Efficient Progressive Compression of 3D Points by Maximizing Tangent-Plane Continuity

2012 ◽  
Author(s):  
Wenfei Jiang ◽  
Jiang Tian ◽  
Kangying Cai ◽  
Fan Zhang ◽  
Tao Luo
Keyword(s):  
Tangent Plane

scholarly journals Path Planning and Replanning for Mobile Robot Navigation on 3D Terrain: An Approach Based on Geodesic

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Kun-Lin Wu ◽  
Ting-Jui Ho ◽  
Sean A. Huang ◽  
Kuo-Hui Lin ◽  
Yueh-Chen Lin ◽  
...  
Keyword(s):  
Mobile Robot ◽  
Robot Navigation ◽  
Tangent Plane ◽  
Intersection Point ◽  
Mobile Robot Navigation ◽  
Complete Manifold ◽  
Geodesic Triangle ◽  
Planning Stage ◽  
3D Terrain ◽  
Bonnet Theorem

In this paper, mobile robot navigation on a 3D terrain with a single obstacle is addressed. The terrain is modelled as a smooth, complete manifold with well-defined tangent planes and the hazardous region is modelled as an enclosing circle with a hazard grade tuned radius representing the obstacle projected onto the terrain to allow efficient path-obstacle intersection checking. To resolve the intersections along the initial geodesic, by resorting to the geodesic ideas from differential geometry on surfaces and manifolds, we present a geodesic-based planning and replanning algorithm as a new method for obstacle avoidance on a 3D terrain without using boundary following on the obstacle surface. The replanning algorithm generates two new paths, each a composition of two geodesics, connected via critical points whose locations are found to be heavily relying on the exploration of the terrain via directional scanning on the tangent plane at the first intersection point of the initial geodesic with the circle. An advantage of this geodesic path replanning procedure is that traversability of terrain on which the detour path traverses could be explored based on the local Gauss-Bonnet Theorem of the geodesic triangle at the planning stage. A simulation demonstrates the practicality of the analytical geodesic replanning procedure for navigating a constant speed point robot on a 3D hill-like terrain.

scholarly journals Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

scholarly journals Nonlocal Estimation of Manifold Structure

2006 ◽  
Vol 18 (10) ◽  
pp. 2509-2528 ◽  
Author(s):  
Yoshua Bengio ◽  
Martin Monperrus ◽  
Hugo Larochelle
Keyword(s):  
Large Class ◽  
Manifold Learning ◽  
Learning Algorithms ◽  
Tangent Plane ◽  
Training Data ◽  
Kernel Learning ◽  
Manifold Structure ◽  
Prediction Function ◽  
Underlying Manifold ◽  
Shared Structure

We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as kernel learning algorithms will suffer from the curse of dimensionality, at the dimension of the true underlying manifold. This observation invites an exploration of nonlocal manifold learning algorithms that attempt to discover shared structure in the tangent planes at different positions. A training criterion for such an algorithm is proposed, and experiments estimating a tangent plane prediction function are presented, showing its advantages with respect to local manifold learning algorithms: it is able to generalize very far from training data (on learning handwritten character image rotations), where local nonparametric methods fail.

Mesh duality and Legendre duality

1990 ◽  
Vol 428 (1875) ◽  
pp. 351-377 ◽  
Keyword(s):  
Piecewise Linear ◽  
Voronoi Tessellation ◽  
Tangent Plane ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Dual Transformation ◽  
Delaunay Mesh ◽  
Definition Of

We describe a sense in which mesh duality is equivalent to Legendre duality. That is, a general pair of meshes, which satisfy a definition of duality for meshes, are shown to be the projection of a pair of piecewise linear functions that are dual to each other in the sense of a Legendre dual transformation. In applications the latter functions can be a tangent plane approximation to a smoother function, and a chordal plane approximation to its Legendre dual. Convex examples include one from meteorology, and also the relation between the Delaunay mesh and the Voronoi tessellation. The latter are shown to be the projections of tangent plane and chordal approximations to the same paraboloid.

Pseudoedge: A Hierarchical Skeletal Modeler for the Design of Structural Components

1991 ◽  
Author(s):  
David L. Bonner ◽  
Mark J. Jakiela ◽  
Masaki Watanabe
Keyword(s):  
Parameter Space ◽  
Local Control ◽  
Tangent Plane ◽  
Design Model ◽  
Shape Features ◽  
Structural Components ◽  
General Shape ◽  
Isolated Singularities ◽  
Traditional Design ◽  
Global And Local

Abstract A new design model for the creation of mechanical components has been developed. In this model, the shape is expressed by its areas of prominence or maximum curvature, for which we use the term pseudoedges. In terms of traditional design, these represent both fillet, chamfer and intersection lines, and more general shape features. The pseudoedges of the model combine with a skeletal shape that is used as a starting form, thereby creating a hierarchy of geometric dependencies that affords both global and local control. The surface is represented by a quilt of parametric Bezier patches, with tangent plane continuity everywhere and only certain isolated singularities. Considerable degrees of deformation are possible, with predictable control and at small computational expense; there is no need for computation of intersections or parameter space trimming of patches.

A Local Approach for Computing Smooth B-spline Surfaces for Arbitrary Quadrilateral Base Meshes

2021 ◽  
pp. 1-28
Author(s):  
Dennis Mosbach ◽  
Katja Schladitz ◽  
Bernd Hamann ◽  
Hans Hagen
Keyword(s):  
Degrees Of Freedom ◽  
Tangent Plane ◽  
Local Approach ◽  
Approximation Process ◽  
Surface Approximation ◽  
B Spline ◽  
Spline Surfaces ◽  
Continuous Surface ◽  
Normal Vectors ◽  
The Individual

Abstract We present a method for approximating surface data of arbitrary topology by a model of smoothly connected B-spline surfaces. Most of the existing solutions for this problem use constructions with limited degrees of freedom or they address smoothness between surfaces in a post-processing step, often leading to undesirable surface behavior in proximity of the boundaries. Our contribution is the design of a local method for the approximation process. We compute a smooth B-spline surface approximation without imposing restrictions on the topology of a quadrilateral base mesh defining the individual B-spline surfaces, the used B-spline knot vectors, or the number of B-spline control points. Exact tangent plane continuity can generally not be achieved for a set of B-spline surfaces for an arbitrary underlying quadrilateral base mesh. Our method generates a set of B-spline surfaces that lead to a nearly tangent plane continuous surface approximation and is watertight, i.e., continuous. The presented examples demonstrate that we can generate B-spline approximations with differences of normal vectors along shared boundary curves of less than one degree. Our approach can also be adapted to locally utilize other approximation methods leading to higher orders of continuity.

Thermodynamic Analysis of Phase Behavior at High Capillary Pressure

SPE Journal ◽  
2018 ◽  
Vol 23 (06) ◽  
pp. 1977-1990 ◽  
Author(s):  
Mohsen Rezaveisi ◽  
Kamy Sepehrnoori ◽  
Gary A. Pope ◽  
Russell T. Johns
Keyword(s):  
Phase Behavior ◽  
Capillary Pressure ◽  
Thermodynamic Analysis ◽  
Phase Stability ◽  
Tangent Plane ◽  
General Purpose ◽  
Distance Method ◽  
Capillary Equilibrium ◽  
Production Behavior ◽  
Reservoir Simulator

Summary High capillary pressure has a significant effect on the phase behavior of fluid mixtures. The capillary pressure is high in unconventional reservoirs because of the small pores in the rock, so understanding the effect of capillary pressure on phase behavior is necessary for reliable modeling of unconventional shale-gas and tight-oil reservoirs. As the main finding of this paper, first we show that the tangent-plane-distance method cannot be used to determine phase stability and present a rigorous thermodynamic analysis of the problem of phase stability with capillary pressure. Second, we demonstrate that there is a maximum capillary pressure (Pcmax) where calculation of capillary equilibrium using bulk-phase thermodynamics is possible and derive the necessary equations to obtain this maximum capillary pressure. We also briefly discuss the implementation of the capillary equilibrium in a general-purpose compositional reservoir simulator. Two simulation case studies for synthetic gas condensate reservoirs were performed to illustrate the influence of capillary pressure on production behavior for the fluids studied.

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