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 gener alisation 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. IntroductionT he 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, 17]). T he 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 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. T he 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 (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 P 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 exist, we don't know whether evolutes of evolutes exist or not. By using relationship between evolutes and involutes of frontals...

Notes on pedal and contrapedal curves of fronts in the Euclidean plane

“…This is because the curvature may diverge at a singular point. Hence, Fukunaga and Takahashi have introduced the evolute, involute, and offset of the front, which are generalizations of the evolute, involute, and offset of a regular curve in the Euclidean plane, respectively. So as to define an evolute, an involute, and an offset of the front, they have used Legendre curves in the unit tangent bundle, the Legendrian Frenet frame, and the Legendrian curvature.…”

“…This frame is well‐defined even if the curve has singular points. Therefore, Fukunaga and Takahashi defined the evolute, involute, and offset (parallel curve) of a curve, which may include singular points by using the Legendrian Frenet frame. Then, Yu et al defined fronts in the Euclidean 2‐sphere and examined the evolutes of such fronts.…”

“…Then, we give some relationships by using the fact that the evolute, involute, and offset of the front are also fronts. [7][8][9] Furthermore, we examine singularities of pedal and contrapedal curves of fronts and also obtain the conditions for a pedal or contrapedal curve to be diffeomorphic to 3/2 cusp or 4/3 cusp. Finally, we present some illustrated examples to support our theory.…”

Involutes of fronts in the Euclidean plane

“…In this case, the involute of the front is a frontal (cf. [10]). We can find other kinds of singularities of the involute, see [1,10,21].…”

Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane

“…There are many applications of envelopes to differential geometry, differential equations and physics, for instance [4,5,7,9,10,15,16,18,20]. An envelope of a family of curves in the plane is a curve that is "tangent" to each member of the family at some point.…”