scholarly journals Propagation of curved folding: the folded annulus with multiple creases exists

2021 ◽  
Author(s):  
Leonardo Alese
Keyword(s):  
Finite Number ◽  
General Relation ◽  
Normal Curvature ◽  
Local Behaviour ◽  
Developable Surfaces ◽  
Planar Domains ◽  
Geometric Descriptors ◽  
Relative Torsion

AbstractIn this paper we consider developable surfaces which are isometric to planar domains and which are piecewise differentiable, exhibiting folds along curves. The paper revolves around the longstanding problem of existence of the so-called folded annulus with multiple creases, which we partially settle by building upon a deeper understanding of how a curved fold propagates to additional prescribed foldlines. After recalling some crucial properties of developables, we describe the local behaviour of curved folding employing normal curvature and relative torsion as parameters and then compute the very general relation between such geometric descriptors at consecutive folds, obtaining novel formulae enjoying a nice degree of symmetry. We make use of these formulae to prove that any proper fold can be propagated to an arbitrary finite number of rescaled copies of the first foldline and to give reasons why problems involving infinitely many foldlines are harder to solve.

Gauss Map Based Curved Origami Discretization

2018 ◽  
Vol 11 (1) ◽  
Author(s):  
Liping Zhang ◽  
Guibing Pang ◽  
Lu Bai ◽  
Tian Ji
Keyword(s):  
Gauss Map ◽  
Surface Curvature ◽  
Normal Direction ◽  
Geometric Constraints ◽  
Normal Curvature ◽  
Curved Surface ◽  
Curved Surfaces ◽  
Gauss Mapping ◽  
Developable Surfaces ◽  
Spherical Curve

This paper addresses the problem of discretizing the curved developable surfaces that are satisfying the equivalent surface curvature change discretizations. Solving basic folding units occurs in such tasks as simulating the behavior of Gauss mapping. The Gauss spherical curves of different developable surfaces are setup under the Gauss map. Gauss map is utilized to investigate the normal curvature change of the curved surface. In this way, spatial curved surfaces are mapped to spherical curves. Each point on the spherical curve represents a normal direction of a ruling line on the curved surface. This leads to the curvature discretization of curved surface being transferred to the normal direction discretization of spherical curves. These developable curved surfaces are then discretized into planar patches to acquire the geometric properties of curved folding such as fold angle, folding direction, folding shape, foldability, and geometric constraints of adjacent ruling lines. It acts as a connection of curved and straight folding knowledge. The approach is illustrated in the context of the Gauss map strategy and the utility of the technique is demonstrated with the proposed principles of Gauss spherical curves. It is applicable to any generic developable surfaces.

Criteria and Methods for Reliable Three-Dimensional Image Reconstruction

1974 ◽  
Vol 32 ◽  
pp. 330-331
Author(s):  
R. A. Crowther
Keyword(s):  
Image Reconstruction ◽  
Finite Number ◽  
Density Distribution ◽  
Electron Micrographs ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Ideal Case ◽  
Dimensional Image ◽  
General Object ◽  
The Ideal

The reconstruction of a three-dimensional image of a specimen from a set of electron micrographs reduces, under certain assumptions about the imaging process in the microscope, to the mathematical problem of reconstructing a density distribution from a set of its plane projections.In the absence of noise we can formulate a purely geometrical criterion, which, for a general object, fixes the resolution attainable from a given finite number of views in terms of the size of the object. For simplicity we take the ideal case of projections collected by a series of m equally spaced tilts about a single axis.

Sensitivity of Stationary Equitable Preferences

2017 ◽  
Author(s):  
Tapan Mitra
Keyword(s):  
Finite Number ◽  
Social Preference ◽  
Social Preferences ◽  
Principal Result ◽  
Fundamental Difficulty ◽  
The Future ◽  
Utility Stream ◽  
Preference Orders ◽  
Infinite Utility Streams

The paper studies the sensitivity implications of the class of monotone social preference orders on infinite utility streams which satisfy the axioms of Equity (Finite Anonymity) and Stationarity (Independent Future). The principal result of this investigation is that representability of such preference orders implies a certain lack of sensitivity to the utility stream of any finite number of generations, which we refer to as ‘insensitivity to the present’. Our result points to a fundamental difficulty in implementing the sustainability principle, which requires intertemporal social preferences to reflect fairly the interests of the generations in the present and in the future.

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

scholarly journals Euler Equations on General Planar Domains

Annals of PDE ◽  
2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Zonglin Han ◽  
Andrej Zlatoš
Keyword(s):  
Euler Equations ◽  
Planar Domains
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