We give a moving frame of a Legendre curve (or, a frontal) in the unite tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. The existence and uniqueness for Legendre curves are holded like as regular plane curves. It is quite useful to analyse the Legendre curves. As applications, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unite tangent bundle.
Bulletin of the Brazilian Mathematical Society manuscript No.
. 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...
The notions of involutes (also known as evolvents) and evolutes were studied by C. Huygens. For a regular plane curve, an involute of it is the trajectory described by the end of stretched string unwinding from a point of the curve. Even if a regular curve, the involute of the curve have singularities. By using a moving frame of the front and the curvature of the Legendre immersion in the unit tangent bundle, we define an involute of the front in the Euclidean plane and discuss properties of them. We also consider about relationship between evolutes and involutes of fronts without inflection points. As a result, we observe that the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral in classical calculus.
We give a homotopy classification of nanophrases with at most four letters. It is an extension of the classification of nanophrases of length 2 with at most four letters, given by the author in a previous paper. As a corollary, we give a stable classification of ordered, pointed, oriented multi-component curves on surfaces with minimal crossing number less than or equal to 2 such that any equivalent curve has no simply closed curves in its components.
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