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Author

Tomonori Fukunaga

Other affiliations: Hokkaido University
Bio: Tomonori Fukunaga is an academic researcher from Kyushu Sangyo University. The author has contributed to research in topics: Homotopy & Curvature. The author has an hindex of 7, co-authored 26 publications receiving 169 citations. Previous affiliations of Tomonori Fukunaga include Hokkaido University.

Papers
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Journal ArticleDOI
TL;DR: In this article, the authors give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of the curve like the curvature of a regular plane curve.
Abstract: We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.

73 citations

Journal ArticleDOI
TL;DR: In this article, the authors defined the evolutes and the involutes of frontals under conditions and gave an existence condition of the evolute with inflection points, which is a generalisation of both evo-forms of regular curves and of fronts.
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. The 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.

43 citations

Journal ArticleDOI
01 Mar 2019
TL;DR: In this paper, the basic invariants and curvatures of a smooth surface with a moving frame are introduced, and the existence and uniqueness of the fundamental invariants of the framed surfaces are established.
Abstract: A framed surface is a smooth surface in the Euclidean space with a moving frame. The framed surfaces may have singularities. We treat smooth surfaces with singular points, that is, singular surfaces more directly. By using the moving frame, the basic invariants and curvatures of the framed surface are introduced. Then we show that the existence and uniqueness for the basic invariants of the framed surfaces. We give properties of framed surfaces and typical examples. Moreover, we construct framed surfaces as one-parameter families of Legendre curves along framed curves. We give a criteria for singularities of framed surfaces by using the curvature of Legendre curves and framed curves.

29 citations

17 Dec 2012
TL;DR: In this article, the authors define an evolute of a front and give properties of such evolute by using a moving frame of the front and the curvature of the Legendre immersion.
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 loci of parallel curves. In general, the evolute of a regular curve have singularities, since such a point is corresponding to a vertex of the regular curve and there are at least four vertices for simple closed curves. If we repeated 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 evolute by using a moving frame of a front and the curvature of the Legendre immersion. As applications, repeated evolutes can be well-defined and these are useful to recognize the shape of curves.

25 citations

Journal ArticleDOI
01 Sep 2016
TL;DR: For a regular plane curve, an involute of it is the trajectory described by the end of a stretched string unwinding from a point of the curve as discussed by the authors, and the involute always has a singularity.
Abstract: For a regular plane curve, an involute of it is the trajectory described by the end of a stretched string unwinding from a point of the curve. Even for a regular curve, the involute always has a singularity. By using a moving frame along 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 give properties of it. We also consider a relationship between evolutes and involutes of fronts without inflection points. As a result, the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral of the curvature of the Legendre immersion.

21 citations


Cited by
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Journal ArticleDOI
TL;DR: In this article, the authors give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of the curve like the curvature of a regular plane curve.
Abstract: We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.

73 citations

Journal ArticleDOI
TL;DR: In this paper, the curvature of a framed curve is defined, similarly to the curvatures of a regular curve and of a Legendre curve in the unit tangent bundle.
Abstract: Abstract A framed curve in the Euclidean space is a curve with a moving frame. It is a generalization not only of regular curves with linear independent condition, but also of Legendre curves in the unit tangent bundle. We define smooth functions for a framed curve, called the curvature of the framed curve, similarly to the curvature of a regular curve and of a Legendre curve. Framed curves may have singularities. The curvature of the framed curve is quite useful to analyse the framed curves and their singularities. In fact, we give the existence and the uniqueness for the framed curves by using their curvature. As applications, we consider a contact between framed curves, and give a relationship between projections of framed space curves and Legendre curves.

52 citations

Journal ArticleDOI
TL;DR: In this article, the authors defined the evolutes and the involutes of frontals under conditions and gave an existence condition of the evolute with inflection points, which is a generalisation of both evo-forms of regular curves and of fronts.
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. The 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.

43 citations