# Dual Marching Cubes: Primal Contouring of Dual Grids

Size: px
Start display at page:

## Transcription

1 Dual Marching Cubes: Primal Contouring of Dual Grids Scott Schaefer and Joe Warren Rice University 6100 Main St. Houston, TX and Abstract We present a method for contouring an implicit function using a grid topologically dual to structured grids such as octrees. By aligning the vertices of the dual grid with the features of the implicit function, we are able to reproduce thin features of the extracted surface without excessive subdivision required by methods such as Marching Cubes or Dual Contouring. Dual Marching Cubes produces a crackfree, adaptive polygonalization of the surface that reproduces sharp features. Our approach maintains the advantage of using structured grids for operations such as CSG while being able to conform to the relevant features of the implicit function yielding much sparser polygonalizations than has been possible using structured grids. 1. Introduction Implicit modeling has become a popular method for extracting surfaces from volumetric information arising from sources such as MRI data and CAT scans. In this technique, a volumetric function is given, whose value typically represents density information or distance to a surface. To extract a surface from this function, we use a level set of. Notice that we can always reduce this problem to examining the zero-contour of the function by simply subtracting. This form of modeling has several advantages over traditional modeling techniques. Since encodes the surface, modifications to the surface (such as CSG operations) can be performed simply by modifying the underlying functions. The topology of the surface is also implicitly defined by so topological modifications are easy because the topology doesn t have to be changed explicitly. Finally, this technique is volumetric and additional information can be gleaned from the underlying function such as distance to the surface simply by evaluating. Since representing functions that contain arbitrary geometry can be difficult, many techniques sample the function on a structured grid. Such grids are typically axis-aligned and have vertices from a uniform sampling of space (uniform grids) or some dyadic sampling (octrees). The main advantage of using these grids is that operations (such as CSG) are easy to perform because the grids between two different implicit surfaces align and the operation can be reduced to an operation on the vertices of the grid. Methods that use structured grids for implicit modeling are quite abundant. Perhaps the most popular is Marching Cubes [9]. Marching Cubes takes as input a uniform grid whose vertices are samples of the function and extracts a surface as the zero-contour. For each cube in the grid, Marching Cubes examines the values at the eight corners of the cube and determines the intersection of the surface with the edges of the cube. Then Marching Cubes provides a lookup table indexed by the sign configuration at the eight corners that yields the topology of the surface inside of that cube. After processing each cube in the grid, the surface is complete. There have been many attempts to extend Marching Cubes from uniform to adaptive grids such as octrees. However, these methods all require some sort of patching between cubes of different resolution [12, 11, 5]. Recently several dual approaches to contouring have been introduced that effectively eliminate this patching problem [2, 10, 4, 13]. In these methods, each cell in the grid is given a representative vertex. For methods such as Dual Contouring, this vertex is placed at sharp features of the surface, which allows the method to reproduce sharp features such as edges and corners. Polygons are then generated by examining minimal edges (an edge that contains no smaller edge) in the octree. Then for each minimal edge the surface passes through, a polygon is generated that connects the vertices of the cells containing that minimal edge. Since the surfaces produced by these methods are topologically dual to the surfaces produced by Marching Cubes, we call these

2 Figure 1. A thin-walled room defined via CSG (upper left). Polygonal approximations were generated by Marching Cubes (lower left, 67K polys), Dual Contouring (lower right, 17K polys) and Dual Marching Cubes (upper right, 440 polys). Using Dual Marching Cubes, the size of the contour mesh is insensitive to the thickness of the walls. methods dual contouring methods and cube-based methods primal contouring methods. However, all of these methods are limited in their expressivity because they are based on structured grids. For instance, to represent very thin features, these methods require that a vertex of the grid lies inside the surface and another, adjacent vertex outside the surface. Since the grids are structured in nature, extracting very small, thin features may require a very fine grid. Figure 1 (top, left) shows a room generated by CSG operations with very thin walls. Three different contouring methods have been used to produce the surface from the CSG tree. Both Marching Cubes and Dual Contouring rely on structured grids and require fine grids (and, consequently, large amounts of polygons) to reproduce the correct topology of the room. While Dual Contouring is a multi-resolution contouring algorithm that can adaptively simplify the surface, the amount of simplification is still limited by the thickness of the walls. Balmelli et al [1] attempt to solve this problem by constructing a uniform grid whose vertices are allowed to conform to the features of the surface using an importance map. While this approach yields better tessellations, it is still limited by uniform sampling and cannot reconstruct sharp features accurately. Furthermore, performing operations such as CSG between two objects becomes difficult because the vertices of the two grids no longer align and resampling must be performed. Varadhan et al [13] provide an alternate solution based off of Dual Contouring [4] and directed distance fields [6]. In that paper the authors provide a technique that can reconstruct a two-sheeted surface in a cell of the octree. This allows their method to reproduce very thin walls. However, their algorithm is still based off of a primal grid and requires the isosurface to intersect the edges of the primal grid in order to be reconstructed faithfully. Contributions We propose a fundamentally different approach to contouring, which we entitle Dual Marching Cubes. The grid that we perform contouring on will be topologically dual to the structured grids used by other techniques. Therefore, we denote this type of grid as a dual grid and structured grids as primal grids. Our underlying data structure is an octree that adaptively samples. To generate a surface, we extract a dual grid to this primal octree. The vertices of this dual grid are positioned at the features of the implicit function inside each cell of the octree. Using this dual grid that conforms to the features of, we generate the surface using a generalized version of Marching Cubes. Contouring on this dual grid yields several advantages, Since we use a structured grid to sample we maintain the advantages associated with structured grids for operations such as CSG. The vertices of our dual grid align with the features of, which allows us to reproduce sharp features similar to the reproduction achieved by [6, 4]. The surfaces produced by this method are adaptive polygonalizations that are crack-free and topologically manifold. Due to the use of a dual grid for contouring, we can reproduce small, thin features in the surface such as walls or tubes without excessive subdivision of the octree. First, we describe the creation of the dual grid that we perform contouring on, which proceeds in two steps: feature isolation and topology creation. Feature isolation takes an octree as input and generates a single vertex for each cell at a feature of inside that cell. After building the vertices of the dual grid, we construct the topology of the grid using a simple extension of the traversal algorithm of Ju et al. [4]. We then describe how Marching Cubes can be applied to this dual grid to generate a surface.

3 2. Feature isolation Given a function and an octree, our goal in feature isolation is to construct a vertex for each cell in the octree that will become a vertex of the dual grid. In contrast to methods such as Dual Contouring, this vertex is not aligned with the features of the surface but with features of the implicit function. Therefore, each vertex will not only contain a position but a scalar value indicating the estimated value of at that vertex. Our approach to detecting features from implicit functions is similar to the technique of Garland [3] for detecting features of surfaces. To determine the vertex that approximates the feature of inside of a cell, we use quadratic error functions (QEFs) (as developed in [3]). To generate these QEFs, we compute tangent planes to the graph of on a grid of points sampled over. At a sample point, the tangent plane to has the equation where and is the gradient of. To build the QEF, we square this equation and sum over all sample points yielding (1) The denominator of this expression normalizes the contribution of each tangent plane to have equal weight. Next, we minimize this quadratic function over the cell to find the vertex of the dual grid. If the minimizer is underdetermined, we follow the method of Lindstrom [7] and use the pseudo-inverse to position the minimizer as close as possible to the center of. Notice that our description of this feature isolation phase does not specify what format must take on. Specifically, our description allows for a wide variety of possible inputs. For instance, uniform scalar grids typically associated with Marching Cubes can be used where each cube in the grid is regarded as a trilinear function so that and are well defined. Alternatively, we can restrict our sampling above to the grid points of this uniform grid and use divided differences to estimate at these vertices. The latter strategy has the advantage of defining a single normal at the grid vertices instead of several discontinuous normals as the former method does. Directed distance fields provide an alternative to uniform, scalar fields and can provide more accurate gradients to help reproduce sharp features. CSG trees are another possible input to our algorithm. CSG performs set operations on solids where the leaves Figure 2. Connectivity of the dual grid (thick black) for a primal quadtree (thin blue). of the CSG trees are solids and interior nodes are operations such as union and intersection. To use such CSG trees in our algorithm, we must convert the trees to scalar val- that contain a well defined gradi- ued functions ent. At the leaves of the CSG tree, we replace the solids with their signed Euclidean distance functions for primitives such as planes, cylinders, etc... At interior nodes, we replace union and intersection operations with Min and Max operations respectively. Evaluation of the CSG tree at a point simply involves evaluation of the distance functions of each leaf at and performing the corresponding operations at the interior nodes of the CSG tree. To compute the gradient, is evaluated at the corresponding point at each of the leaves of the CSG tree and passed upwards during the Min/Max operations along with the value of the function at that point. Figures 1 and 6 were each generated by evaluating a CSG tree Octree construction Our current description of feature isolation has assumed that an octree is present to partition space into cells. However, sometimes it is desirable to approximate to a given tolerance. To perform this approximation, we present a top-down octree construction algorithm. This octree construction algorithm proceeds by starting with a single cell as the octree. We use the sampling algorithm above to sample finely on a uniform grid over (a random sampling strategy could also be used). Next, we construct a QEF for using those sample points and minimize the QEF. If the error from equation 1 is greater than, then we subdivide the cell into eight sub-cells and proceed recursively. The procedure stops when all of the leaves of the octree have error less thanand constructs a minimal octree to approximate.

5 Figure 5. Contoured sphere without sliver elimination (left) and with sliver elimination (right). 4. Contouring dual grids After generating the dual grid, we extract the surface using a simple extension of Marching Cubes to these dual grids. Marching Cubes was originally designed to operate only on axis-aligned cubes. However, the cells of our dual grid have arbitrary vertices in space. Our solution is to notice that each cell in the dual grid generated by the recursive algorithm in section 3 is topologically equivalent to a cube (although some vertices may be repeated due to the multiresolution structure of the octree). Using the scalar values at the vertices of the dual grid, we compute edge intersection points for each edge that contains a sign change. Then we use the look-up table provided by Marching Cubes to generate the topology of the surface interior to that cell. Since the vertices of the dual grid lie on the features of, the contours of this dual grid also exhibit sharp features similar to those produced by Extended Marching Cubes and Dual Contouring. The principle advantage of Dual Marching Cubes over these methods is that the underlying octree used to produce an equivalent contour is much sparser. To understand this phenomena, we consider a 2D example. Figure 4 (left) contains the distance function for two closely space, parallel lines shown on the right of the figure. These two linear functions are formed by taking the Max of the distance function for each line. The left of the figure shows the dual grid of which consists of four quads where the height of the vertex is formed by the scalar value. Contouring each of these quads yields four line segments independent of the spacing of the two parallel lines. In methods such as EMC and Dual Contouring, the density of the primal grid used to resolve the two separate contour lines of this function depends on the separation distance of the two contours. With Dual Marching Cubes, the grid used in contouring the function adapts to features of the distance function as opposed to features of the contour. This distinction often avoids the need for high levels of refinement to separate closely spaced features of the contour. As a more complex 3D example, consider the thinwalled room of figure 1. This room is designed as a sequence of CSG operations based on planar primitives. The lower left mesh is the Marching Cubes contour of this distance function on a fine grid to reproduce the thin walls (note the rounding of sharp edges). The lower right mesh, produced by Dual Contouring, exactly reproduces the shape of the room but cannot simplify the numerous flat regions in the mesh due to the thinness of the room s walls. Our method computes an octree to approximate the CSG tree using the method of section 2.1. The resulting surface on the upper right was generated by Dual Marching Cubes by contouring the dual grid of that octree. This surface exactly reproduces the shape of the room but uses far fewer polygons. Note that as the thickness of the walls of this room decreases, the number of polygons used to contour this room using Marching Cubes and Dual Contouring increases dramatically while the number of polygons produced by Dual Marching Cubes remains roughly constant. One drawback with using Marching Cubes to contour these dual grids is that it tends to produce numerous sliver polygons when the vertices of the dual grid lie close to the plane. The solution to this problem is to position the vertices of the dual grid to lie exactly on when possible. The effect on the resulting contour is to eliminate most sliver polygons. To achieve this positioning, we first compute the minimizer of the restricted QEF. If the residual at is less than, is used as the vertex for that cube. Otherwise, we perform feature isolation as before. Figure 5 shows the contours generated by Dual Marching Cubes illustrating sliver elimination. Figure 6 depicts an example of a rocket constructed as a sequence of CSG operations and the approximating models using Dual Marching Cubes, Marching Cubes and Dual Contouring. The rocket is an interesting example because the thin fins on the bottom of the shape exhibit five-fold symmetry. While primal grids perform well when features are axis aligned, these fins do not conform to parameter lines on the primal grid. Consequently, methods such as Marching Cubes and Dual Contouring have a difficult time reproducing these thin features. On the other hand, Dual Marching Cubes constructs a dual grid that conforms to the fins and can reconstruct them accurately despite the fact that the fins do not align with the octree. Figure 7 illustrates adaptive surface extraction using a horse and a dragon. On the horse, the polygons are more coarse on the body than along the neck and the legs of the horse. Despite the fact that the legs are thin compared to the rest of the horse, they are reproduced accurately. The dragon also shows an adaptive polygonalization where its face and feet contain finer polygons than the rest of its body. Though we have simply run Marching Cubes on these models without any multi-resolution extensions, the structure

6 Figure 7. Adaptive surfaces extracted using Dual Marching Cubes. Figure 6. CSG model of a rocket (upper left) and models approximating the shape using the same number of polygons. Dual Marching Cubes (upper right), Marching Cubes (lower left) and Dual Contouring (lower right). of the dual grid naturally generates these multi-resolution contours. Furthermore, since we use Marching Cubes to contour these models, the resulting surface is topologically manifold. 5. Implementation When implementing this method, we abstract out the different possible inputs to our algorithm as a function that we can evaluate to determine the scalar value and gradient at our sample points. Since we are estimating sharp features using QEFs, the same problems arise for volumes as found in surface methods. For instance, when minimizing equation 1, the feature vertex may lie outside of the cube that generated it. In this case, we minimize the QEF over the boundary of the cube as in Lindstrom [8]. Also, notice that there is nothing that requires we place vertices at the features of the distance function. While this at the placement reproduces sharp features, one could modify the feature isolation phase to simply sample center of each cube. The resulting algorithm is a contouring algorithm similar to [10] except no special cases are generated in adaptive configurations and a manifold surface is always produced. As far as performance is concerned, contouring dual grids takes no longer than contouring primal grids. Since dual grids conform better to the features of, sparser grids can be used to extract surfaces, which can actually decrease the time taken to contour. However, dual grid generation is still somewhat slow. Feature isolation took seconds for figure 1, seconds for figure 6 and several minutes for figure 7 using the octree construction algorithm from section 2.1 on a GHz Pentium. One way of solving this problem is to realize that most of the dual grid does not directly contribute to the extracted surface. Only cells of the dual grid that contain a sign change will contain a piece of the surface. Therefore, it may be possible to develop an algorithm that restricts the feature isolation and topology creation phases to only those cells that contain a piece of the extracted surface. We anticipate the speed of this optimized method to be comparable with other popular methods. 6. Future Work We believe that the concept of dual grid generation has applications outside of contouring. In fact, this grid generation actually extracts a piecewise linear approximation to a given function over a cubical domain. With this

### Dual Contouring of Hermite Data

Dual Contouring of Hermite Data Tao Ju, Frank Losasso, Scott Schaefer, Joe Warren Rice University Figure 1: A temple undergoing destructive modifications. Both models were generated by dual contouring

### Surface Reconstruction from a Point Cloud with Normals

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

### A 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

### Feature Sensitive Surface Extraction from Volume Data

Feature Sensitive Surface Extraction from Volume Data Leif P. Kobbelt Mario Botsch Ulrich Schwanecke Hans-Peter Seidel Computer Graphics Group, RWTH-Aachen Computer Graphics Group, MPI Saarbrücken Figure

### Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science - Technion. An Example.

An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: Wire-Frame Representation Object is represented as as a set of points

### Model 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

### Segmentation 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

### H.Calculating Normal Vectors

Appendix H H.Calculating Normal Vectors This appendix describes how to calculate normal vectors for surfaces. You need to define normals to use the OpenGL lighting facility, which is described in Chapter

### Parallel 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

### Computer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include

### Curves and Surfaces. Goals. How do we draw surfaces? How do we specify a surface? How do we approximate a surface?

Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.1-10.6] Goals How do we draw surfaces? Approximate with polygons Draw polygons

### Improved Billboard Clouds for Extreme Model Simplification

Improved Billboard Clouds for Extreme Model Simplification I.-T. Huang, K. L. Novins and B. C. Wünsche Graphics Group, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland,

### Delaunay 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

### Lecture 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.

### The Essentials of CAGD

The Essentials of CAGD Chapter 2: Lines and Planes Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/essentials-cagd c 2000 Farin & Hansford

### Constrained 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

### Arrangements And Duality

Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Volume visualization I Elvins

Volume visualization I Elvins 1 surface fitting algorithms marching cubes dividing cubes direct volume rendering algorithms ray casting, integration methods voxel projection, projected tetrahedra, splatting

### 6.1 Application of Solid Models In mechanical engineering, a solid model is used for the following applications:

CHAPTER 6 SOLID MODELING 6.1 Application of Solid Models In mechanical engineering, a solid model is used for the following applications: 1. Graphics: generating drawings, surface and solid models 2. Design:

### Proc. First Users Conference of the National Library of Medicine's Visible Human Project, active contours free forms Triangle meshes may be imp

Proc. First Users Conference of the National Library of Medicine's Visible Human Project, 1996 1 Creation of Finite Element Models of Human Body based upon Tissue Classied Voxel Representations M. Muller,

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### COMPARING TECHNIQUES FOR TETRAHEDRAL MESH GENERATION

COMPARING TECHNIQUES FOR TETRAHEDRAL MESH GENERATION M. A. S. Lizier Instituto de Ciências Matemáticas e de Computação - Universidade de São Paulo, Brazil lizier@icmc.usp.br J. F. Shepherd Sandia National

### Introduction 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

### Utah Core Curriculum for Mathematics

Core Curriculum for Mathematics correlated to correlated to 2005 Chapter 1 (pp. 2 57) Variables, Expressions, and Integers Lesson 1.1 (pp. 5 9) Expressions and Variables 2.2.1 Evaluate algebraic expressions

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### Efficient 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

### Accurate Minkowski Sum Approximation of Polyhedral Models

Accurate Minkowski Sum Approximation of Polyhedral Models Gokul Varadhan Dinesh Manocha University of North Carolina at Chapel Hill {varadhan,dm}@cs.unc.edu Brake Hub Rod Brake Hub Rod Bunny Bunny Offset

### A unified representation for interactive 3D modeling

A unified representation for interactive 3D modeling Dragan Tubić, Patrick Hébert, Jean-Daniel Deschênes and Denis Laurendeau Computer Vision and Systems Laboratory, University Laval, Québec, Canada [tdragan,hebert,laurendeau]@gel.ulaval.ca

### Pre-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

### Whole Numbers and Integers (44 topics, no due date)

Course Name: PreAlgebra into Algebra Summer Hwk Course Code: GHMKU-KPMR9 ALEKS Course: Pre-Algebra Instructor: Ms. Rhame Course Dates: Begin: 05/30/2015 End: 12/31/2015 Course Content: 302 topics Whole

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

### Faculty of Computer Science Computer Graphics Group. Final Diploma Examination

Faculty of Computer Science Computer Graphics Group Final Diploma Examination Communication Mechanisms for Parallel, Adaptive Level-of-Detail in VR Simulations Author: Tino Schwarze Advisors: Prof. Dr.

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### From 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

### CREATION OF OVERLAY TILINGS THROUGH DUALIZATION OF REGULAR NETWORKS. Douglas Zongker

CREATION OF OVERLAY TILINGS THROUGH DUALIZATION OF REGULAR NETWORKS Douglas Zongker Department of Computer Science & Engineering University of Washington Box 352350 Seattle, Washington 98195-2350 USA dougz@cs.washington.edu

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### Introduction to SolidWorks Software

Introduction to SolidWorks Software Marine Advanced Technology Education Design Tools What is SolidWorks? SolidWorks is design automation software. In SolidWorks, you sketch ideas and experiment with different

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### DXF Import and Export for EASE 4.0

DXF Import and Export for EASE 4.0 Page 1 of 9 DXF Import and Export for EASE 4.0 Bruce C. Olson, Dr. Waldemar Richert ADA Copyright 2002 Acoustic Design Ahnert EASE 4.0 allows both the import and export

### Activity 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

### 5. GEOMETRIC MODELING

5. GEOMETRIC MODELING Types of Curves and Their Mathematical Representation Types of Surfaces and Their Mathematical Representation Types of Solids and Their Mathematical Representation CAD/CAM Data Exchange

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### angle Definition and illustration (if applicable): a figure formed by two rays called sides having a common endpoint called the vertex

angle a figure formed by two rays called sides having a common endpoint called the vertex area the number of square units needed to cover a surface array a set of objects or numbers arranged in rows and

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### SECONDARY 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,

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### HowTo 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

### Example Degree Clip Impl. Int Sub. 1 3 2.5 1 10 15 2 3 1.8 1 5 6 3 5 1 1.7 3 5 4 10 1 na 2 4. Table 7.1: Relative computation times

Chapter 7 Curve Intersection Several algorithms address the problem of computing the points at which two curves intersect. Predominant approaches are the Bézier subdivision algorithm [LR80], the interval

Adaptive Fourier-Based Surface Reconstruction Oliver Schall, Alexander Belyaev, and Hans-Peter Seidel Computer Graphics Group Max-Planck-Institut für Informatik Stuhlsatzenhausweg 85, 66123 Saarbrücken,

### Homework Exam 1, Geometric Algorithms, 2016

Homework Exam 1, Geometric Algorithms, 2016 1. (3 points) Let P be a convex polyhedron in 3-dimensional space. The boundary of P is represented as a DCEL, storing the incidence relationships between the

### Intersection 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

### Interactive 3D Medical Visualization: A Parallel Approach to Surface Rendering 3D Medical Data

Interactive 3D Medical Visualization: A Parallel Approach to Surface Rendering 3D Medical Data Terry S. Yoo and David T. Chen Department of Computer Science University of North Carolina Chapel Hill, NC

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Honors Geometry Final Exam Study Guide

2011-2012 Honors Geometry Final Exam Study Guide Multiple Choice Identify the choice that best completes the statement or answers the question. 1. In each pair of triangles, parts are congruent as marked.

### Constructing Geometric Solids

Constructing Geometric Solids www.everydaymathonline.com Objectives To provide practice identifying geometric solids given their properties; and to guide the construction of polyhedrons. epresentations

### A 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

### MATHS LEVEL DESCRIPTORS

MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and

### We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant

### Tutorial: 3D Pipe Junction Using Hexa Meshing

Tutorial: 3D Pipe Junction Using Hexa Meshing Introduction In this tutorial, you will generate a mesh for a three-dimensional pipe junction. After checking the quality of the first mesh, you will create

### Customer 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

### Topographic Change Detection Using CloudCompare Version 1.0

Topographic Change Detection Using CloudCompare Version 1.0 Emily Kleber, Arizona State University Edwin Nissen, Colorado School of Mines J Ramón Arrowsmith, Arizona State University Introduction CloudCompare

### Data Visualization (DSC 530/CIS )

Data Visualization (DSC 530/CIS 602-01) Volume Rendering Dr. David Koop Fields Tables Networks & Trees Fields Geometry Clusters, Sets, Lists Items Items (nodes) Grids Items Items Attributes Links Positions

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

### Computer Animation: Art, Science and Criticism

Computer Animation: Art, Science and Criticism Tom Ellman Harry Roseman Lecture 4 Parametric Curve A procedure for distorting a straight line into a (possibly) curved line. The procedure lives in a black

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Lines That Pass Through Regions

: Student Outcomes Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe

### Consolidated Visualization of Enormous 3D Scan Point Clouds with Scanopy

Consolidated Visualization of Enormous 3D Scan Point Clouds with Scanopy Claus SCHEIBLAUER 1 / Michael PREGESBAUER 2 1 Institute of Computer Graphics and Algorithms, Vienna University of Technology, Austria

### Volumetric 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.

### Topological Treatment of Platonic, Archimedean, and Related Polyhedra

Forum Geometricorum Volume 15 (015) 43 51. FORUM GEOM ISSN 1534-1178 Topological Treatment of Platonic, Archimedean, and Related Polyhedra Tom M. Apostol and Mamikon A. Mnatsakanian Abstract. Platonic

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### Multiresolution 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

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### Prentice Hall Mathematics: Course 1 2008 Correlated to: Arizona Academic Standards for Mathematics (Grades 6)

PO 1. Express fractions as ratios, comparing two whole numbers (e.g., ¾ is equivalent to 3:4 and 3 to 4). Strand 1: Number Sense and Operations Every student should understand and use all concepts and

### NEW MEXICO Grade 6 MATHEMATICS STANDARDS

PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

### Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in

### Human tissue geometrical modelling

Human tissue geometrical modelling Přemysl Kršek Department of Computer Graphics and Multimedia Faculty of Information Technology Brno University of Technology Božetěchova 2 612 66 Brno, Czech Republic

### Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

### Mean Value Coordinates

Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its

### A typical 3D modeling system involves the phases of 1. Individual range image acquisition from different viewpoints.

Efficient Model Creation of Large Structures based on Range Segmentation 2nd International Symposium on 3D Data Processing, Visualization & Transmission, September 2004, Thessaloniki, Greece. Ioannis Stamos

### Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

### Using 3 D SketchUp Design Software. Building Blocks Tools

Using 3 D SketchUp Design Software Building Blocks Tools Introduction Google s SketchUp is an easy to use drawing program capable of building 3 dimensional (or 3 D) objects (SketchUp also does 2 D easily).

### Decide how many topics you wish to revise at a time (let s say 10).

1 Minute Maths for the Higher Exam (grades B, C and D topics*) Too fast for a first-time use but... brilliant for topics you have already understood and want to quickly revise. for the Foundation Exam

### Expression. Variable Equation Polynomial Monomial Add. Area. Volume Surface Space Length Width. Probability. Chance Random Likely Possibility Odds

Isosceles Triangle Congruent Leg Side Expression Equation Polynomial Monomial Radical Square Root Check Times Itself Function Relation One Domain Range Area Volume Surface Space Length Width Quantitative

### Simplification of Nonconvex Tetrahedral Meshes

Simplification of Nonconvex Tetrahedral Meshes Martin Kraus and Thomas Ertl Visualization and Interactive Systems Group, Institut für Informatik, Universität Stuttgart, Breitwiesenstr. 20 22, 70565 Suttgart,

### 3D Distance from a Point to a Triangle

3D Distance from a Point to a Triangle Mark W. Jones Technical Report CSR-5-95 Department of Computer Science, University of Wales Swansea February 1995 Abstract In this technical report, two different

### Vector 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

### Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test

Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are

Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation

### CHAPTER 1 Splines and B-splines an Introduction

CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the

### Freehand Sketching. Sections

3 Freehand Sketching Sections 3.1 Why Freehand Sketches? 3.2 Freehand Sketching Fundamentals 3.3 Basic Freehand Sketching 3.4 Advanced Freehand Sketching Key Terms Objectives Explain why freehand sketching

### 1. Relational database accesses data in a sequential form. (Figures 7.1, 7.2)

Chapter 7 Data Structures for Computer Graphics (This chapter was written for programmers - option in lecture course) Any computer model of an Object must comprise three different types of entities: 1.

### Perfume Packaging. Ch 5 1. Chapter 5: Solids and Nets. Chapter 5: Solids and Nets 279. The Charles A. Dana Center. Geometry Assessments Through

Perfume Packaging Gina would like to package her newest fragrance, Persuasive, in an eyecatching yet cost-efficient box. The Persuasive perfume bottle is in the shape of a regular hexagonal prism 10 centimeters

### Hierarchical Visualization of Large-scale Unstructured Hexahedral Volume Data

Hierarchical Visualization of Large-scale Unstructured Hexahedral Volume Data Jaya Sreevalsan-Nair Lars Linsen Benjamin A. Ahlborn Michael S. Green Bernd Hamann Center for Image Processing and Integrated

### Computational 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,

### *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles.

Students: 1. Students understand and compute volumes and areas of simple objects. *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Review

### Greater Nanticoke Area School District Math Standards: Grade 6

Greater Nanticoke Area School District Math Standards: Grade 6 Standard 2.1 Numbers, Number Systems and Number Relationships CS2.1.8A. Represent and use numbers in equivalent forms 43. Recognize place

### Solving Geometric Problems with the Rotating Calipers *

Solving Geometric Problems with the Rotating Calipers * Godfried Toussaint School of Computer Science McGill University Montreal, Quebec, Canada ABSTRACT Shamos [1] recently showed that the diameter of