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.

Motion Analysis and Simulation of Complex Curved Surface WEDM System

2008 ◽  
Vol 392-394 ◽  
pp. 151-155
Author(s):  
Tong Wang ◽  
K. Jiang ◽  
Shu Qiang Xie ◽  
Shuang Shuang Hao
Keyword(s):  
Electrical Discharge Machining ◽  
Three Dimensional ◽  
Simulation Method ◽  
Curved Surface ◽  
Polar Coordinates ◽  
Curved Surfaces ◽  
Machining Quality ◽  
Motion Parameters ◽  
Dimensional Graph ◽  
Analysis Of Motion

In this paper, the characteristics and general laws of cutting complex curved surface by wire electrical discharge machining (WEDM) system are studied. Based on analysis of motion parameters the universal mathematical model of polar coordinates is derived. Moreover, the simulation of WEDM system is introduced, which is carried out by using language Visual C++ and the three dimensional graph software OpenGL.This simulation method is helpful in improving machining quality and productivity of complex curved surfaces, and is fundation for establishing CAD/CAPP/CAM technology in WEDM.

Digital Planimetry With a New Adaptive Calibration Procedure Results in Accurate and Precise Wound Area Measurement at Curved Surfaces

2020 ◽  
pp. 193229682095934
Author(s):  
Piotr Foltynski ◽  
Piotr Ladyzynski
Keyword(s):  
Systematic Error ◽  
Calibration Procedure ◽  
Curved Surface ◽  
Area Measurement ◽  
Curved Surfaces ◽  
Wound Area ◽  
Optical Scanner ◽  
Adaptive Calibration ◽  
Relative Errors ◽  
Commercial Device

Background: The purpose of this study was to determine the accuracy of wound area measurement at a curved surface using a digital planimetry (DP) with the newly proposed adaptive calibration. Methods: Forty wound shapes were printed and placed at the side surfaces of cylinders with diameters of 9.4 and 6.2 cm. Area measurements were carried out using a commercial device SilhouetteMobile (Aranz, New Zealand) and the planimetric app Planimator. Planimetric area measurements were carried out using 2 one-dimensional calibration markers placed above and below the wound shape. The method of adaptive calibration for DP was described. Reference area values of wound shapes were obtained by pixel counting on digital scans made with an optical scanner. Relative errors (REs) and relative differences (RDs) for area measurements were analyzed. Results: The median of REs for the DP with adaptive calibration (DPwAC) was equal to 0.60% and was significantly smaller than the median for the SilhouetteMobile device (SMD) (2.65%), and significantly smaller than the median for the DP (2.23%). The SD of RDs for the DPwAC of 0.87% was considerably lower than for the SMD (6.45%), and for the DP without adaptive calibration (2.51%). The mean of RDs for the DPwAC (0.082%) was not significantly different from zero, which means that the systematic error was not present for the DPwAC. Conclusions: The use of the adaptive calibration in DP to measure the areas at curved surface resulted in a significant increase of accuracy and precision, and removal of systematic error. The DPwAC revealed 4.4 times lower error and 7.4 times higher precision of area measurement at curved surfaces than the SMD.

The three kinds of degree distributions and nash equilibrium on the limiting random network

2019 ◽  
Vol 20 (05) ◽  
pp. 2050033
Author(s):  
Dong Han ◽  
Min Xia
Keyword(s):  
Nash Equilibrium ◽  
Random Network ◽  
Sufficient Conditions ◽  
Curved Surface ◽  
High Dimensional ◽  
Game Model ◽  
Curved Surfaces

A generalized dynamically evolving random network and a game model taking place on the evolving network are presented. We show that there exists a high-dimensional critical curved surface of the parameters related the probabilities of adding or removing vertices or edges such that the evolving network may exhibit three kinds of degree distributions as the time goes to infinity when the parameters belong to the super-critical, critical and sub-critical curved surfaces, respectively. Some sufficient conditions are given for the existence of a regular Nash equilibrium which depends on the three kinds of degree distributions in the game model on the limiting random network.

scholarly journals Developable mechanisms on developable surfaces

2019 ◽  
Vol 4 (27) ◽  
pp. eaau5171 ◽  
Author(s):  
Todd G. Nelson ◽  
Trent K. Zimmerman ◽  
Spencer P. Magleby ◽  
Robert J. Lang ◽  
Larry L. Howell
Keyword(s):  
Mechanical Systems ◽  
Relative Orientation ◽  
Developable Surface ◽  
Curved Surfaces ◽  
Surface Type ◽  
Complex Tasks ◽  
Rigid Link ◽  
Developable Surfaces ◽  
Flat Sheet ◽  
Wide Range

The trend toward smaller mechanism footprints and volumes, while maintaining the ability to perform complex tasks, presents the opportunity for exploration of hypercompact mechanical systems integrated with curved surfaces. Developable surfaces are shapes that a flat sheet can take without tearing or stretching, and they represent a wide range of manufactured surfaces. This work introduces “developable mechanisms” as devices that emerge from or conform to developable surfaces. They are made possible by aligning hinge axes with developable surface ruling lines to enable mobility. Because rigid-link motion depends on the relative orientation of hinge axes and not link geometry, links can take the shape of the corresponding developable surface. Mechanisms are classified by their associated surface type, and these relationships are defined and demonstrated by example. Developable mechanisms show promise for meeting unfilled needs using systems not previously envisioned.

scholarly journals Abutting Objects Warp the Three-Dimensional Curvature of Modally Completing Surfaces

i-Perception ◽  
2020 ◽  
Vol 11 (2) ◽  
pp. 204166952090355 ◽  
Author(s):  
Peter U. Tse
Keyword(s):  
Visual Processing ◽  
Binocular Disparity ◽  
Three Dimensional ◽  
Surface Curvature ◽  
Amodal Completion ◽  
Three Dimensions ◽  
Curved Surfaces ◽  
Open Surface ◽  
Modal Completion

Binocular disparity can give rise to the perception of open surfaces or closed curved surfaces (volumes) that appear to vary smoothly across discrete depths. Here I build on my recent papers by providing examples where modally completing surfaces not only fill in from one depth layer’s visible contours to another layer’s visible contours within virtual contours in an analog manner, but where modally completing surface curvature is altered by the interpolation of an abutting object perceived to be connected to or embedded within that modally completing surface. Seemingly minor changes in such an abutting object can flip the interpretation of distal regions, for example, turning a distant edge (where a surface ends) into rim (where a surface bends to occlude itself) or turning an open surface into a closed one. In general, the interpolated modal surface appears to deform, warp, or bend in three-dimensions to accommodate the abutting object. These demonstrations cannot be easily explained by existing models of visual processing or modal completion and drive home the implausibility of localistic accounts of modal or amodal completion that are based, for example, solely on extending contours in space until they meet behind an occluder or in front of “pacmen.” These demonstrations place new constraints on the holistic surface and volume generation processes that construct our experience of a three-dimensional world of surfaces and objects under normal viewing conditions.

Study on the Fuzzy-PID Policy for Force Control in Free-Form Surfaces Polishing by Robots

2011 ◽  
Vol 101-102 ◽  
pp. 422-426
Author(s):  
Xiao Ling Su ◽  
Jian Ming Zhan
Keyword(s):  
Force Control ◽  
Normal Direction ◽  
Curved Surface ◽  
Free Form ◽  
Work Piece ◽  
Fuzzy Pid ◽  
Polishing Force ◽  
The Right ◽  
Free Form Surfaces ◽  
Polishing Tool

When a robot is used to polish or finish a curved surface, both feed movement and contact force have to be controlled at the same time so that the polishing tool would machine its work-piece at the right position in right posture with required force. In this paper, a passive wrist system is developed to adapt the shape of the machining curved surface by changing its posture along with the surface. And under the Fuzzy-PID policy, polishing force is controlled at a stable value in the normal direction of the named machining point while the polishing tool moving along the curved surface by multi-point machining. It means that the passive wrist system and the model of the surroundings could be used in force controlling when robots polish free-form surfaces with multi-point machining by a grinding ring.

Nanostructuring Curved Surfaces Using a Flexible Stamp

2009 ◽  
Author(s):  
Bahador Farshchian ◽  
JaeJong Lee ◽  
Sunggook Park
Keyword(s):  
Hot Embossing ◽  
Curved Surface ◽  
Curved Surfaces ◽  
Pmma Substrate ◽  
Pdms Stamp ◽  
Flat Surfaces ◽  
Force Microscopy ◽  
Atomic Force ◽  
Metallic Object ◽  
Planar Substrates

We report on a simple and effective process that allows direct imprinting of micro- and nanostructures on non-flat surfaces. A thin polydimethylsiloxane (PDMS) stamp having micro/nanogratings was placed between a metallic bar with a trapezoidal cross section or a metallic pellet and a flat polymethyl methacrylate (PMMA) substrate, followed by hot embossing at 200°C. During the hot embossing process, the metallic bar/pellet is pushed into the PMMA sheet forming a millimeter scale channel or a curved surface. Due to the presence of the PDMS stamp between the metallic object and the substrate, micro/nanostructures are produced into the channel or over the curved surface. With this method, we have successfully demonstrated micro- and nanostructures down to 300 nm wide gratings on non-flat substrates, as confirmed by scanning electron microscopy and atomic force microscopy. The process so developed will fill the gap in current micro- and nanofabrication technologies in that most of the technologies allow for patterning only on planar substrates.

Structural Optimization of Woven Fabric with Curved Surface

2012 ◽  
Vol 443-444 ◽  
pp. 408-411
Author(s):  
Yan Fang Wang ◽  
Xing Feng Guo
Keyword(s):  
Structural Optimization ◽  
Shear Deformation ◽  
Three Dimensional ◽  
Single Layer ◽  
Woven Fabrics ◽  
Woven Fabric ◽  
Curved Surface ◽  
Theoretical Formula ◽  
Curved Surfaces

The woven fabric with curved surfaces is a kind of single layer woven fabrics, which was produced to smoothly fit three-dimensional solids. The warp or weft of the winding fabric bend were normally made with different lengths, which may result in shear deformation in many cases and accordingly twisting the structure of the fabric after fitted onto the solid. In order to solve the problem mentioned above, a theoretical formula was used to calculate the optimal intervals of the pick-spacing and an improved structure thus was developed in this study.

scholarly journals Large elastic deformations of isotropic materials. V. The problem of flexure

1949 ◽  
Vol 195 (1043) ◽  
pp. 463-473 ◽  
Keyword(s):  
Energy Function ◽  
Incompressible Material ◽  
Curved Surface ◽  
Curved Surfaces ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Axis Ii ◽  
Strain Invariants ◽  
Deformed State ◽  
Derivatives Of

A cuboid of highly elastic incompressible material, whose stored-energy function W is a function of the strain invariants, has its edges parallel to the axes x, y and z of a rectangular Cartesian co-ordinate system. It can be bent so that: (i) every plane, initially normal to the x -axis, becomes part of the curved surface of a cylinder whose axis is the z -axis; (ii) every plane, initially normal to the y -axis, becomes a plane containing the z -axis; (iii) there is no displacement parallel to the z -axis. It is found that such a state of flexure can be maintained by the application of surface tractions only, and these are calculated explicitly in terms of the derivatives of W with respect to the strain invariants. The surface tractions are normal to the surfaces on which they act, in their deformed state. Those acting on the surfaces initially normal to the x -axis are uniform over each of these surfaces. The assumption is then made that the stored-energy function W has the form, originally suggested by Mooney (1940), for rubber, W = C 1 ( λ 2 1 + λ 2 2 + λ 2 3 -3) + C 2 ( λ 2 2 λ 2 3 + λ 2 3 λ 2 1 + λ 2 1 λ 2 2 -3), where C 1 and C 2 are physical constants for the material and λ 1 , λ 2 , λ 3 are the principal extension ratios. For this case—and therefore for the incompressible neo-Hookean material (Rivlin 1948 a, b, c ), which is obtained from this by putting C 2 = 0—it is found that the flexure can be maintained without the application of surface tractions to the curved surface, provided that 2( a 1 - a 2 ) ( r 1 r 2 ) ½ = r 2 1 - r 2 2 , where ( a 1 - a 2 ) is the initial dimension of the cuboid, parallel to the x -axis, and r 1 and r 2 are the radii of the curved surfaces. When this condition is satisfied, the system of surface tractions applied to a boundary initially normal to the y -axis is equivalent to a couple M , proportional to ( C 1 + C 2 ). It is also found that the surface tractions applied to a boundary normal to the z -axis has a resultant F 2 proportional to ( C 1 - C 2 ).

Design of Milling Cutter for Machining Helical Curved Surfaces

2010 ◽  
Vol 44-47 ◽  
pp. 748-753 ◽  
Author(s):  
Jian Jun Yang ◽  
Qin Wu ◽  
Jun Gong ◽  
Zhi Yuan Rui
Keyword(s):  
Curved Surface ◽  
Screw Compressor ◽  
Curved Surfaces ◽  
Milling Cutter ◽  
Rolling Surface ◽  
Elementary Theories

The article focused on the design of milling cutter for machining helical curved surfaces. First, the characteristics of 3th generation profile in screw compressor were presented; then the equation of touch line between milling cutter and workpiece was constituted based on the envelop principle from milling cutter rolling surface and spiral curved surface; and elementary theories and methods were put forward for designing contour of milling cutter in machining spiral curved surfaces, finally the calculate equation was validated through the machining example by Matlab.

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