Tangential developable surfaces as bonnet surfaces

We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry. Dedication Dedicated to Siuping Ho for all her invaluable support and encouragement.     

Piecewise smooth developable surfaces

A.V. Pogorelov introduced developable surfaces with regularity (twice differentiability) violated along separate lines. In particular, the surface may not be smooth at all points of these lines (which form edges in this case). It is assumed that each point of the surface under consideration that belongs to a curvilinear edge (as well as any other interior point of this surface) has a neighborhood isometric to a Euclidean disk. In this paper we study the behavior of a developable surface near its curvilinear edge. It is proved that if two smooth pieces of a developable surface are adjacent along a curvilinear edge, then the spatial location of one of them in ?³ is uniquely determined by that of the other.     

Inextensible flows of curves and developable surfaces

The flow of a curve or surface is said to be inextensible if, in the former case, the arclength is preserved, and in the latter case, if the intrinsic curvature is preserved. Physically, inextensible curve and surface flows are characterized by the absence of any strain energy induced from the motion. In this paper we investigate inextensible flows of curves and developable surfaces in ${\mathbb{R}}^3$. Necessary and sufficient conditions for an inextensible curve flow are first expressed as a partial differential equation involving the curvature and torsion. We then derive the corresponding equations for the inextensible flow of a developable surface, and show that it suffices to describe its evolution in terms of two inextensible curve flows.     

Design and shape adjustment of developable surfaces

This paper presents two direct explicit methods of computer-aided design for developable surfaces. The developable surfaces are designed by using control planes with C-Bézier basis functions. The shape of developable surfaces can be adjusted by using a control parameter. When the parameter takes on different values, a family of developable surfaces can be constructed and they keep the characteristics of Bézier surfaces. The thesis also discusses the properties of designed developable surfaces and presents geometric construction algorithms, including the de Casteljau algorithm, the Farin–Boehm construction for G² continuity, and the G² Beta restricted condition algorithm. The techniques for the geometric design of developable surfaces in this paper have all the characteristics of existing approaches for curves design, but go beyond the limitations of traditional approaches in designing developable surfaces and resolve problems frequently encountered in engineering by adjusting the position and shape of developable surfaces.     

Reconstruction of quasi developable surfaces from ribbon curves

This paper deals with the acquisition and reconstruction of physical surfaces by mean of a ribbon device equipped with micro-sensors, providing geodesic curves running on the surface. The whole process involves the reconstruction of these 3D ribbon curves together with their global treatment so as to produce a consistent network for the geodesic surface interpolation by filling methods based on triangular Coons-like approaches. However, the ribbon curves follow their own way, subdividing thus the surface into arbitrary n-sided patches. We present here a method for the reconstruction of quasi developable surfaces from such n-sided curvilinear boundary curves acquired with the ribbon device.     

A generalization of the Pogorelov-Stocker theorem on complete developable surfaces

The well-known Pogorelov theorem stating the cylindricity of any C ¹-smooth, complete, developable surface of bounded exterior curvature in ℝ³ was generalized by Stocker to C ²-smooth surfaces with a more general notion of completeness. We extend Stocker’s result to C ¹-smooth surfaces that are normal developable in the Burago-Shefel’ sense. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 247–252, 2006.     

An intrinsic characterization of isometric pluriharmonic immersions with codimension one

We give an intrinsic characterization of isometric pluriharmonic immersions of Kähler manifolds into semi-Euclidean spaces with real codimension one, which is a generalization of the Ricci-Curbastro theorem.Research partly supported by the Grants-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan.     

On the extrinsic curvature and the extrinsic structure of normal developable C 1 surfaces

It is proved that any normal C ¹ surface developable in the sense of Shefel has zero extrinsic curvature in the sense of Pogorelov. A condition under which such a surface has a standard line of striction is obtained.     

Design and G connection of developable surfaces through Bézier geodesics

For potential application in shoemaking and garment manufacture industries, the G¹ connection of (1, k) developable surfaces with abutting geodesic is important. In this paper, we discuss the developable surface which contains a given 3D Bézier curve as geodesic and prove the corresponding conclusions in detail. Primarily we study G¹ connection of developable surfaces through abutting cubic Bézier geodesics and give some examples.     

A characterization of intrinsic reciprocity

This paper studies a game-theoretic model in which players have preferences over their strategies. These preferences vary with the strategic context. The paper further assumes that each player has an ordering over an opponent’s strategies that describes the niceness of these strategies. It introduces a condition that insures that the weight on an opponent’s utility increases if and only if the opponent chooses a nicer strategy.     

Minimizability of developable Riemannian foliations

Let (M,F){(M,\mathcal{F})} be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino’s commuting sheaf of (M,F){(M,\mathcal{F})} vanish if (M,F){(M,\mathcal{F})} is developable and π ₁ M is of polynomial growth. By theorems of álvarez López in (álvarez López, Ann. Global Anal. Geom., 10:179–194, 1992) and (álvarez López, Ann. Pol. Math., 64:253–265, 1996), our result implies that (M,F){(M,\mathcal{F})} is minimizable under the same conditions. As a corollary, we show that (M,F){(M,\mathcal{F})} is minimizable if F{\mathcal{F}} is of codimension 2 and π ₁ M is of polynomial growth.     

Bending of a piecewise developable surface

We substantiate in detail the possibility of one-parameter bending of a developable surface possessing a stationary curvilinear edge and stationary rectilinear generators. We construct particular examples of developable surfaces that possess a curvilinear edge and admit one-parameter bendings. We also give examples of closed piecewise developable surfaces that admit one-parameter bendings.