NORMAL-MAP BETWEEN NORMAL-COMPATIBLE MANIFOLDS

2007 ◽  
Vol 17 (05) ◽  
pp. 403-421 ◽  
Author(s):  
FREDERIC CHAZAL ◽  
ANDRE LIEUTIER ◽  
JAREK ROSSIGNAC
Keyword(s):  
Hausdorff Distance ◽  
Arbitrary Dimension ◽  
The Other ◽  
Fréchet Distance ◽  
Feature Size ◽  
Normal Map ◽  
Frechet Distance ◽  
Minimum Feature Size

Consider two (n−1)-dimensional manifolds, S and S′ in ℝn. We say that they are normal-compatible when the closest projection of each one onto the other is a homeomorphism. We give a tight condition under which S and S′ are normal-compatible. It involves the minimum feature size of S and of S′ and the Hausdorff distance between them. Furthermore, when S and S′ are normal-compatible, their Frechet distance is equal to their Hausdorff distance. Our results hold for arbitrary dimension n.

BALL-MAP: HOMEOMORPHISM BETWEEN COMPATIBLE SURFACES

2010 ◽  
Vol 20 (03) ◽  
pp. 285-306 ◽  
Author(s):  
FRÉDÉRIC CHAZAL ◽  
ANDRÉ LIEUTIER ◽  
JAREK ROSSIGNAC ◽  
BRIAN WHITED
Keyword(s):  
3D Modeling ◽  
Hausdorff Distance ◽  
Feature Size ◽  
Ambient Isotopy ◽  
Minimum Feature Size

Homeomorphisms between curves and between surfaces are fundamental to many applications of 3D modeling, graphics, and animation. They define how to map a texture from one object to another, how to morph between two shapes, and how to measure the discrepancy between shapes or the variability in a class of shapes. Previously proposed maps between two surfaces, S and S′, suffer from two drawbacks: (1) it is difficult to formally define a relation between S and S′ which guarantees that the map will be bijective and (2) mapping a point x of S to a point x′ of S′ and then mapping x′ back to S does in general not yield x, making the map asymmetric. We propose a new map, called ball-map, that is symmetric. We define simple and precise conditions for the ball-map to be a homeomorphism. We show that these conditions apply when the minimum feature size of each surface exceeds their Hausdorff distance. The ball-map, BM S,S′, between two such manifolds, S and S′, maps each point x of S to a point x′ = BM s,s′(x) of S′. BM S′,S is the inverse of BM S,S′, hence BM is symmetric. We also show that, when S and S′ are Ck (n - 1)-manifolds in ℝn, BM S,S′ is a Ck-1 diffeomorphism and defines a Ck-1 ambient isotopy that smoothly morphs between S to S′. In practice, the ball-map yields an excellent map for transferring parameterizations and textures between ball compatible curves or surfaces. Furthermore, it may be used to define a morph, during which each point x of S travels to the corresponding point x′ of S′ along a broken line that is normal to S at x and to S′ at x′.

scholarly journals The Topological Correctness of PL Approximations of Isomanifolds

2021 ◽  
Author(s):  
Jean-Daniel Boissonnat ◽  
Mathijs Wintraecken
Keyword(s):  
Smooth Function ◽  
Piecewise Linear ◽  
Arbitrary Dimension ◽  
Ambient Space ◽  
Zero Set ◽  
Fréchet Distance ◽  
Frechet Distance ◽  
Topological Correctness ◽  
Arbitrary Dimensions

AbstractIsomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function $$f: {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d-n}$$ f : R d → R d - n . A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation $$\mathcal {T}$$ T of the ambient space $${\mathbb {R}}^d$$ R d . In this paper, we give conditions under which the PL approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine and thick triangulation $$\mathcal {T}$$ T . This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary.

scholarly journals Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

2021 ◽  
Author(s):  
Lie Fu ◽  
Robert Laterveer ◽  
Charles Vial
Keyword(s):  
Projective Variety ◽  
Arbitrary Dimension ◽  
Smooth Projective Variety ◽  
The Other ◽  
Fano Variety ◽  
Fano Varieties ◽  
Intersection Product ◽  
Canonical Class ◽  
Main Input

AbstractGiven a smooth projective variety, a Chow–Künneth decomposition is called multiplicative if it is compatible with the intersection product. Following works of Beauville and Voisin, Shen and Vial conjectured that hyper-Kähler varieties admit a multiplicative Chow–Künneth decomposition. In this paper, based on the mysterious link between Fano varieties with cohomology of K3 type and hyper-Kähler varieties, we ask whether Fano varieties with cohomology of K3 type also admit a multiplicative Chow–Künneth decomposition, and provide evidence by establishing their existence for cubic fourfolds and Küchle fourfolds of type c7. The main input in the cubic hypersurface case is the Franchetta property for the square of the Fano variety of lines; this was established in our earlier work in the fourfold case and is generalized here to arbitrary dimension. On the other end of the spectrum, we also give evidence that varieties with ample canonical class and with cohomology of K3 type might admit a multiplicative Chow–Künneth decomposition, by establishing this for two families of Todorov surfaces.

Simulation and Characterization of Nanoparticle Thermal Conductivity for a Microscale Selective Laser Sintering System

2021 ◽  
Author(s):  
Joshua Grose ◽  
Obehi G. Dibua ◽  
Dipankar Behera ◽  
Chee S. Foong ◽  
Michael Cullinan
Keyword(s):  
Thermal Conductivity ◽  
Selective Laser Sintering ◽  
Primary Source ◽  
Laser Sintering ◽  
Metal Nanoparticle ◽  
Feature Size ◽  
Cad Models ◽  
Minimum Feature Size ◽  
Geometric Arrangements ◽  
Thermal Simulations

Abstract Additive Manufacturing (AM) technologies are often restricted by the minimum feature size of parts they can repeatably build. The microscale selective laser sintering (μ-SLS) process, which is capable of producing single micron resolution parts, addresses this issue directly. However, the unwanted dissipation of heat within the powder bed of a μ-SLS device during laser sintering is a primary source of error that limits the minimum feature size of the producible parts. A particle scale thermal model is needed to characterize the thermal properties of the nanoparticles undergoing sintering and allow for the prediction of heat affected zones (HAZ) and the improvement of final part quality. Thus, this paper presents a method for the determination of the effective thermal conductivity of metal nanoparticle beds in a microscale selective laser sintering process using finite element simulations in ANSYS. CAD models of nanoparticle groups at various timesteps during sintering are developed from Phase Field Modeling (PFM) output data, and steady state thermal simulations are performed on each group. The complete simulation framework developed in this work is adaptable to particle groups of variable sizes and geometric arrangements. Results from the thermal models are used to estimate the thermal conductivity of the copper nanoparticles as a function of sintering duration.

scholarly journals A Method for Fabricating Arrays of Nanopatterns with the Feature Size beyond Diffraction Limit

2008 ◽  
Vol 2008 ◽  
pp. 1-4 ◽  
Author(s):  
Shuhong Li ◽  
Lifang Shi ◽  
Xiaochun Dong ◽  
Chunlei Du ◽  
Yudong Zhang
Keyword(s):  
Time Domain ◽  
Finite Difference Time Domain ◽  
Diffraction Limit ◽  
Evanescent Waves ◽  
The Other ◽  
Optical Intensity ◽  
Polystyrene Spheres ◽  
Feature Size ◽  
Contact Points ◽  
Difference Time

A convenient lithographic technique is proposed in this paper, which can be used to produce subdiffraction-limit arrays of nanopatterns over large areas (about several square centimeters). An array of polystyrene spheres (PS) is arranged on the surface of a layer of silver which has a thickness of about tens of nanometers. With the normal illumination light of wavelength 365 nm perpendicular to the substrate, PS can generate an array of optical patterns with high intensity at their contact points with silver. By designing the silver slab, the evanescent waves that carry subwavelength information about the optical patterns are substantially enhanced, while propagating components are restrained. In the photoresist which is on the other side of silver, the optical intensity is redistributed and subdiffraction-limit patterns are obtained after exposure and development. Simulation by finite-difference time-domain (FDTD) and experiments were carried out to verify the technique. The results show that by using PS with diameter of 600 nm, nanopatterns with dimension of less than 80 nm can be obtained.

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