatively, another way to construct the involute of a curve is to replace the taut string by a line segment that is tangent to the curve on one end, while the other end traces out the involute. The length of the line segment is changed by an amount equal to the arc length traversed by the tangent point as it moves along the curve.
In the previous papers [8,9], we defined the evolutes and the involutes of fronts without inflection points and gave properties of them. In §2, we review the evolutes and the involutes of regular curves and of fronts. We introduce the moving frame along Legendre curves and the curvature of Legendre curves (cf. [7]). Moreover, we also gave properties of the evolutes and the involutes of fronts, for more detail see [8,9]. For a Legendre immersion without inflection points, the evolute and the involute of the front are also fronts without inflection points. It follows that we can repeat the evolute and the involute of fronts without inflection points. We gave the n-th form of evolutes and involutes of fronts without inflection points for all n ∈ N in [8,9]. The evolute and the involute of the front without inflection points are corresponding to the differential and the integral of the curvatures of the Legendre immersions.
The evolutes of fronts can not be defined at inflection points. On the other hand, the involutes of fronts can be defined at inflection points. In this case, the involute is a frontal at inflection points. In this paper, we consider evolutes and involutes of frontals under conditions. In §3, we define evolutes and involutes of frontals by extending to the evolutes and the involutes of fronts. These definitions are generalisations of evolutes and involutes of regular curves and of fronts. Even if evolutes of frontals exists, we don't know whether evolutes of evolutes exists or not. By using relationship between evolutes and involutes of frontals, we give an existence condition for the n-th evolute. In §4, we give examples of Legendre curves and evolutes of frontals. These examples are useful to understand properties and results.
We shall assume throughout the whole paper that all maps and manifolds are C ∞ unless the contrary is explicitly stated.
Acknowledgement. The second author was supported by a Grant-in-Aid for Young Scientists (B) No. 23740041.

Regular plane curves
Let I be an interval or R and let R 2 be the Euclidean plane with the inner product a · b = a 1 b 1 + a 2 b 2 , where a = (a 1 , a 2 ) and b = (b 1 , b 2 ) ∈ R 2 . Suppose that γ : I → R 2 is a regular plane curve, that is,γ(t) = (dγ/dt)(t) ̸ = 0 for any t ∈ I. We have the unit tangent vector t(t) =γ(t)/|γ(t)| and the unit normal vector n(t) = J(t(t)), where |γ(t)| = √γ (t) ·γ(t) and J is the anti-clockwise rotation by π/2 on R 2 . Then we have the Frenet formula (ṫ (t) n(t) where the curvature is given by Note that the curvature κ(t) is independent on the choice of a parametrisation. In this paper, we consider evolutes and involutes of plane curves.

Definition 2.2
The involute Inv(γ, t 0 ) : I → R 2 of a regular plane curve γ at t 0 ∈ I is given by The following properties are also well-known in the classical differential geometry of curves: (2) If t and t 0 are regular points of Ev(γ) and not inflection points of γ, then Inv(Ev(γ), t 0 )(t) = γ(t) + (1/κ(t 0 ))n(t).
In fact, the Legendre curve whose associated curvature of the Legendre curve is (ℓ, β), is given by the form ) .
Remark that the definition of the inflection point of the frontal is a generalisation of the definition of the inflection point of a regular curve (cf. [7]).

Definition 2.11
We say that a Legendre curve (γ, ν) : Proof. Assume the Legendre curve (γ, ν) is a part of a circle. There exist a smooth function θ : I → R and constants r, a, b ∈ R such that Since By the existence and uniqueness Theorems 2.7 and 2.8, the converse is holded. 2 Note that a part of a circle may have singular points and inflection points.

Definition 2.13
The evolute Ev(γ) : I → R 2 of the front γ is given by Definition 2.14 The involute Inv(γ, t 0 ) : I → R 2 of the front γ at t 0 ∈ I is given by Proposition 2.15 ([9, Proposition 2.14]) Under the above notations, we have the following.
The following results give the relationships between singular points of γ and the properties of the evolutes and involutes. (2) Suppose that t 0 is a singular point of both γ and Ev(γ). Then γ is diffeomorphic to the 4/3 cusp at t 0 if and only if Ev(γ) is diffeomorphic to the 3/2 cusp at t 0 .
(2) (Inv n (γ, t 0 ), J −n (ν)) : I → R 2 × S 1 is a Legendre immersion with the curvature (ℓ, β −n ), where the n-th involute of the front γ at t 0 is given by and J −n is n-times operation of J −1 .

Evolutes and involutes of frontals
Let (γ, ν) : I → R 2 × S 1 be a Legendre curve with the curvature (ℓ, β). We can define the involute of the frontal as the same form of the involute of the front.

Definition 3.1
The involute Inv(γ, t 0 ) : I → R 2 of the frontal γ at t 0 ∈ I is given by On the other hand, Proposition 2.16 suggests that we may define an evolute of the frontal under existence and uniqueness conditions.

Definition 3.2
The evolute Ev(γ) : I → R 2 of the frontal γ is given by if there exists a unique smooth function α : In this case, we say that the evolute Ev(γ) exists.
The uniqueness condition is well-known as a topological condition. Let (γ, ν) : I → R 2 × S 1 be a Legendre curve with the curvature (ℓ, β). In this paper, we assume that L = {t ∈ I | ℓ(t) ̸ = 0} is a dense subset of I. This condition follows that if such a smooth function α exists, then the uniqueness condition is satisfied by Lemma 3.3.

Remark 3.4
If the inflection points ℓ(t) = 0 are isolated, then the condition that L is a dense subset of I is satisfied.
(2) The involute of the frontal Inv(γ, t 0 ) is also a frontal for each t 0 ∈ I. More precisely, ) . Proof.
2 By Proposition 3.5, if t 0 is an inflection point of a Legendre curve (γ, ν), then t 0 is also an inflection point of both the evolute if exists, and the involute of the frontal. Moreover, t 0 is a singular point of the involute of the frontal. The important difference between the evolute and the involute of the frontal is that we can always repeat the involute of the frontal but can not repeat the evolute of the frontal in general. Proposition 3.6 Let (γ, ν) : I → R 2 × S 1 be a Legendre curve with the curvature of the Legendre curve (ℓ, β).

2) If the evolute Ev(γ) of the frontal exists and β(t) = α(t)ℓ(t), then Inv(Ev
Proof. (1) We denote the curvature of the involute of the frontal by (ℓ −1 (t), β −1 (t)). Since the form of the curvature of the involute of the frontal in Proposition 3.5 (2), ℓ −1 (t) = ℓ(t) and By definition of the evolute of the frontal, it holds that (2) By definition of the involute of the frontal, it holds that 2 By a direct calculation, we have the following Lemma.  (

Example
We give examples of evolutes of frontals. These are useful to understand the phenomena and results.