(Discrete) Differential Geometry

Size: px
Start display at page:

Download "(Discrete) Differential Geometry"

Transcription

1 (Discrete) Differential Geometry

2 Motivation Understand the structure of the surface Properties: smoothness, curviness, important directions How to modify the surface to change these properties What properties are preserved for different modifications The math behind the scenes for many geometry processing applications 2

3 Motivation Smoothness Mesh smoothing Curvature Adaptive simplification Parameterization 3

4 Motivation Triangle shape Remeshing Principal directions Quad remeshing 4

5 Differential Geometry MP M.P. docarmo: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976 Leonard Euler ( ) Carl Friedrich Gauss ( ) 5

6 Parametric Curves velocity of particle on trajectory 6

7 Parametric Curves A Simple Example t a α 1 (t) = (a cos(t), a sin(t)) t [0,2π] 2 ] α 2 (t) = (a cos(2t), a sin(2t)) t [0,π] 7

8 Parametric Curves A Simple Example t α'(t) α(t) = (cos(t), sin(t)) α'(t) = (-sin(t) sin(t), cos(t)) α'(t) - direction of movement α'(t) - speed of movement 8

9 Parametric Curves A Simple Example α 2 '(t) t α 1 '(t) α 1 (t) = (cos(t), sin(t)) α 2 (t) = (cos(2t), sin(2t)) Same direction, different speed 9

10 Length of a Curve Let and 10

11 Length of a Curve Chord length Euclidean norm 11

12 Length of a Curve Chord length Arc length 12

13 Examples Straight line x(t) = (t,t), t [0, ) x(t) = (2t,2t), 2t) t [0, ) ) x(t) = (t 2,t 2 ), t [0, ) y(t) x(t) ( ) x(t) Circle x(t) = (cos(t),sin(t)),(t)), t [0,2π) ) y(t) x(t) x(t) 13

14 Examples α(t) = (a cos(t), a sin(t)), t [0,2π] α'(t) = (-a a sin(t), a cos(t)) Many possible parameterizations Length of the curve does not depend on parameterization! 14

15 Arc Length Parameterization Re-parameterization Arc length parameterization parameter value s for x(s) equals length of curve from x(a) to x(s) 15

16 Curvature x(t) a curve parameterized by arc length The curvature of x at t: the tangent vector at t the change in the tangent tvector attt R(t) = 1/κ (t) is the radius of curvature at t 16

17 Examples Straight line α(s) = us + v, u,v R 2 α'(s) = u α''(s) = 0 α''(s) = 0 Circle α(s) = (a cos(s/a), a sin(s/a)), s [0,2πa] α'(s) = (-sin(s/a), cos(s/a)) α''(s) = (-cos(s/a)/a, -sin(s/a)/a) α''(s) = 1/a 17

18 The Normal Vector x'(t) = T(t) - tangent vector x'(t) ( ) x'(t) - velocity x''(t) x''(t) = T'(t) - normal direction x''(t) -curvature x'(t) If x''(t) 0, define N(t) = T'(t)/ T'(t) Then x''(t) = T'(t) = κ(t)n(t) 18 x''(t)

19 The Osculating Plane The plane determined by the unit tangent and normal vectors T(s) and N(s) is called the osculating plane at s N N T T 19

20 The Binormal Vector For points s, s.t. κ(s) 0, the binormal vector B(s) is defined as: B(s) ) = T(s) ) N(s) ) T B B N N The binormal vector defines the osculating plane T 20

21 The Frenet Frame tangent normal binormal 21

22 The Frenet Frame {T(s), N(s), ) B(s)} form an orthonormal basis for R 3 called the Frenet frame How does the frame change when the particle il moves? What are T', N', B' in terms of T, N, B? 22

23 The Frenet Frame Frenet-Serret formulas curvature torsion (arc-length parameterization) 23

24 Curvature and Torsion Curvature: Deviation from straight line Torsion: Deviation from planarity Independent of parameterization intrinsic properties of the curve Invariant under rigid (translation+rotation) motion Define curve uniquely up to rigid motion 24

25 Curvature and Torsion Planes defined by x and two vectors: osculating plane: vectors T and N normal plane: vectors N and B rectifying plane: vectors T and B Osculating circle second order contact with curve center x radius C R 25

26 Differential Geometry: Surfaces 26

27 Differential Geometry: Surfaces Continuous surface Normal vector assuming regular parameterization, i.e. 27

28 Normal Curvature If x and x are orthogonal: u v 28

29 Normal Curvature 29

30 Surface Curvature Principal Curvatures maximum curvature minimum curvature Mean Curvature Gaussian Curvature 30

31 Principal Curvature Euler s Theorem: Planes of principal curvature are orthogonal and independent of parameterization. 31

32 Curvature 32

33 Surface Classification Isotropic k 1 = k 2 > 0 k 1 = k 2 = 0 Equal in all directions spherical planar Anisotropic Distinct principal directions k 1 > 0 k 2 > 0 k 1 > 0 k 2 =0 k 1 > 0 k 2 < 0 elliptic parabolic hyperbolic K > 0 K = 0 K < 0 developable 33

34 Principal Directions k2 k1 34

35 Gauss-Bonnet Theorem For ANY closed manifold surface with Euler number χ=2-2g: = 4 K( ) = K( ) K( ) = π 35

36 Gauss-Bonnet Theorem Example Sphere k 1 = k 2 = 1/r 1 2 K = k 1 k 2 = 1/r 2 Manipulate sphere New positive + negative curvature Cancel out! 36

37 Fundamental Forms First fundamental form Second fundamental form 37

38 Fundamental Forms I and II allow to measure length, angles, area, curvature arc element area element 38

39 Fundamental Forms Normal curvature = curvature of the normal curve at point Can be expressed in terms of fundamental forms as 39

40 Intrinsic Geometry Properties of the surface that only depend on the first fundamental form length angles Gaussian curvature (Theorema Egregium) 40

41 Laplace Operator Laplace operator gradient operator 2nd partial derivatives R scalar function in Euclidean space divergence operator Cartesian coordinates 41

42 Laplace-Beltrami Operator Extension to functions on manifolds Laplace- Beltrami gradient operator scalar function on manifold divergence operator 42

43 Laplace-Beltrami Operator For coordinate function(s) Laplace- Beltrami gradient operator mean curvature surface coordinate divergence normal function operator 43

44 Discrete Differential Operators Assumption: Meshes are piecewise linear approximations of smooth surfaces Approach: Approximate differential properties at point v as finite differences over local mesh neighborhood N(v) v = mesh vertex N d (v) = d-ring neighborhood Disclaimer: many possible discretizations, none is perfect v N 1 (v) 44

45 Functions on Meshes Function f givenatmeshverticesf(v vertices i ) = f(x i ) = f i Linear interpolation to triangle 1 B i (x) x i x k x j 45

46 Gradient of a Function 1 B i (x) Steepest ascent direction perpendicular to opposite edge x i x k x j Constant in the triangle 46

47 Gradient of a Function 47

48 Laplace operator: Discrete Laplace-Beltrami First Approach In 2D: On a grid finite it differences discretization: 48

49 Discrete Laplace-Beltrami Uniform Discretization v N 1 (v) Normalized: 49

50 Discrete Laplace-Beltrami Second dapproach Laplace-Beltrami operator: Compute integral around vertex Divergence theorem 50

51 Discrete Laplace-Beltrami Cotangent Formula Plugging in expression for gradients gives: 51

52 The Averaging g Region Edge midpoint θ Barycentric cell Voronoi cell Mixed cell c i = barycenter of triangle c i = circumcenter of triangle 52

53 The Averaging Region Mixed Cell If θ < π/2, c i is the circumcenter of the triangle (v i, v, v i+1 ) θ If θ π/2, c i is the midpoint of the edge (v i, v i+1 ) 53

54 Discrete Normal n ( x ) 54

55 Discrete Curvatures Mean curvature Gaussian curvature Principal curvatures 55

56 Example: Mean Curvature 56

57 Example: Gaussian Curvature original smoothed Discrete Gauss-Bonnet (Descartes) theorem: 57

58 References Discrete Differential-Geometry Operators for Triangulated 2- Manifolds, Meyer et al., 02 Restricted Delaunay triangulations and normal cycle, Cohen-Steiner et al., SoCG 03 On the convergence ence of metric and geometric properties of polyhedral surfaces, Hildebrandt et al., 06 Discrete Laplace operators: No free lunch, Wardetzkyet tal., SGP 07 58

Shape Modeling and Geometry Processing

Shape Modeling and Geometry Processing 252-0538-00L, Spring 2018 Shape Modeling and Geometry Processing Discrete Differential Geometry Differential Geometry Motivation Formalize geometric properties of shapes Roi Poranne # 2 Differential Geometry

More information

05 - Surfaces. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Daniele Panozzo

05 - Surfaces. Acknowledgements: Olga Sorkine-Hornung. CSCI-GA Geometric Modeling - Daniele Panozzo 05 - Surfaces Acknowledgements: Olga Sorkine-Hornung Reminder Curves Turning Number Theorem Continuous world Discrete world k: Curvature is scale dependent is scale-independent Discrete Curvature Integrated

More information

CS 523: Computer Graphics, Spring Shape Modeling. Differential Geometry of Surfaces

CS 523: Computer Graphics, Spring Shape Modeling. Differential Geometry of Surfaces CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry of Surfaces Andrew Nealen, Rutgers, 2011 2/22/2011 Differential Geometry of Surfaces Continuous and Discrete Motivation Smoothness

More information

CS 523: Computer Graphics, Spring Differential Geometry of Surfaces

CS 523: Computer Graphics, Spring Differential Geometry of Surfaces CS 523: Computer Graphics, Spring 2009 Shape Modeling Differential Geometry of Surfaces Andrew Nealen, Rutgers, 2009 3/4/2009 Recap Differential Geometry of Curves Andrew Nealen, Rutgers, 2009 3/4/2009

More information

Motivation. Parametric Curves (later Surfaces) Outline. Tangents, Normals, Binormals. Arclength. Advanced Computer Graphics (Fall 2010)

Motivation. Parametric Curves (later Surfaces) Outline. Tangents, Normals, Binormals. Arclength. Advanced Computer Graphics (Fall 2010) Advanced Computer Graphics (Fall 2010) CS 283, Lecture 19: Basic Geometric Concepts and Rotations Ravi Ramamoorthi http://inst.eecs.berkeley.edu/~cs283/fa10 Motivation Moving from rendering to simulation,

More information

Shape Modeling. Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell. CS 523: Computer Graphics, Spring 2011

Shape Modeling. Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell. CS 523: Computer Graphics, Spring 2011 CS 523: Computer Graphics, Spring 2011 Shape Modeling Differential Geometry Primer Smooth Definitions Discrete Theory in a Nutshell 2/15/2011 1 Motivation Geometry processing: understand geometric characteristics,

More information

Tutorial 4. Differential Geometry I - Curves

Tutorial 4. Differential Geometry I - Curves 23686 Numerical Geometry of Images Tutorial 4 Differential Geometry I - Curves Anastasia Dubrovina c 22 / 2 Anastasia Dubrovina CS 23686 - Tutorial 4 - Differential Geometry I - Curves Differential Geometry

More information

Surfaces: notes on Geometry & Topology

Surfaces: notes on Geometry & Topology Surfaces: notes on Geometry & Topology 1 Surfaces A 2-dimensional region of 3D space A portion of space having length and breadth but no thickness 2 Defining Surfaces Analytically... Parametric surfaces

More information

04 - Normal Estimation, Curves

04 - Normal Estimation, Curves 04 - Normal Estimation, Curves Acknowledgements: Olga Sorkine-Hornung Normal Estimation Implicit Surface Reconstruction Implicit function from point clouds Need consistently oriented normals < 0 0 > 0

More information

Digital Geometry Processing Differential Geometry

Digital Geometry Processing Differential Geometry Digial Geomery Processing Differenial Geomery Moivaion Undersand he srucure of he surface Differenial Geomery Properies: smoohness, curviness, imporan direcions How o modify he surface o change hese properies

More information

Laplacian Operator and Smoothing

Laplacian Operator and Smoothing Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung, Mario Botsch, and Daniele Panozzo Applications in Geometry Processing Smoothing Parameterization Remeshing

More information

Digital Geometry Processing Parameterization I

Digital Geometry Processing Parameterization I Problem Definition Given a surface (mesh) S in R 3 and a domain find a bective F: S Typical Domains Cutting to a Disk disk = genus zero + boundary sphere = closed genus zero Creates artificial boundary

More information

Differential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder]

Differential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder] Differential Geometry: Circle Patterns (Part 1) [Discrete Conformal Mappinngs via Circle Patterns. Kharevych, Springborn and Schröder] Preliminaries Recall: Given a smooth function f:r R, the function

More information

Construction and smoothing of triangular Coons patches with geodesic boundary curves

Construction and smoothing of triangular Coons patches with geodesic boundary curves Construction and smoothing of triangular Coons patches with geodesic boundary curves R. T. Farouki, (b) N. Szafran, (a) L. Biard (a) (a) Laboratoire Jean Kuntzmann, Université Joseph Fourier Grenoble,

More information

Space Curves of Constant Curvature *

Space Curves of Constant Curvature * Space Curves of Constant Curvature * 2-11 Torus Knot of constant curvature. See also: About Spherical Curves Definition via Differential Equations. Space Curves that 3DXM can exhibit are mostly given in

More information

Differential Geometry MAT WEEKLY PROGRAM 1-2

Differential Geometry MAT WEEKLY PROGRAM 1-2 15.09.014 WEEKLY PROGRAM 1- The first week, we will talk about the contents of this course and mentioned the theory of curves and surfaces by giving the relation and differences between them with aid of

More information

What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape

What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape Geometry Processing What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape What is Geometry Processing? Understanding the math of 3D shape and applying

More information

Lectures in Discrete Differential Geometry 3 Discrete Surfaces

Lectures in Discrete Differential Geometry 3 Discrete Surfaces Lectures in Discrete Differential Geometry 3 Discrete Surfaces Etienne Vouga March 19, 2014 1 Triangle Meshes We will now study discrete surfaces and build up a parallel theory of curvature that mimics

More information

Computing Curvature CS468 Lecture 8 Notes

Computing Curvature CS468 Lecture 8 Notes Computing Curvature CS468 Lecture 8 Notes Scribe: Andy Nguyen April 4, 013 1 Motivation In the previous lecture we developed the notion of the curvature of a surface, showing a bunch of nice theoretical

More information

Review 1. Richard Koch. April 23, 2005

Review 1. Richard Koch. April 23, 2005 Review Richard Koch April 3, 5 Curves From the chapter on curves, you should know. the formula for arc length in section.;. the definition of T (s), κ(s), N(s), B(s) in section.4. 3. the fact that κ =

More information

DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU /858B Fall 2017

DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU /858B Fall 2017 DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-458/858B Fall 2017 LECTURE 10: DISCRETE CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-458/858B

More information

WWW links for Mathematics 138A notes

WWW links for Mathematics 138A notes WWW links for Mathematics 138A notes General statements about the use of Internet resources appear in the document listed below. We shall give separate lists of links for each of the relevant files in

More information

Introduction to geometry

Introduction to geometry 1 2 Manifolds A topological space in which every point has a neighborhood homeomorphic to (topological disc) is called an n-dimensional (or n-) manifold Introduction to geometry The German way 2-manifold

More information

Fast marching methods

Fast marching methods 1 Fast marching methods Lecture 3 Alexander & Michael Bronstein tosca.cs.technion.ac.il/book Numerical geometry of non-rigid shapes Stanford University, Winter 2009 Metric discretization 2 Approach I:

More information

Math 348 Differential Geometry of Curves and Surfaces

Math 348 Differential Geometry of Curves and Surfaces Math 348 Differential Geometry of Curves and Surfaces Lecture 3 Curves in Calculus Xinwei Yu Sept. 12, 2017 CAB 527, xinwei2@ualberta.ca Department of Mathematical & Statistical Sciences University of

More information

Justin Solomon MIT, Spring 2017

Justin Solomon MIT, Spring 2017 http://www.alvinomassage.com/images/knot.jpg Justin Solomon MIT, Spring 2017 Some materials from Stanford CS 468, spring 2013 (Butscher & Solomon) What is a curve? A function? Not a curve Jams on accelerator

More information

Curvature. Corners. curvature of a straight segment is zero more bending = larger curvature

Curvature. Corners. curvature of a straight segment is zero more bending = larger curvature Curvature curvature of a straight segment is zero more bending = larger curvature Corners corner defined by large curvature value (e.g., a local maxima) borders (i.e., edges in gray-level pictures) can

More information

Geometry Processing TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

Geometry Processing TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA Geometry Processing What is Geometry Processing? Understanding the math of 3D shape What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape What is Geometry

More information

3. The three points (2, 4, 1), (1, 2, 2) and (5, 2, 2) determine a plane. Which of the following points is in that plane?

3. The three points (2, 4, 1), (1, 2, 2) and (5, 2, 2) determine a plane. Which of the following points is in that plane? Math 4 Practice Problems for Midterm. A unit vector that is perpendicular to both V =, 3, and W = 4,, is (a) V W (b) V W (c) 5 6 V W (d) 3 6 V W (e) 7 6 V W. In three dimensions, the graph of the equation

More information

Iain Claridge. Surface Curvature

Iain Claridge. Surface Curvature Iain Claridge Surface Curvature Siddhartha Chaudhuri http://www.cse.iitb.ac.in/~cs749 Curves and surfaces in 3D For our purposes: A curve is a map α : ℝ ℝ3 a b I (or from some subset I of ℝ) α(a) α(b)

More information

Preliminary Mathematics of Geometric Modeling (3)

Preliminary Mathematics of Geometric Modeling (3) Preliminary Mathematics of Geometric Modeling (3) Hongxin Zhang and Jieqing Feng 2006-11-27 State Key Lab of CAD&CG, Zhejiang University Differential Geometry of Surfaces Tangent plane and surface normal

More information

Planar quad meshes from relative principal curvature lines

Planar quad meshes from relative principal curvature lines Planar quad meshes from relative principal curvature lines Alexander Schiftner Institute of Discrete Mathematics and Geometry Vienna University of Technology 15.09.2007 Alexander Schiftner (TU Vienna)

More information

Face-based Estimations of Curvatures on Triangle Meshes

Face-based Estimations of Curvatures on Triangle Meshes Journal for Geometry and Graphics Volume 12 (2008), No. 1, 63 73. Face-based Estimations of Curvatures on Triangle Meshes Márta Szilvási-Nagy Dept. of Geometry, Budapest University of Technology and Economics

More information

Background for Surface Integration

Background for Surface Integration Background for urface Integration 1 urface Integrals We have seen in previous work how to define and compute line integrals in R 2. You should remember the basic surface integrals that we will need to

More information

Mesh Processing Pipeline

Mesh Processing Pipeline Mesh Smoothing 1 Mesh Processing Pipeline... Scan Reconstruct Clean Remesh 2 Mesh Quality Visual inspection of sensitive attributes Specular shading Flat Shading Gouraud Shading Phong Shading 3 Mesh Quality

More information

Curves. Chapter 1. Geometric quantities As we study planar and space curves, we follow the philosophy that a

Curves. Chapter 1. Geometric quantities As we study planar and space curves, we follow the philosophy that a Chapter 1 Curves Curves, in the context of geometry, are one-dimensional lines which may be bent. Such a geometric object can describe a thin wire in space or the path of a moving object. Abstractly speaking

More information

GAUSS-BONNET FOR DISCRETE SURFACES

GAUSS-BONNET FOR DISCRETE SURFACES GAUSS-BONNET FOR DISCRETE SURFACES SOHINI UPADHYAY Abstract. Gauss-Bonnet is a deep result in differential geometry that illustrates a fundamental relationship between the curvature of a surface and its

More information

PG TRB MATHS /POLYTECNIC TRB MATHS NATIONAL ACADEMY DHARMAPURI

PG TRB MATHS /POLYTECNIC TRB MATHS NATIONAL ACADEMY DHARMAPURI PG TRB MATHS /POLYTECNIC TRB MATHS CLASSES WILL BE STARTED ON JULY 7 th Unitwise study materials and question papers available contact: 8248617507, 7010865319 PG TRB MATHS DIFFERENTIAL GEOMETRY TOTAL MARKS:100

More information

Measuring Lengths The First Fundamental Form

Measuring Lengths The First Fundamental Form Differential Geometry Lia Vas Measuring Lengths The First Fundamental Form Patching up the Coordinate Patches. Recall that a proper coordinate patch of a surface is given by parametric equations x = (x(u,

More information

MATH 2023 Multivariable Calculus

MATH 2023 Multivariable Calculus MATH 2023 Multivariable Calculus Problem Sets Note: Problems with asterisks represent supplementary informations. You may want to read their solutions if you like, but you don t need to work on them. Set

More information

Surface Parameterization

Surface Parameterization Surface Parameterization A Tutorial and Survey Michael Floater and Kai Hormann Presented by Afra Zomorodian CS 468 10/19/5 1 Problem 1-1 mapping from domain to surface Original application: Texture mapping

More information

The World Is Not Flat: An Introduction to Modern Geometry

The World Is Not Flat: An Introduction to Modern Geometry The World Is Not Flat: An to The University of Iowa September 15, 2015 The story of a hunting party The story of a hunting party What color was the bear? The story of a hunting party Overview Gauss and

More information

Mathematically, the path or the trajectory of a particle moving in space in described by a function of time.

Mathematically, the path or the trajectory of a particle moving in space in described by a function of time. Module 15 : Vector fields, Gradient, Divergence and Curl Lecture 45 : Curves in space [Section 45.1] Objectives In this section you will learn the following : Concept of curve in space. Parametrization

More information

Trajectory planning in Cartesian space

Trajectory planning in Cartesian space Robotics 1 Trajectory planning in Cartesian space Prof. Alessandro De Luca Robotics 1 1 Trajectories in Cartesian space! in general, the trajectory planning methods proposed in the joint space can be applied

More information

A GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES

A GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES A GEOMETRIC APPROACH TO CURVATURE ESTIMATION ON TRIANGULATED 3D SHAPES Mohammed Mostefa Mesmoudi, Leila De Floriani, Paola Magillo Department of Computer Science and Information Science (DISI), University

More information

274 Curves on Surfaces, Lecture 5

274 Curves on Surfaces, Lecture 5 274 Curves on Surfaces, Lecture 5 Dylan Thurston Notes by Qiaochu Yuan Fall 2012 5 Ideal polygons Previously we discussed three models of the hyperbolic plane: the Poincaré disk, the upper half-plane,

More information

COMPUTING CURVATURE AND CURVATURE NORMALS ON SMOOTH LOGICALLY CARTESIAN SURFACE MESHES

COMPUTING CURVATURE AND CURVATURE NORMALS ON SMOOTH LOGICALLY CARTESIAN SURFACE MESHES COMPUTING CURVATURE AND CURVATURE NORMALS ON SMOOTH LOGICALLY CARTESIAN SURFACE MESHES by John Thomas Hutchins A thesis submitted in partial fulfillment of the requirements for the degree of Master of

More information

Voronoi Diagram. Xiao-Ming Fu

Voronoi Diagram. Xiao-Ming Fu Voronoi Diagram Xiao-Ming Fu Outlines Introduction Post Office Problem Voronoi Diagram Duality: Delaunay triangulation Centroidal Voronoi tessellations (CVT) Definition Applications Algorithms Outlines

More information

Parameterization. Michael S. Floater. November 10, 2011

Parameterization. Michael S. Floater. November 10, 2011 Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point

More information

Geometry and Gravitation

Geometry and Gravitation Chapter 15 Geometry and Gravitation 15.1 Introduction to Geometry Geometry is one of the oldest branches of mathematics, competing with number theory for historical primacy. Like all good science, its

More information

Fathi El-Yafi Project and Software Development Manager Engineering Simulation

Fathi El-Yafi Project and Software Development Manager Engineering Simulation An Introduction to Geometry Design Algorithms Fathi El-Yafi Project and Software Development Manager Engineering Simulation 1 Geometry: Overview Geometry Basics Definitions Data Semantic Topology Mathematics

More information

Workbook. MAT 397: Calculus III

Workbook. MAT 397: Calculus III Workbook MAT 397: Calculus III Instructor: Caleb McWhorter Name: Summer 2017 Contents Preface..................................................... 2 1 Spatial Geometry & Vectors 3 1.1 Basic n Euclidean

More information

Final Review II: Examples

Final Review II: Examples Final Review II: Examles We go through last year's nal. This should not be treated as a samle nal. Question. Let (s) cos s; cos s; sin s. a) Prove that s is the arc length arameter. b) Calculate ; ; T

More information

Parameterization of Meshes

Parameterization of Meshes 2-Manifold Parameterization of Meshes What makes for a smooth manifold? locally looks like Euclidian space collection of charts mutually compatible on their overlaps form an atlas Parameterizations are

More information

CS 177 Homework 1. Julian Panetta. October 22, We want to show for any polygonal disk consisting of vertex set V, edge set E, and face set F:

CS 177 Homework 1. Julian Panetta. October 22, We want to show for any polygonal disk consisting of vertex set V, edge set E, and face set F: CS 177 Homework 1 Julian Panetta October, 009 1 Euler Characteristic 1.1 Polyhedral Formula We want to show for any polygonal disk consisting of vertex set V, edge set E, and face set F: V E + F = 1 First,

More information

Impulse Gauss Curvatures 2002 SSHE-MA Conference. Howard Iseri Mansfield University

Impulse Gauss Curvatures 2002 SSHE-MA Conference. Howard Iseri Mansfield University Impulse Gauss Curvatures 2002 SSHE-MA Conference Howard Iseri Mansfield University Abstract: In Riemannian (differential) geometry, the differences between Euclidean geometry, elliptic geometry, and hyperbolic

More information

Discrete Surfaces. David Gu. Tsinghua University. Tsinghua University. 1 Mathematics Science Center

Discrete Surfaces. David Gu. Tsinghua University. Tsinghua University. 1 Mathematics Science Center Discrete Surfaces 1 1 Mathematics Science Center Tsinghua University Tsinghua University Discrete Surface Discrete Surfaces Acquired using 3D scanner. Discrete Surfaces Our group has developed high speed

More information

ON THE GEODESIC TORSION OF A TANGENTIAL INTERSECTION CURVE OF TWO SURFACES IN R Introduction

ON THE GEODESIC TORSION OF A TANGENTIAL INTERSECTION CURVE OF TWO SURFACES IN R Introduction Acta Math. Univ. Comenianae Vol. LXXXII, (3), pp. 77 89 77 ON THE GEODESIC TORSION OF A TANGENTIAL INTERSECTION CURVE OF TWO SURFACES IN R 3 B. UYAR DÜLDÜL and M. ÇALIŞKAN Abstract. In this paper, we find

More information

Parameterization of triangular meshes

Parameterization of triangular meshes Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to

More information

Greedy Routing with Guaranteed Delivery Using Ricci Flow

Greedy Routing with Guaranteed Delivery Using Ricci Flow Greedy Routing with Guaranteed Delivery Using Ricci Flow Jie Gao Stony Brook University Joint work with Rik Sarkar, Xiaotian Yin, Wei Zeng, Feng Luo, Xianfeng David Gu Greedy Routing Assign coordinatesto

More information

SOLVING PARTIAL DIFFERENTIAL EQUATIONS ON POINT CLOUDS

SOLVING PARTIAL DIFFERENTIAL EQUATIONS ON POINT CLOUDS SOLVING PARTIAL DIFFERENTIAL EQUATIONS ON POINT CLOUDS JIAN LIANG AND HONGKAI ZHAO Abstract. In this paper we present a general framework for solving partial differential equations on manifolds represented

More information

Estimating normal vectors and curvatures by centroid weights

Estimating normal vectors and curvatures by centroid weights Computer Aided Geometric Design 21 (2004) 447 458 www.elsevier.com/locate/cagd Estimating normal vectors and curvatures by centroid weights Sheng-Gwo Chen, Jyh-Yang Wu Department of Mathematics, National

More information

Research in Computational Differential Geomet

Research in Computational Differential Geomet Research in Computational Differential Geometry November 5, 2014 Approximations Often we have a series of approximations which we think are getting close to looking like some shape. Approximations Often

More information

ME 115(b): Final Exam, Spring

ME 115(b): Final Exam, Spring ME 115(b): Final Exam, Spring 2011-12 Instructions 1. Limit your total time to 5 hours. That is, it is okay to take a break in the middle of the exam if you need to ask me a question, or go to dinner,

More information

MOTION OF A LINE SEGMENT WHOSE ENDPOINT PATHS HAVE EQUAL ARC LENGTH. Anton GFRERRER 1 1 University of Technology, Graz, Austria

MOTION OF A LINE SEGMENT WHOSE ENDPOINT PATHS HAVE EQUAL ARC LENGTH. Anton GFRERRER 1 1 University of Technology, Graz, Austria MOTION OF A LINE SEGMENT WHOSE ENDPOINT PATHS HAVE EQUAL ARC LENGTH Anton GFRERRER 1 1 University of Technology, Graz, Austria Abstract. The following geometric problem originating from an engineering

More information

ON THE GEODESIC TORSION OF A TANGENTIAL INTERSECTION CURVE OF TWO SURFACES IN R Introduction

ON THE GEODESIC TORSION OF A TANGENTIAL INTERSECTION CURVE OF TWO SURFACES IN R Introduction ON THE GEODESIC TORSION OF A TANGENTIAL INTERSECTION CURVE OF TWO SURFACES IN R 3 B. UYAR DÜLDÜL and M. ÇALIŞKAN Abstract. In this paper, we find the unit tangent vector and the geodesic torsion of the

More information

Let and be a differentiable function. Let Then be the level surface given by

Let and be a differentiable function. Let Then be the level surface given by Module 12 : Total differential, Tangent planes and normals Lecture 35 : Tangent plane and normal [Section 35.1] > Objectives In this section you will learn the following : The notion tangent plane to a

More information

Let be a function. We say, is a plane curve given by the. Let a curve be given by function where is differentiable with continuous.

Let be a function. We say, is a plane curve given by the. Let a curve be given by function where is differentiable with continuous. Module 8 : Applications of Integration - II Lecture 22 : Arc Length of a Plane Curve [Section 221] Objectives In this section you will learn the following : How to find the length of a plane curve 221

More information

Greedy Routing in Wireless Networks. Jie Gao Stony Brook University

Greedy Routing in Wireless Networks. Jie Gao Stony Brook University Greedy Routing in Wireless Networks Jie Gao Stony Brook University A generic sensor node CPU. On-board flash memory or external memory Sensors: thermometer, camera, motion, light sensor, etc. Wireless

More information

Curvature-Adaptive Remeshing with Feature Preservation of Manifold Triangle Meshes with Boundary

Curvature-Adaptive Remeshing with Feature Preservation of Manifold Triangle Meshes with Boundary Curvature-Adaptive Remeshing with Feature Preservation of Manifold Triangle Meshes with Boundary Master s Project Tanja Munz Master of Science Computer Animation and Visual Effects 24th August, 2015 Abstract

More information

Geometric Modeling Assignment 3: Discrete Differential Quantities

Geometric Modeling Assignment 3: Discrete Differential Quantities Geometric Modeling Assignment : Discrete Differential Quantities Acknowledgements: Julian Panetta, Olga Diamanti Assignment (Optional) Topic: Discrete Differential Quantities with libigl Vertex Normals,

More information

Updated: August 24, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University

Updated: August 24, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University Updated: August 24, 216 Calculus III Section 1.2 Math 232 Calculus III Brian Veitch Fall 215 Northern Illinois University 1.2 Calculus with Parametric Curves Definition 1: First Derivative of a Parametric

More information

GD - Differential Geometry

GD - Differential Geometry Coordinating unit: 200 - FME - School of Mathematics and Statistics Teaching unit: 749 - MAT - Department of Mathematics Academic year: Degree: 2017 BACHELOR'S DEGREE IN MATHEMATICS (Syllabus 2009). (Teaching

More information

Curvatures of Smooth and Discrete Surfaces

Curvatures of Smooth and Discrete Surfaces Curvatures of Smooth and iscrete Surfaces John M. Sullivan The curvatures of a smooth curve or surface are local measures of its shape. Here we consider analogous quantities for discrete curves and surfaces,

More information

Tangent & normal vectors The arc-length parameterization of a 2D curve The curvature of a 2D curve

Tangent & normal vectors The arc-length parameterization of a 2D curve The curvature of a 2D curve Local Analysis of 2D Curve Patches Topic 4.2: Local analysis of 2D curve patches Representing 2D image curves Estimating differential properties of 2D curves Tangent & normal vectors The arc-length parameterization

More information

Solution 2. ((3)(1) (2)(1), (4 3), (4)(2) (3)(3)) = (1, 1, 1) D u (f) = (6x + 2yz, 2y + 2xz, 2xy) (0,1,1) = = 4 14

Solution 2. ((3)(1) (2)(1), (4 3), (4)(2) (3)(3)) = (1, 1, 1) D u (f) = (6x + 2yz, 2y + 2xz, 2xy) (0,1,1) = = 4 14 Vector and Multivariable Calculus L Marizza A Bailey Practice Trimester Final Exam Name: Problem 1. To prepare for true/false and multiple choice: Compute the following (a) (4, 3) ( 3, 2) Solution 1. (4)(

More information

Midterm Review II Math , Fall 2018

Midterm Review II Math , Fall 2018 Midterm Review II Math 2433-3, Fall 218 The test will cover section 12.5 of chapter 12 and section 13.1-13.3 of chapter 13. Examples in class, quizzes and homework problems are the best practice for the

More information

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv.

MAC2313 Final A. a. The vector r u r v lies in the tangent plane of S at a given point. b. S f(x, y, z) ds = R f(r(u, v)) r u r v du dv. MAC2313 Final A (5 pts) 1. Let f(x, y, z) be a function continuous in R 3 and let S be a surface parameterized by r(u, v) with the domain of the parameterization given by R; how many of the following are

More information

Numerical Treatment of Geodesic Differential. Equations on a Surface in

Numerical Treatment of Geodesic Differential. Equations on a Surface in International Mathematical Forum, Vol. 8, 2013, no. 1, 15-29 Numerical Treatment of Geodesic Differential Equations on a Surface in Nassar H. Abdel-All Department of Mathematics, Faculty of Science Assiut

More information

Modern Differential Geometry ofcurves and Surfaces

Modern Differential Geometry ofcurves and Surfaces K ALFRED GRAY University of Maryland Modern Differential Geometry ofcurves and Surfaces /, CRC PRESS Boca Raton Ann Arbor London Tokyo CONTENTS 1. Curves in the Plane 1 1.1 Euclidean Spaces 2 1.2 Curves

More information

Lecture 11 Differentiable Parametric Curves

Lecture 11 Differentiable Parametric Curves Lecture 11 Differentiable Parametric Curves 11.1 Definitions and Examples. 11.1.1 Definition. A differentiable parametric curve in R n of class C k (k 1) is a C k map t α(t) = (α 1 (t),..., α n (t)) of

More information

Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry

Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry Introduction p. 1 What Is Geometric Modeling? p. 1 Computer-aided geometric design Solid modeling Algebraic geometry Computational geometry Representation Ab initio design Rendering Solid modelers Kinematic

More information

Möbius Transformations in Scientific Computing. David Eppstein

Möbius Transformations in Scientific Computing. David Eppstein Möbius Transformations in Scientific Computing David Eppstein Univ. of California, Irvine School of Information and Computer Science (including joint work with Marshall Bern from WADS 01 and SODA 03) Outline

More information

Estimating affine-invariant structures on triangle meshes

Estimating affine-invariant structures on triangle meshes Estimating affine-invariant structures on triangle meshes Thales Vieira Mathematics, UFAL Dimas Martinez Mathematics, UFAM Maria Andrade Mathematics, UFS Thomas Lewiner École Polytechnique Invariant descriptors

More information

Spline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1

Spline Curves. Spline Curves. Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Spline Curves Prof. Dr. Hans Hagen Algorithmic Geometry WS 2013/2014 1 Problem: In the previous chapter, we have seen that interpolating polynomials, especially those of high degree, tend to produce strong

More information

Optimizing triangular meshes to have the incrircle packing property

Optimizing triangular meshes to have the incrircle packing property Packing Circles and Spheres on Surfaces Ali Mahdavi-Amiri Introduction Optimizing triangular meshes to have p g g the incrircle packing property Our Motivation PYXIS project Geometry Nature Geometry Isoperimetry

More information

Shape Interrogation. 1 Introduction

Shape Interrogation. 1 Introduction Shape Interrogation Stefanie Hahmann 1, Alexander Belyaev 2, Laurent Busé 3, Gershon Elber 4, Bernard Mourrain 3, and Christian Rössl 3 1 Laboratoire Jean Kuntzmann, Institut National Polytechnique de

More information

Visualizing Quaternions

Visualizing Quaternions Visualizing Quaternions Andrew J. Hanson Computer Science Department Indiana University Siggraph 1 Tutorial 1 GRAND PLAN I: Fundamentals of Quaternions II: Visualizing Quaternion Geometry III: Quaternion

More information

Further Graphics. A Brief Introduction to Computational Geometry

Further Graphics. A Brief Introduction to Computational Geometry Further Graphics A Brief Introduction to Computational Geometry 1 Alex Benton, University of Cambridge alex@bentonian.com Supported in part by Google UK, Ltd Terminology We ll be focusing on discrete (as

More information

CS 4620 Final Exam. (a) Is a circle C 0 continuous?

CS 4620 Final Exam. (a) Is a circle C 0 continuous? CS 4620 Final Exam Wednesday 9, December 2009 2 1 2 hours Prof. Doug James Explain your reasoning for full credit. You are permitted a double-sided sheet of notes. Calculators are allowed but unnecessary.

More information

Invariant Measures of Convex Sets. Eitan Grinspun, Columbia University Peter Schröder, Caltech

Invariant Measures of Convex Sets. Eitan Grinspun, Columbia University Peter Schröder, Caltech Invariant Measures of Convex Sets Eitan Grinspun, Columbia University Peter Schröder, Caltech What will we measure? Subject to be measured, S an object living in n-dim space convex, compact subset of R

More information

Fuzzy Voronoi Diagram

Fuzzy Voronoi Diagram Fuzzy Voronoi Diagram Mohammadreza Jooyandeh and Ali Mohades Khorasani Mathematics and Computer Science, Amirkabir University of Technology, Hafez Ave., Tehran, Iran mohammadreza@jooyandeh.info,mohades@aut.ac.ir

More information

Pythagorean - Hodograph Curves: Algebra and Geometry Inseparable

Pythagorean - Hodograph Curves: Algebra and Geometry Inseparable Rida T. Farouki Pythagorean - Hodograph Curves: Algebra and Geometry Inseparable With 204 Figures and 15 Tables 4y Springer Contents 1 Introduction 1 1.1 The Lure of Analytic Geometry 1 1.2 Symbiosis of

More information

Discrete Differential Geometry: An Applied Introduction

Discrete Differential Geometry: An Applied Introduction Discrete Differential Geometry: An Applied Introduction Eitan Grinspun with Mathieu Desbrun, Konrad Polthier, Peter Schröder, & Ari Stern 1 Differential Geometry Why do we care? geometry of surfaces Springborn

More information

Geometric structures on 2-orbifolds

Geometric structures on 2-orbifolds Geometric structures on 2-orbifolds Section 1: Manifolds and differentiable structures S. Choi Department of Mathematical Science KAIST, Daejeon, South Korea 2010 Fall, Lectures at KAIST S. Choi (KAIST)

More information

Dr. Allen Back. Nov. 21, 2014

Dr. Allen Back. Nov. 21, 2014 Dr. Allen Back of Nov. 21, 2014 The most important thing you should know (e.g. for exams and homework) is how to setup (and perhaps compute if not too hard) surface integrals, triple integrals, etc. But

More information

Unwarping paper: A differential geometric approach

Unwarping paper: A differential geometric approach Unwarping paper: A differential geometric approach Nail Gumerov, Ali Zandifar, Ramani Duraiswami and Larry S. Davis March 28, 2003 Outline Motivation Statement of the Problem Previous Works Definition

More information

CGT 581 G Geometric Modeling Curves

CGT 581 G Geometric Modeling Curves CGT 581 G Geometric Modeling Curves Bedrich Benes, Ph.D. Purdue University Department of Computer Graphics Technology Curves What is a curve? Mathematical definition 1) The continuous image of an interval

More information

CS 468 (Spring 2013) Discrete Differential Geometry

CS 468 (Spring 2013) Discrete Differential Geometry CS 468 (Spring 2013) Discrete Differential Geometry 1 Math Review Lecture 14 15 May 2013 Discrete Exterior Calculus Lecturer: Justin Solomon Scribe: Cassidy Saenz Before we dive into Discrete Exterior

More information

Notes on Spherical Geometry

Notes on Spherical Geometry Notes on Spherical Geometry Abhijit Champanerkar College of Staten Island & The Graduate Center, CUNY Spring 2018 1. Vectors and planes in R 3 To review vector, dot and cross products, lines and planes

More information