Geometric and Topological Multi-Resolution of n-dimensional Solids
|
|
- Gordon Gray
- 7 years ago
- Views:
Transcription
1 Geometric and Topological Multi-Resolution of n-dimensional Solids Luiz Velho IMPA - Instituto de Matematica Pura e Aplicada
2 Acknowledgements PhD. Thesis Work Esdras Medeiros Filho (IMPA) Co-Supervision Helio Lopes (PUC-Rio) Collaboration Thomas Lewiner (PUC-Rio) Luiz Velho October,
3 Outline Context and Motivation Mathematical Background Sampling Solid Objects Geometric / Topological Multiresolution Adaptive Multi-Triangulation Structure Examples and Applications Future Work Luiz Velho October,
4 Motivation Why Solids? Most Objects are NOT Hollow Matter Inside! [Kopf et al, 2007] An Empty Glass is Full of Air Inside Matters ;-) [C-SAFE, 2005] Luiz Velho October,
5 3D Applications Modeling CSG Animation Deformation Simulation [Romeiro et al, 2005] [de Goes et al, 2007] Fluids Scientific Visualization [de Goes et al, 2007] Luiz Velho October,
6 Image Analysis Segmentation 2D Applications [Felzenszwalb et al, 2004] Image Compression Adaptive Triangulation [Demaret et al, 2000] Medical Imaging Contouring [Lewiner et al, 2006] Luiz Velho October,
7 Related Work Geometric Multiresolution Multi-Triangulations Adaptive Meshes Etc.. [Marroquim et al, 2005] Topology Control Simplification Filtration / Persistence Etc.. [Zomorodian et al, 2005] Luiz Velho October,
8 What is Missing? Unified Geometric Topological Framework For Example: Simplest Topology / Highest Resolution Only Holes with Size Greater than X Fine Boundary and Coarse Interior * Why it is difficult? Continuous Geometry x Discrete Topology Representation / Data Structure Adaptation, Computation, etc [Romeiro et al, 2005] Luiz Velho October,
9 Game Plan Characteristic Function Multiresolution Hierachy Adaptation * Obs: Attributes! Luiz Velho October,
10 Relations with M.M. Morphological Scale-Spaces Mathematical Morphology Dilation / Erosion Binary Image (Characteristic Function) * Discrete Representation Level Sets Diffusion Interface Curve (Boundary) * Continuous Front Evolution Luiz Velho October,
11 Key Ideas Our Framework Solid Topology + Stochastic Point Sampling Geometric / Topological Operators Variable-Resolution Structure Advantages Integrate Geometry and Attributes Natural Notion of Scale General Adaptation Luiz Velho October,
12 Contributions Solid Poisson-Disk Sampling Scale Space for Characteristic Function α-filtration Multiresolution of α-solids G/T Multi-Triangulation Adaptation with Stellar and Handle Operators Luiz Velho October,
13 Stochastic Sampling/Reconstruction Poisson-Disk Distribution Random Uniform Sampling Irregular Pattern Points r-distant, at least r Reconstruction without Aliasing Combined Low-Pass Filter * Natural Scale / Resolution Luiz Velho October,
14 Sampling Solid Regions * Poisson-Disk Sampling is only Defined for Restriction to Must take into account does not follow geometry Luiz Velho October,
15 Two-Steps Algorithm (1) 1. Sample the Boundary 2. Sample the Interior minus r-tubular Neighborhood r * Stratified Sampling - Feature Sensitive Implementation - Dart Throwing - Quad-Tree Acceleration Luiz Velho October,
16 Multiresolution Sampling Spaces Nested Poisson-Disk Scale Spaces Disk Radius Multiresolution Hierarchy P 1 P P 0-1 Luiz Velho October,
17 Multiresolution Solid Sampling Apply Algorithm (1) for Tagged Samples: Boundary / Interior Resolution Level S 4 S 2 S 1 Luiz Velho October,
18 Structuring and Reconstruction Piecewise-Linear Approximation α-solids (α-shape + Regularization) * Subset of Delaunay Triangulation circumradius < α Luiz Velho October,
19 Looking for the Right Operators Operations on Simplicial Complexes Building Change Resolution Change Topology Types of Operators Topological Handlebody Theory Geometric Stellar Theory Luiz Velho October,
20 Handle Operators Change Topology Connected Components create Boundary destroy glue unglue Luiz Velho October,
21 Stellar Operators Change Combinatorial Structure Resolution split Structure weld flip Luiz Velho October,
22 Integrated Framework Combinatorial Manifold Operators Handle Stellar Properties Atomic Minimal Set Consistent (by construction) * Effective Abstractions Luiz Velho October,
23 Reconstruction from α-sampling Advancing Front Triangulation Ball-Pivoting Algorithm (2) while (points to process) while (e=candidate edge) p = ball_pivot e p create σ p glue σ p if (t=find seed) create σ t * Obs: Roll α-ball in Luiz Velho October,
24 Topological Guarantees Theorem (Amenta, 1999): If P r is an r-sample for r < k.lfs(s), then the crust is homeomorphic to S. * Applies to α-solids, (α = r) medial axis LFS(q) q r-sample r-solid Luiz Velho October,
25 Quality Guarantees Theorem (Medeiros et al, 2007): Let P α be an α-sampling of S, then the aspect ratio of C α (P α ) is bounded by. Worst case: a = b = r = α r Luiz Velho October,
26 Sampling Independence Different α-samplings Aspect Ratio Distribution Luiz Velho October,
27 α-filtration Defining the Multiresolution of Solids Ball-Pivoting at Each Level How to Move Gradually Between Levels? Change radius: 2 j < α < 2 j+1 Apply Stellar and Handle Operators * g(α) defines an Order for Samples Construction Strategies: Refinement Simplification Luiz Velho October,
28 Top-Down Construction Point Insertion Algorithm (3) while (samples to insert) if (inside) split σ p else create σ p glue σ p while (not Delaunay) switch (condition) case 1: flip e case 2: destroy σ t Luiz Velho October,
29 Bottom-Up Construction Point Removal Algorithm (4) while (samples to remove) while (valence 3) flip e if (inside) weld σ t if (boundary) create σ p weld σ p Luiz Velho October,
30 Coarse Grain Global Resolution Levels Luiz Velho October,
31 Intra Levels Fine Grain Local Luiz Velho October,
32 Properties Theorem (Uniqueness) : Given S α and g, the α-filtration generates a unique family of triangulations T α Theorem (Symmetry) : In α-filtration, the refinement sequence is the inverse of the simplification sequence insert p 1 split t 1 flip e 1 insert p 2 create t 2 glue t 2... insert p n split t n weld t n remov p n... unglue t 2 destroy t 2 remov p 2 flip e 1 weld t 1 remov p 1 Luiz Velho October,
33 G/T Multi-Triangulation Operators (stellar / handle) make only local changes (geometry / topology) Changes are weakly interdependent! Multi-Triangulation Structure Encodes Hierarchy Dependencies Representation for All Possible Meshes! Luiz Velho October,
34 Lattice Geometric Hierarchy Interior Decomposition Luiz Velho October,
35 Graph Topological Hierarchy Boundaries Luiz Velho October,
36 Geometric Dependencies Incident Faces at Same Level weld flip Topological All Edges must be Interior weld flip Luiz Velho October,
37 Mesh (half-edge) Data Structure h0 h1 h1 h3 h2 v h edge face vertex Multi-triangulation face-vertex edge-edge int-bd edge Luiz Velho October,
38 Adaptation Spatially Variant Function Domains Geometric Boundary (Shape) Interior (Attributes) Structure Components (Topology) Combinatory (Quality) Luiz Velho October,
39 Covering Elements Partition the Mesh in Regions simp.face ref.vertex int.edge int.edge int.edge bd.edge Priority Queues Sort Regions based on Adaptation Function refinement simp-faces int-edges int-edges simplification ref-vertices int-edges bd-edges Luiz Velho October,
40 Maintaining the Queues Algorithm (5) while (top_val > target) pop queue apply transition update elements Same Algorithm for All Tasks! Refinement / Simplification / Improvement Transition includes Dependencies Update re-evaluates Changes Luiz Velho October,
41 Algorithm (6) Dynamic Adaptation T 0 = base_triangulation initialize priority_queues while (adapting) read parameters evaluate f on T k simplify T k refine T k improve T k Algorithm (5) * Obs: Conservative! Luiz Velho October,
42 Adaptation Criteria Single Criteria (defaults for others) Multiple Criteria (must solve conflicts) Applications Examples Attribute Resolution Topology Granularity Geometry Detail Luiz Velho October,
43 Area of Triangles Attribute Example Luiz Velho October,
44 Topological Example Size of Holes Luiz Velho October,
45 Future Work Applications Extend to 3D Final Remarks Conclusions Theoretical Results Computationally Efficient Effective in Practice Luiz Velho October,
46 Thanks! Luiz Velho October,
A FRAMEWORK FOR REAL-TIME TERRAIN VISUALIZATION WITH ADAPTIVE SEMI- REGULAR MESHES
A FRAMEWORK FOR REAL-TIME TERRAIN VISUALIZATION WITH ADAPTIVE SEMI- REGULAR MESHES Lourena Rocha, Sérgio Pinheiro, Marcelo B. Vieira, Luiz Velho IMPA - Instituto Nacional de Matemática Pura e Aplicada
More informationSECONDARY STORAGE TERRAIN VISUALIZATION IN A CLIENT-SERVER ENVIRONMENT: A SURVEY
SECONDARY STORAGE TERRAIN VISUALIZATION IN A CLIENT-SERVER ENVIRONMENT: A SURVEY Kai Xu and Xiaofang Zhou School of Information Technology and Electrical Engineering The University of Queensland, Brisbane,
More informationModel Repair. Leif Kobbelt RWTH Aachen University )NPUT $ATA 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS !NALYSIS OF SURFACE QUALITY
)NPUT $ATA 2ANGE 3CAN #!$ 4OMOGRAPHY 2EMOVAL OF TOPOLOGICAL AND GEOMETRICAL ERRORS!NALYSIS OF SURFACE QUALITY 3URFACE SMOOTHING FOR NOISE REMOVAL 0ARAMETERIZATION 3IMPLIFICATION FOR COMPLEXITY REDUCTION
More informationEfficient Storage, Compression and Transmission
Efficient Storage, Compression and Transmission of Complex 3D Models context & problem definition general framework & classification our new algorithm applications for digital documents Mesh Decimation
More informationConstrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume *
Constrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume * Xiaosong Yang 1, Pheng Ann Heng 2, Zesheng Tang 3 1 Department of Computer Science and Technology, Tsinghua University, Beijing
More informationParallel Simplification of Large Meshes on PC Clusters
Parallel Simplification of Large Meshes on PC Clusters Hua Xiong, Xiaohong Jiang, Yaping Zhang, Jiaoying Shi State Key Lab of CAD&CG, College of Computer Science Zhejiang University Hangzhou, China April
More informationHow To Create A Surface From Points On A Computer With A Marching Cube
Surface Reconstruction from a Point Cloud with Normals Landon Boyd and Massih Khorvash Department of Computer Science University of British Columbia,2366 Main Mall Vancouver, BC, V6T1Z4, Canada {blandon,khorvash}@cs.ubc.ca
More informationGeometry and Topology from Point Cloud Data
Geometry and Topology from Point Cloud Data Tamal K. Dey Department of Computer Science and Engineering The Ohio State University Dey (2011) Geometry and Topology from Point Cloud Data WALCOM 11 1 / 51
More informationComputational Geometry. Lecture 1: Introduction and Convex Hulls
Lecture 1: Introduction and convex hulls 1 Geometry: points, lines,... Plane (two-dimensional), R 2 Space (three-dimensional), R 3 Space (higher-dimensional), R d A point in the plane, 3-dimensional space,
More informationBOĞAZİÇİ UNIVERSITY COMPUTER ENGINEERING
Parallel l Tetrahedral Mesh Refinement Mehmet Balman Computer Engineering, Boğaziçi University Adaptive Mesh Refinement (AMR) A computation ti technique used to improve the efficiency i of numerical systems
More informationLecture 7 - Meshing. Applied Computational Fluid Dynamics
Lecture 7 - Meshing Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Outline Why is a grid needed? Element types. Grid types.
More informationTetrahedron Meshes. Linh Ha
Tetrahedron Meshes Linh Ha Overview Fundamental Tetrahedron meshing Meshing polyhedra Tetrahedral shape Delaunay refinements Removing silver Variational Tetrahedral Meshing Meshing Polyhedra Is that easy?
More informationExpress Introductory Training in ANSYS Fluent Lecture 1 Introduction to the CFD Methodology
Express Introductory Training in ANSYS Fluent Lecture 1 Introduction to the CFD Methodology Dimitrios Sofialidis Technical Manager, SimTec Ltd. Mechanical Engineer, PhD PRACE Autumn School 2013 - Industry
More informationOff-line Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park
NSF GRANT # 0727380 NSF PROGRAM NAME: Engineering Design Off-line Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park Atul Thakur
More informationMultiresolution Terrain Database Visualization
Multiresolution Terrain Database Visualization Kai Xu School of Information Technology and Electrical Engineering The University of Queensland, Brisbane, QLD 4072, Australia kaixu@itee.uq.edu.au Abstract.
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationPre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More informationRepresenting Geography
3 Representing Geography OVERVIEW This chapter introduces the concept of representation, or the construction of a digital model of some aspect of the Earth s surface. The geographic world is extremely
More informationDigital Imaging and Communications in Medicine (DICOM) Supplement 132: Surface Segmentation Storage SOP Class
5 10 Digital Imaging and Communications in Medicine (DICOM) Supplement 132: Surface Segmentation Storage SOP Class 15 20 Prepared by: DICOM Standards Committee, Working Group 17 (3D) and 24 (Surgery) 25
More informationRobust Blind Watermarking Mechanism For Point Sampled Geometry
Robust Blind Watermarking Mechanism For Point Sampled Geometry Parag Agarwal Balakrishnan Prabhakaran Department of Computer Science, University of Texas at Dallas MS EC 31, PO Box 830688, Richardson,
More informationME6130 An introduction to CFD 1-1
ME6130 An introduction to CFD 1-1 What is CFD? Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat and mass transfer, chemical reactions, and related phenomena by solving numerically
More informationTopological Data Analysis Applications to Computer Vision
Topological Data Analysis Applications to Computer Vision Vitaliy Kurlin, http://kurlin.org Microsoft Research Cambridge and Durham University, UK Topological Data Analysis quantifies topological structures
More informationCharacterizing the Performance of Dynamic Distribution and Load-Balancing Techniques for Adaptive Grid Hierarchies
Proceedings of the IASTED International Conference Parallel and Distributed Computing and Systems November 3-6, 1999 in Cambridge Massachusetts, USA Characterizing the Performance of Dynamic Distribution
More informationIntersection of a Line and a Convex. Hull of Points Cloud
Applied Mathematical Sciences, Vol. 7, 213, no. 13, 5139-5149 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.213.37372 Intersection of a Line and a Convex Hull of Points Cloud R. P. Koptelov
More informationA fast and robust algorithm to count topologically persistent holes in noisy clouds
A fast and robust algorithm to count topologically persistent holes in noisy clouds Vitaliy Kurlin Durham University Department of Mathematical Sciences, Durham, DH1 3LE, United Kingdom vitaliy.kurlin@gmail.com,
More informationDual Marching Cubes: Primal Contouring of Dual Grids
Dual Marching Cubes: Primal Contouring of Dual Grids Scott Schaefer and Joe Warren Rice University 6100 Main St. Houston, TX 77005 sschaefe@rice.edu and jwarren@rice.edu Abstract We present a method for
More informationIntroduction to ANSYS
Lecture 3 Introduction to ANSYS Meshing 14. 5 Release Introduction to ANSYS Meshing 2012 ANSYS, Inc. March 27, 2014 1 Release 14.5 Introduction to ANSYS Meshing What you will learn from this presentation
More information3D Model of the City Using LiDAR and Visualization of Flood in Three-Dimension
3D Model of the City Using LiDAR and Visualization of Flood in Three-Dimension R.Queen Suraajini, Department of Civil Engineering, College of Engineering Guindy, Anna University, India, suraa12@gmail.com
More informationEuclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li. Advised by: Dave Mount. May 22, 2014
Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li Advised by: Dave Mount May 22, 2014 1 INTRODUCTION In this report we consider the implementation of an efficient
More informationTOWARDS AN AUTOMATED HEALING OF 3D URBAN MODELS
TOWARDS AN AUTOMATED HEALING OF 3D URBAN MODELS J. Bogdahn a, V. Coors b a University of Strathclyde, Dept. of Electronic and Electrical Engineering, 16 Richmond Street, Glasgow G1 1XQ UK - jurgen.bogdahn@strath.ac.uk
More informationSegmentation of building models from dense 3D point-clouds
Segmentation of building models from dense 3D point-clouds Joachim Bauer, Konrad Karner, Konrad Schindler, Andreas Klaus, Christopher Zach VRVis Research Center for Virtual Reality and Visualization, Institute
More informationMultiphysics Software Applications in Reverse Engineering
Multiphysics Software Applications in Reverse Engineering *W. Wang 1, K. Genc 2 1 University of Massachusetts Lowell, Lowell, MA, USA 2 Simpleware, Exeter, United Kingdom *Corresponding author: University
More informationIntroduction to ANSYS ICEM CFD
Workshop 8.2 3D Pipe Junction 14.5 Release Introduction to ANSYS ICEM CFD 2012 ANSYS, Inc. April 1, 2013 1 Release 14.5 3D Pipe Junction 3D Pipe Junction This is a simple 4-way pipe intersection with two
More informationHowTo Rhino & ICEM. 1) New file setup: choose Millimeter (automatically converts to Meters if imported to ICEM)
HowTo Rhino & ICEM Simple 2D model 1) New file setup: choose Millimeter (automatically converts to Meters if imported to ICEM) 2) Set units: File Properties Units: Model units: should already be Millimeters
More informationSurface bundles over S 1, the Thurston norm, and the Whitehead link
Surface bundles over S 1, the Thurston norm, and the Whitehead link Michael Landry August 16, 2014 The Thurston norm is a powerful tool for studying the ways a 3-manifold can fiber over the circle. In
More informationMultiresolution 3D Rendering on Mobile Devices
Multiresolution 3D Rendering on Mobile Devices Javier Lluch, Rafa Gaitán, Miguel Escrivá, and Emilio Camahort Computer Graphics Section Departament of Computer Science Polytechnic University of Valencia
More informationwalberla: Towards an Adaptive, Dynamically Load-Balanced, Massively Parallel Lattice Boltzmann Fluid Simulation
walberla: Towards an Adaptive, Dynamically Load-Balanced, Massively Parallel Lattice Boltzmann Fluid Simulation SIAM Parallel Processing for Scientific Computing 2012 February 16, 2012 Florian Schornbaum,
More informationCONVERGE Features, Capabilities and Applications
CONVERGE Features, Capabilities and Applications CONVERGE CONVERGE The industry leading CFD code for complex geometries with moving boundaries. Start using CONVERGE and never make a CFD mesh again. CONVERGE
More informationCreation of an Unlimited Database of Virtual Bone. Validation and Exploitation for Orthopedic Devices
Creation of an Unlimited Database of Virtual Bone Population using Mesh Morphing: Validation and Exploitation for Orthopedic Devices Najah Hraiech 1, Christelle Boichon 1, Michel Rochette 1, 2 Thierry
More information1. Abstract 2. Introduction 3. Algorithms and Techniques
MS PROJECT Virtual Surgery Piyush Soni under the guidance of Dr. Jarek Rossignac, Brian Whited Georgia Institute of Technology, Graphics, Visualization and Usability Center Atlanta, GA piyush_soni@gatech.edu,
More informationReflection and Refraction
Equipment Reflection and Refraction Acrylic block set, plane-concave-convex universal mirror, cork board, cork board stand, pins, flashlight, protractor, ruler, mirror worksheet, rectangular block worksheet,
More informationA Generalized Marching Cubes Algorithm Based On Non-Binary Classifications
Konrad-Zuse-Zentrum fu r Informationstechnik Berlin Takustraße 7 D-14195 Berlin-Dahlem Germany HANS-CHRISTIAN HEGE DETLEV STALLING MARTIN SEEBASS MALTE ZOCKLER A Generalized Marching Cubes Algorithm Based
More informationShortcut sets for plane Euclidean networks (Extended abstract) 1
Shortcut sets for plane Euclidean networks (Extended abstract) 1 J. Cáceres a D. Garijo b A. González b A. Márquez b M. L. Puertas a P. Ribeiro c a Departamento de Matemáticas, Universidad de Almería,
More informationTriangulation by Ear Clipping
Triangulation by Ear Clipping David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 1998-2016. All Rights Reserved. Created: November 18, 2002 Last Modified: August 16, 2015 Contents
More informationBig Ideas in Mathematics
Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards
More informationDelaunay Based Shape Reconstruction from Large Data
Delaunay Based Shape Reconstruction from Large Data Tamal K. Dey Joachim Giesen James Hudson Ohio State University, Columbus, OH 4321, USA Abstract Surface reconstruction provides a powerful paradigm for
More informationMultiresolution Tetrahedral Meshes: an Analysis and a Comparison
Multiresolution Tetrahedral Meshes: an Analysis and a Comparison Emanuele Danovaro, Leila De Floriani Dipartimento di Informatica e Scienze dell Informazione, Università di Genova, Via Dodecaneso 35, 16146
More informationFEAWEB ASP Issue: 1.0 Stakeholder Needs Issue Date: 03/29/2000. 04/07/2000 1.0 Initial Description Marco Bittencourt
)($:(%$63 6WDNHKROGHU1HHGV,VVXH 5HYLVLRQ+LVWRU\ 'DWH,VVXH 'HVFULSWLRQ $XWKRU 04/07/2000 1.0 Initial Description Marco Bittencourt &RQILGHQWLDO DPM-FEM-UNICAMP, 2000 Page 2 7DEOHRI&RQWHQWV 1. Objectives
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationCourse Description. Spring 2004 1
Spring 2004 1 Course Description AE4375: advanced treatment for undergrads; focus on learning and aplying CAD to engineering; CAD modeling projects. AE6380: graduate course on CAD focusing on how tools
More informationCOMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS
International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 7, Issue 2, March-April 2016, pp. 30 52, Article ID: IJARET_07_02_004 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=7&itype=2
More informationPrentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009
Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level
More informationTennessee Mathematics Standards 2009-2010 Implementation. Grade Six Mathematics. Standard 1 Mathematical Processes
Tennessee Mathematics Standards 2009-2010 Implementation Grade Six Mathematics Standard 1 Mathematical Processes GLE 0606.1.1 Use mathematical language, symbols, and definitions while developing mathematical
More informationABAQUS/CAE Tutorial: Analysis of an Aluminum Bracket
H. Kim FEA Tutorial 1 ABAQUS/CAE Tutorial: Analysis of an Aluminum Bracket Hyonny Kim last updated: August 2004 In this tutorial, you ll learn how to: 1. Sketch 2D geometry & define part. 2. Define material
More informationA Short Introduction to Computer Graphics
A Short Introduction to Computer Graphics Frédo Durand MIT Laboratory for Computer Science 1 Introduction Chapter I: Basics Although computer graphics is a vast field that encompasses almost any graphical
More informationNeural Gas for Surface Reconstruction
Neural Gas for Surface Reconstruction Markus Melato, Barbara Hammer, Kai Hormann IfI Technical Report Series IfI-07-08 Impressum Publisher: Institut für Informatik, Technische Universität Clausthal Julius-Albert
More informationComputer Graphics AACHEN AACHEN AACHEN AACHEN. Public Perception of CG. Computer Graphics Research. Methodological Approaches - - - - - - - - - -
Public Perception of CG Games Computer Graphics Movies Computer Graphics Research algorithms & data structures fundamental continuous & discrete mathematics optimization schemes 3D reconstruction global
More informationAn Overview of the Finite Element Analysis
CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry
More informationCyber Graphics. Abstract. 1. What is cyber graphics? 2. An incrementally modular abstraction hierarchy of shape invariants
Preprint of the Keynote Paper: Tosiyasu L. Kunii, Cyber Graphics, Proceedings of the First International Symposium on Cyber Worlds (CW2002), November 6-8 2002 Tokyo, Japan, in press, IEEE Computer Society
More informationDatum > Curve KIM,ME,NIU
Datum > Curve Intersect First create at least one quilt on the surface of the model. Feature > Surface (> New) > Copy (do not use offset that creates a surface off the solid surface even with zero offset)
More informationTerrain Database. Kai Xu, Xiaofang Zhou, Ke Deng, Heng Tao Shen, and Xuemin Lin. Abstract
Direct Mesh: Visualization for Multiresolution 1 Terrain Database Kai Xu, Xiaofang Zhou, Ke Deng, Heng Tao Shen, and Xuemin Lin Abstract Multiresolution Triangular Mesh (MTM) models are widely used in
More informationLECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS
LECTURE NOTES ON GENERALIZED HEEGAARD SPLITTINGS TOSHIO SAITO, MARTIN SCHARLEMANN AND JENNIFER SCHULTENS 1. Introduction These notes grew out of a lecture series given at RIMS in the summer of 2001. The
More informationFrom Scattered Samples to Smooth Surfaces
From Scattered Samples to Smooth Surfaces Kai Hormann 1 California Institute of Technology (a) (b) (c) (d) Figure 1: A point cloud with 4,100 scattered samples (a), its triangulation with 7,938 triangles
More informationPro/ENGINEER Wildfire 4.0 Basic Design
Introduction Datum features are non-solid features used during the construction of other features. The most common datum features include planes, axes, coordinate systems, and curves. Datum features do
More informationCAD/CAE interoperability, an automatic generation of Analysis Model based on idealization of CAD geometry
CAD/CAE interoperability, an automatic generation of Analysis Model based on idealization of CAD geometry M. HAMDI a, N. AIFAOUI a, B. LOUHICHI a, A. BENAMARA a a. Laboratoire de Génie Mécanique Lab-Ma-05,
More informationActivity Set 4. Trainer Guide
Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES
More informationVisualization and Astronomy
Visualization and Astronomy Prof.dr. Jos Roerdink Institute for Mathematics and Computing Science University of Groningen URL: www.cs.rug.nl/svcg/ Scientific Visualization & Computer Graphics Group Institute
More informationPart-Based Recognition
Part-Based Recognition Benedict Brown CS597D, Fall 2003 Princeton University CS 597D, Part-Based Recognition p. 1/32 Introduction Many objects are made up of parts It s presumably easier to identify simple
More informationSPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM
SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationRaster Data Structures
Raster Data Structures Tessellation of Geographical Space Geographical space can be tessellated into sets of connected discrete units, which completely cover a flat surface. The units can be in any reasonable
More information39 Symmetry of Plane Figures
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationOracle8i Spatial: Experiences with Extensible Databases
Oracle8i Spatial: Experiences with Extensible Databases Siva Ravada and Jayant Sharma Spatial Products Division Oracle Corporation One Oracle Drive Nashua NH-03062 {sravada,jsharma}@us.oracle.com 1 Introduction
More informationHydrodynamic Limits of Randomized Load Balancing Networks
Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli
More informationCustomer Training Material. Lecture 4. Meshing in Mechanical. Mechanical. ANSYS, Inc. Proprietary 2010 ANSYS, Inc. All rights reserved.
Lecture 4 Meshing in Mechanical Introduction to ANSYS Mechanical L4-1 Chapter Overview In this chapter controlling meshing operations is described. Topics: A. Global Meshing Controls B. Local Meshing Controls
More informationCurrent Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary
Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:
More informationCAMI Education linked to CAPS: Mathematics
- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to
More informationABAQUS Tutorial. 3D Modeling
Spring 2011 01/21/11 ABAQUS Tutorial 3D Modeling This exercise intends to demonstrate the steps you would follow in creating and analyzing a simple solid model using ABAQUS CAE. Introduction A solid undergoes
More informationanimation animation shape specification as a function of time
animation animation shape specification as a function of time animation representation many ways to represent changes with time intent artistic motion physically-plausible motion efficiency control typically
More informationNorth Carolina Math 2
Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4.
More informationGeometric Modelling & Curves
Geometric Modelling & Curves Geometric Modeling Creating symbolic models of the physical world has long been a goal of mathematicians, scientists, engineers, etc. Recently technology has advanced sufficiently
More informationVirtual Diesel Particulate Filters: Simulation of the Structure, Exhaust Gas Flow and Particle Deposition
Virtual Diesel Particulate Filters: Simulation of the Structure, Exhaust Gas Flow and Particle Deposition Dr. Stefan Rief, Dipl.-Math. Kilian Schmidt Andreas Wiegmann, PhD Department Flow and Material
More informationAPP INVENTOR. Test Review
APP INVENTOR Test Review Main Concepts App Inventor Lists Creating Random Numbers Variables Searching and Sorting Data Linear Search Binary Search Selection Sort Quick Sort Abstraction Modulus Division
More informationCFD analysis for road vehicles - case study
CFD analysis for road vehicles - case study Dan BARBUT*,1, Eugen Mihai NEGRUS 1 *Corresponding author *,1 POLITEHNICA University of Bucharest, Faculty of Transport, Splaiul Independentei 313, 060042, Bucharest,
More informationMedial Axis Construction and Applications in 3D Wireless Sensor Networks
Medial Axis Construction and Applications in 3D Wireless Sensor Networks Su Xia, Ning Ding, Miao Jin, Hongyi Wu, and Yang Yang Presenter: Hongyi Wu University of Louisiana at Lafayette Outline Introduction
More informationP. Lu, Sh. Huang and K. Jiang
416 Rev. Adv. Mater. Sci. 33 (2013) 416-422 P. Lu, Sh. Huang and K. Jiang NUMERICAL ANALYSIS FOR THREE-DIMENSIONAL BULK METAL FORMING PROCESSES WITH ARBITRARILY SHAPED DIES USING THE RIGID/VISCO-PLASTIC
More informationSolid Edge ST3 Advances the Future of 3D Design
SUMMARY AND OPINION Solid Edge ST3 Advances the Future of 3D Design A Product Review White Paper Prepared by Collaborative Product Development Associates, LLC for Siemens PLM Software The newest release
More informationCATIA Wireframe & Surfaces TABLE OF CONTENTS
TABLE OF CONTENTS Introduction... 1 Wireframe & Surfaces... 2 Pull Down Menus... 3 Edit... 3 Insert... 4 Tools... 6 Generative Shape Design Workbench... 7 Bottom Toolbar... 9 Tools... 9 Analysis... 10
More informationCFD SIMULATION OF SDHW STORAGE TANK WITH AND WITHOUT HEATER
International Journal of Advancements in Research & Technology, Volume 1, Issue2, July-2012 1 CFD SIMULATION OF SDHW STORAGE TANK WITH AND WITHOUT HEATER ABSTRACT (1) Mr. Mainak Bhaumik M.E. (Thermal Engg.)
More informationLine Segment Intersection
Chapter 1 Line Segment Intersection (with material from [1], [3], and [5], pictures are missing) 1.1 Interval case We first think of having n intervals (e.g., the x-range of horizontal line segments) and
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationVector storage and access; algorithms in GIS. This is lecture 6
Vector storage and access; algorithms in GIS This is lecture 6 Vector data storage and access Vectors are built from points, line and areas. (x,y) Surface: (x,y,z) Vector data access Access to vector
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationA COOL AND PRACTICAL ALTERNATIVE TO TRADITIONAL HASH TABLES
A COOL AND PRACTICAL ALTERNATIVE TO TRADITIONAL HASH TABLES ULFAR ERLINGSSON, MARK MANASSE, FRANK MCSHERRY MICROSOFT RESEARCH SILICON VALLEY MOUNTAIN VIEW, CALIFORNIA, USA ABSTRACT Recent advances in the
More informationRAF Application. The Origin of Life. E. Bingham 1 L. Hutton-Smith 2 E. Rolls 2. Oxford Summer School in Computational Biology, 2013
The Origin of Life E. Bingham 1 L. Hutton-Smith 2 E. Rolls 2 1 Department of Mathematics University of North Carolina 2 Mathematical Institute University of Oxford Oxford Summer School in Computational
More informationCATIA V5R21 - FACT SHEET
CATIA V5R21 - FACT SHEET Introduction What s New at a Glance Overview Detailed Description INTRODUCTION CATIA V5 is the leading solution for product success. It addresses all manufacturing organizations;
More informationCatalan Numbers. Thomas A. Dowling, Department of Mathematics, Ohio State Uni- versity.
7 Catalan Numbers Thomas A. Dowling, Department of Mathematics, Ohio State Uni- Author: versity. Prerequisites: The prerequisites for this chapter are recursive definitions, basic counting principles,
More informationVolumetric Meshes for Real Time Medical Simulations
Volumetric Meshes for Real Time Medical Simulations Matthias Mueller and Matthias Teschner Computer Graphics Laboratory ETH Zurich, Switzerland muellerm@inf.ethz.ch, http://graphics.ethz.ch/ Abstract.
More information