Introduction to Geometric Algebra
|
|
- Theresa Porter
- 6 years ago
- Views:
Transcription
1 Introduction to Geometric Algebra Lecture 1 Why Geometric Algebra? Professor Leandro Augusto Frata Fernandes laffernandes@ic.uff.br Lecture notes available in Geometric Problems Geometric data Lines, planes, circles, spheres, etc. Transformations, translation, scaling, etc. Other operations Intersection, basis orthogonalization, etc. Linear Algebra is the standard framework 2 1
2 Using Vectors to Encode Geometric Data Directions Points 3 Using Vectors to Encode Geometric Data Directions Points Drawback HOMEWORK The semantic difference between a direction vector and a point vector is not encoded in the vector type itself. R. N. Goldman (1985), Illicit expressions in vector algebra, ACM Trans. Graph., vol. 4, no. 3, pp
3 Assembling Geometric Data Straight lines Point vector Direction vector Planes Normal vector Distance from origin Spheres Center point Radius 5 Assembling Geometric Data Straight lines Point vector Direction vector Planes Normal vector Distance from origin Spheres Center point Radius 6 3
4 Assembling Geometric Data Straight lines Point vector Direction vector Planes Normal vector Distance from origin Spheres Center point Radius 7 Assembling Geometric Data Straight lines Point vector Direction vector Planes Normal vector Distance from origin Spheres Center point Radius Drawback The factorization of geometric elements prevents their use as computing primitives. 8 4
5 Intersection of Two Geometric Elements A specialized equation for each pair of elements Straight line Straight line Straight line Plane Straight line Sphere Plane Sphere etc. Special cases must be handled explicitly Straight line Circle may return o Empty set o One point o Points pair 9 Intersection of Two Geometric Elements A specialized equation for each pair of elements Straight line Straight line Straight line Plane Straight line Sphere Plane Sphere etc. Special cases must be handled explicitly Straight line Circle may return o Empty set o One point o Points pair Plücker Coordinates Linear Algebra Extension An alternative to represent flat geometric elements Points, lines and planes as elementary types Allow the development of more general solution Not fully compatible with transformation matrices J. Stolfi (1991) Oriented projective geometry. Academic Press Professional, Inc. 10 5
6 Using Matrices do Encode Transformations Projective Affine Similitude Isotropic Scaling Linear Scaling Reflection Shear Perspective 11 Rigid Body / Euclidean Transforms Preserves distances Preserves angles 12 6
7 Similitudes / Similarity Transforms Preservers angles Similitudes Isotropic Scaling 13 Linear Transformations L(p + q) = L(p) + L(q) L(α p) = α L(p) Similitudes Isotropic Scaling Linear Scaling Reflection Shear 14 7
8 Affine Transformations Preserves parallel lines Affine Similitudes Isotropic Scaling Linear Scaling Reflection Shear 15 Projective Transformations Preserves lines Projective Affine Similitudes Isotropic Scaling Linear Scaling Reflection Shear Perspective 16 8
9 Using Matrices do Encode Transformations Projective Affine Similitude Isotropic Scaling Linear Scaling Reflection Shear Perspective 17 Drawbacks of Transformation Matrices Non-uniform scaling affects point vectors and normal vectors differently HOMEWORK 90 > X 18 9
10 Drawbacks of Transformation Matrices Non-uniform scaling affects point vectors and normal vectors differently matrices Not suitable for interpolation Encode the rotation axis and angle in a costly way 19 Drawbacks of Transformation Matrices Non-uniform scaling affects point vectors and normal vectors differently matrices Not suitable for interpolation Quaternion Encode the rotation axis and angle in a costly way Linear Algebra Extension Represent and interpolate rotations consistently Can be combined with isotropic scaling Not well connected with other transformations Not compatible with Plücker coordinates Defined only in 3-D W. R. Hamilton (1844) On a new species of imaginary quantities connected with the theory of quaternions. In Proc. of the Royal Irish Acad., vol. 2, pp
11 Linear Algebra Standard language for geometric problems Well-known limitations Aggregates different formalisms to obtain complete solutions Matrices Plücker coordinates Quaternion Jumping back and forth between formalisms requires custom and ad hoc conversions 21 Geometric Algebra High-level framework for geometric operations Geometric elements as primitives for computation Naturally generalizes and integrates Plücker coordinates Quaternion Complex numbers Extends the same solution to Higher dimensions All kinds of geometric elements 22 11
12 Motivational Example 1. Create the circle through points c 1, c 2 and c 3 2. Create a straight line L through points a 1 and a 2 3. Rotate the circle around the line and show n rotation steps 4. Create a plane through point p and with normal vector n Dual plane Adapted from L. Dorst, D. Fontijine, S. Mann. Geometric algebra for computer science. Morgan Kaufmann Publishers, Reflect the whole situation with the line and the circlers in the plane 23 Motivational Example 1. Create the circle through points c 1, c 2 and c 3 2. Create a straight line L through points a 1 and a 2 3. Rotate the circle around the line and show n rotation steps The reflected line becomes a circle. The reflected rotation becomes a scaled rotation around the circle. Adapted from L. Dorst, D. Fontijine, S. Mann. Geometric algebra for computer science. Morgan Kaufmann Publishers, Create a plane sphere through point pp and with normal center cvector n The only thing that is different is that the plane was changed Dual sphere plane by the sphere. 5. Reflect the whole situation with the line and the circlers in the plane sphere 24 12
13 Why I never heard about GA before? Geometric algebra is a new formalism Paul Dirac ( ) William R. Hamilton ( ) Hermann G. Grassmann ( ) William K. Clifford ( ) David O. Hestenes (1933-) Wolfgang E. Pauli ( ) 1840s s, 1920s 2001 D. Hestenes (2001) Old wine in new bottles..., in Geometric algebra with applications in science and engineering, chapter 1, pp. 3-17, Birkhäuser, Boston. 25 Differences Between Algebras Clifford algebra Developed in nongeometric directions Permits us to construct elements by a universal addition Arbitrary multivectors may be important Geometric algebra The geometrically significant part of Clifford algebra Only permits exclusively multiplicative constructions o The only elements that can be added are scalars, vectors, pseudovectors, and pseudoscalars Only blades and versors are important 26 13
Introduction to Geometric Algebra Lecture I
Introduction to Geometric Algebra Lecture I Leandro A. F. Fernandes laffernandes@inf.ufrgs.br Manuel M. Oliveira oliveira@inf.ufrgs.br CG UFRGS Geometric problems Geometric data Lines, planes, circles,
More informationIntroduction to Geometric Algebra Lecture VI
Introduction to Geometric Algebra Lecture VI Leandro A. F. Fernandes laffernandes@inf.ufrgs.br Manuel M. Oliveira oliveira@inf.ufrgs.br Visgraf - Summer School in Computer Graphics - 2010 CG UFRGS Lecture
More informationIntroduction to Geometric Algebra Lecture V
Introduction to Geometric Algebra Lecture V Leandro A. F. Fernandes laffernandes@inf.ufrgs.br Manuel M. Oliveira oliveira@inf.ufrgs.br Visgraf - Summer School in Computer Graphics - 2010 CG UFRGS Lecture
More informationAPPENDIX A CLIFFORD ALGEBRA
1 APPENDIX A CLIFFORD ALGEBRA Clifford algebra (CA), or geometric algebra, is a powerful mathematical tool which allows for a direct and intuitive solution of geometric problems in fields as computer graphics,
More informationIntroduction to Geometric Algebra
Introduction to Geometric Algebra Lecture 6 Intersection and Union of Subspaces Professor Leandro Augusto Frata Fernandes laffernandes@ic.uff.br Lecture notes available in http://www.ic.uff.br/~laffernandes/teaching/2013.1/topicos_ag
More informationCoordinate Free Perspective Projection of Points in the Conformal Model Using Transversions
Coordinate Free Perspective Projection of Points in the Conformal Model Using Transversions Stephen Mann Abstract Goldman presented a method for computing a versor form of the perspective projection of
More informationGeometric Algebra for Computer Graphics
John Vince Geometric Algebra for Computer Graphics 4u Springer Contents Preface vii 1 Introduction 1 1.1 Aims and objectives of this book 1 1.2 Mathematics for CGI software 1 1.3 The book's structure 2
More informationFrom Grassmann s vision to Geometric Algebra Computing
From Grassmann s vision to Geometric Algebra Computing Dietmar Hildenbrand 1. Introduction What mathematicians often call Clifford algebra is called geometric algebra if the focus is on the geometric meaning
More informationGeometric Algebra. 8. Conformal Geometric Algebra. Dr Chris Doran ARM Research
Geometric Algebra 8. Conformal Geometric Algebra Dr Chris Doran ARM Research Motivation Projective geometry showed that there is considerable value in treating points as vectors Key to this is a homogeneous
More information2D Euclidean Geometric Algebra Matrix Representation
2D Euclidean Geometric Algebra Matrix Representation Kurt Nalt March 29, 2015 Abstract I present the well-known matrix representation of 2D Euclidean Geometric Algebra, and suggest a literal geometric
More informationMath background. 2D Geometric Transformations. Implicit representations. Explicit representations. Read: CS 4620 Lecture 6
Math background 2D Geometric Transformations CS 4620 Lecture 6 Read: Chapter 2: Miscellaneous Math Chapter 5: Linear Algebra Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector
More informationUnified Mathematics (Uni-Math)
Unified Mathematics (Uni-Math) with Geometric Algebra (GA) David Hestenes Arizona State University For geometry, you know, is the gateway to science, and that gate is so low and small that you can enter
More informationVector Algebra Transformations. Lecture 4
Vector Algebra Transformations Lecture 4 Cornell CS4620 Fall 2008 Lecture 4 2008 Steve Marschner 1 Geometry A part of mathematics concerned with questions of size, shape, and relative positions of figures
More informationANALYSIS OF POINT CLOUDS Using Conformal Geometric Algebra
ANALYSIS OF POINT CLOUDS Using Conformal Geometric Algebra Dietmar Hildenbrand Research Center of Excellence for Computer Graphics, University of Technology, Darmstadt, Germany Dietmar.Hildenbrand@gris.informatik.tu-darmstadt.de
More informationGEOMETRIC TOOLS FOR COMPUTER GRAPHICS
GEOMETRIC TOOLS FOR COMPUTER GRAPHICS PHILIP J. SCHNEIDER DAVID H. EBERLY MORGAN KAUFMANN PUBLISHERS A N I M P R I N T O F E L S E V I E R S C I E N C E A M S T E R D A M B O S T O N L O N D O N N E W
More information2D Transforms. Lecture 4 CISC440/640 Spring Department of Computer and Information Science
2D Transforms Lecture 4 CISC440/640 Spring 2015 Department of Computer and Information Science Where are we going? A preview of assignment #1 part 2: The Ken Burns Effect 2 Where are we going? A preview
More informationGraphics Pipeline 2D Geometric Transformations
Graphics Pipeline 2D Geometric Transformations CS 4620 Lecture 8 1 Plane projection in drawing Albrecht Dürer 2 Plane projection in drawing source unknown 3 Rasterizing triangles Summary 1 evaluation of
More informationMotivation. 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 informationFoundations of Geometric Algebra Computing
Foundations of Geometric Algebra Computing 26.10.2012 Dr.-Ing. Dietmar Hildenbrand LOEWE Priority Program Cocoon Technische Universität Darmstadt Achtung Änderung! Die Übung findet Montags jeweils 11:40
More informationAdvanced Geometric Approach for Graphics and Visual Guided Robot Object Manipulation
Advanced Geometric Approach for Graphics and Visual Guided Robot Object Manipulation Dietmar Hildenbrand Interactive Graphics Systems Group University of Technology Darmstadt, Germany dhilden@gris.informatik.tu-darmstadt.de
More informationLecture 5 2D Transformation
Lecture 5 2D Transformation What is a transformation? In computer graphics an object can be transformed according to position, orientation and size. Exactly what it says - an operation that transforms
More informationTransformations. Examples of transformations: shear. scaling
Transformations Eamples of transformations: translation rotation scaling shear Transformations More eamples: reflection with respect to the y-ais reflection with respect to the origin Transformations Linear
More informationAdvanced Computer Graphics Transformations. Matthias Teschner
Advanced Computer Graphics Transformations Matthias Teschner Motivation Transformations are used To convert between arbitrary spaces, e.g. world space and other spaces, such as object space, camera space
More informationGeometric transformations assign a point to a point, so it is a point valued function of points. Geometric transformation may destroy the equation
Geometric transformations assign a point to a point, so it is a point valued function of points. Geometric transformation may destroy the equation and the type of an object. Even simple scaling turns a
More informationCS354 Computer Graphics Rotations and Quaternions
Slide Credit: Don Fussell CS354 Computer Graphics Rotations and Quaternions Qixing Huang April 4th 2018 Orientation Position and Orientation The position of an object can be represented as a translation
More informationTranslations. Geometric Image Transformations. Two-Dimensional Geometric Transforms. Groups and Composition
Geometric Image Transformations Algebraic Groups Euclidean Affine Projective Bovine Translations Translations are a simple family of two-dimensional transforms. Translations were at the heart of our Sprite
More informationGeometric transformations in 3D and coordinate frames. Computer Graphics CSE 167 Lecture 3
Geometric transformations in 3D and coordinate frames Computer Graphics CSE 167 Lecture 3 CSE 167: Computer Graphics 3D points as vectors Geometric transformations in 3D Coordinate frames CSE 167, Winter
More information6.837 LECTURE 7. Lecture 7 Outline Fall '01. Lecture Fall '01
6.837 LECTURE 7 1. Geometric Image Transformations 2. Two-Dimensional Geometric Transforms 3. Translations 4. Groups and Composition 5. Rotations 6. Euclidean Transforms 7. Problems with this Form 8. Choose
More informationToday. Today. Introduction. Matrices. Matrices. Computergrafik. Transformations & matrices Introduction Matrices
Computergrafik Matthias Zwicker Universität Bern Herbst 2008 Today Transformations & matrices Introduction Matrices Homogeneous Affine transformations Concatenating transformations Change of Common coordinate
More informationSEMINARI CHIMICI. Dr. Eckhard Hitzer Department of Applied Physics University of Fukui - Japan
SEMINARI CHIMICI Data mercoledì 3 Marzo 2010, ore 14.15-16.15 giovedì 4 Marzo 2010, ore 14.15-16.15 Dipartimento di Chimica Strutturale e Stereochimica Inorganica Aula H dcssi.istm.cnr.it Via Venezian,
More informationPRIMITIVES INTERSECTION WITH CONFORMAL 5D GEOMETRY
PRIMITIVES INTERSECTION WITH CONFORMAL 5D GEOMETRY Eduardo Roa eduroam@ldc.usb.ve Víctor Theoktisto vtheok@usb.ve Laboratorio de Computación Gráfica e Interacción Universidad Simón Bolívar, Caracas-VENEZUELA.
More informationFundamentals of 3D. Lecture 3: Debriefing: Lecture 2 Rigid transformations Quaternions Iterative Closest Point (+Kd-trees)
INF555 Fundamentals of 3D Lecture 3: Debriefing: Lecture 2 Rigid transformations Quaternions Iterative Closest Point (+Kd-trees) Frank Nielsen nielsen@lix.polytechnique.fr Harris-Stephens' combined corner/edge
More informationHomogeneous coordinates, lines, screws and twists
Homogeneous coordinates, lines, screws and twists In lecture 1 of module 2, a brief mention was made of homogeneous coordinates, lines in R 3, screws and twists to describe the general motion of a rigid
More informationTransformations in Ray Tracing. MIT EECS 6.837, Durand and Cutler
Transformations in Ray Tracing Linear Algebra Review Session Tonight! 7:30 9 PM Last Time: Simple Transformations Classes of Transformations Representation homogeneous coordinates Composition not commutative
More informationGeometric Transformations
Geometric Transformations CS 4620 Lecture 9 2017 Steve Marschner 1 A little quick math background Notation for sets, functions, mappings Linear and affine transformations Matrices Matrix-vector multiplication
More informationTransformations Week 9, Lecture 18
CS 536 Computer Graphics Transformations Week 9, Lecture 18 2D Transformations David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 3 2D Affine Transformations
More informationS U N G - E U I YO O N, K A I S T R E N D E R I N G F R E E LY A VA I L A B L E O N T H E I N T E R N E T
S U N G - E U I YO O N, K A I S T R E N D E R I N G F R E E LY A VA I L A B L E O N T H E I N T E R N E T Copyright 2018 Sung-eui Yoon, KAIST freely available on the internet http://sglab.kaist.ac.kr/~sungeui/render
More informationKatu Software Algebra 1 Geometric Sequence
Katu Software Algebra 1 Geometric Sequence Read Book Online: Katu Software Algebra 1 Geometric Sequence Download or read online ebook katu software algebra 1 geometric sequence in any format for any devices.
More information2D/3D Geometric Transformations and Scene Graphs
2D/3D Geometric Transformations and Scene Graphs Week 4 Acknowledgement: The course slides are adapted from the slides prepared by Steve Marschner of Cornell University 1 A little quick math background
More informationRotation parameters for model building and stable parameter inversion in orthorhombic media Cintia Lapilli* and Paul J. Fowler, WesternGeco.
otation parameters for model building and stable parameter inversion in orthorhombic media Cintia Lapilli* and Paul J Fowler, WesternGeco Summary Symmetry groups commonly used to describe seismic anisotropy
More informationCHAPTER 1 Graphics Systems and Models 3
?????? 1 CHAPTER 1 Graphics Systems and Models 3 1.1 Applications of Computer Graphics 4 1.1.1 Display of Information............. 4 1.1.2 Design.................... 5 1.1.3 Simulation and Animation...........
More information2D transformations: An introduction to the maths behind computer graphics
2D transformations: An introduction to the maths behind computer graphics Lecturer: Dr Dan Cornford d.cornford@aston.ac.uk http://wiki.aston.ac.uk/dancornford CS2150, Computer Graphics, Aston University,
More informationXPM 2D Transformations Week 2, Lecture 3
CS 430/585 Computer Graphics I XPM 2D Transformations Week 2, Lecture 3 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel
More informationGeometric Algebra for Computer Science
Geometric Algebra for Computer Science Geometric Algebra for Computer Science An Object-oriented Approach to Geometry LEO DORST DANIEL FONTIJNE STEPHEN MANN AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK
More informationGraphics and Interaction Transformation geometry and homogeneous coordinates
433-324 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationHumanoid Robotics. Projective Geometry, Homogeneous Coordinates. (brief introduction) Maren Bennewitz
Humanoid Robotics Projective Geometry, Homogeneous Coordinates (brief introduction) Maren Bennewitz Motivation Cameras generate a projected image of the 3D world In Euclidian geometry, the math for describing
More informationCOMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationVisual Recognition: Image Formation
Visual Recognition: Image Formation Raquel Urtasun TTI Chicago Jan 5, 2012 Raquel Urtasun (TTI-C) Visual Recognition Jan 5, 2012 1 / 61 Today s lecture... Fundamentals of image formation You should know
More informationCMSC 425: Lecture 6 Affine Transformations and Rotations
CMSC 45: Lecture 6 Affine Transformations and Rotations Affine Transformations: So far we have been stepping through the basic elements of geometric programming. We have discussed points, vectors, and
More informationXPM 2D Transformations Week 2, Lecture 3
CS 430/585 Computer Graphics I XPM 2D Transformations Week 2, Lecture 3 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel
More informationCS4620/5620. Professor: Kavita Bala. Cornell CS4620/5620 Fall 2012 Lecture Kavita Bala 1 (with previous instructors James/Marschner)
CS4620/5620 Affine and 3D Transformations Professor: Kavita Bala 1 Announcements Updated schedule on course web page 2 Prelim days finalized and posted Oct 11, Nov 29 No final exam, final project will
More information2D Object Definition (1/3)
2D Object Definition (1/3) Lines and Polylines Lines drawn between ordered points to create more complex forms called polylines Same first and last point make closed polyline or polygon Can intersect itself
More informationInverse kinematics computation in computer graphics and robotics using conformal geometric algebra
This is page 1 Printer: Opaque this Inverse kinematics computation in computer graphics and robotics using conformal geometric algebra Dietmar Hildenbrand, Julio Zamora and Eduardo Bayro-Corrochano ABSTRACT
More informationVisualizing 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 informationAnimation. Keyframe animation. CS4620/5620: Lecture 30. Rigid motion: the simplest deformation. Controlling shape for animation
Keyframe animation CS4620/5620: Lecture 30 Animation Keyframing is the technique used for pose-to-pose animation User creates key poses just enough to indicate what the motion is supposed to be Interpolate
More informationAdvanced Rendering Techniques
Advanced Rendering Techniques Lecture Rendering Pipeline (Part I) Professor Leandro Augusto Frata Fernandes laffernandes@ic.uff.br Lecture notes available in http://www.ic.uff.br/~laffernandes/teaching/0./topicos_rendering
More informationRida Farouki from the University of California at Davis (UCD), USA, for inviting me to work with them. The work on random multivector variables and
Preface Geometry is a pervasive mathematical concept that appears in many places and in many disguises. The representation of geometric entities, of their unions and intersections, and of their transformations
More informationLast week. Machiraju/Zhang/Möller/Fuhrmann
Last week Machiraju/Zhang/Möller/Fuhrmann 1 Geometry basics Scalar, point, and vector Vector space and affine space Basic point and vector operations Sided-ness test Lines, planes, and triangles Linear
More informationQuaternions and Dual Coupled Orthogonal Rotations in Four-Space
Quaternions and Dual Coupled Orthogonal Rotations in Four-Space Kurt Nalty January 8, 204 Abstract Quaternion multiplication causes tensor stretching) and versor turning) operations. Multiplying by unit
More informationGame Mathematics. (12 Week Lesson Plan)
Game Mathematics (12 Week Lesson Plan) Lesson 1: Set Theory Textbook: Chapter One (pgs. 1 15) We begin the course by introducing the student to a new vocabulary and set of rules that will be foundational
More informationMetric Rectification for Perspective Images of Planes
789139-3 University of California Santa Barbara Department of Electrical and Computer Engineering CS290I Multiple View Geometry in Computer Vision and Computer Graphics Spring 2006 Metric Rectification
More informationDD2429 Computational Photography :00-19:00
. Examination: DD2429 Computational Photography 202-0-8 4:00-9:00 Each problem gives max 5 points. In order to pass you need about 0-5 points. You are allowed to use the lecture notes and standard list
More informationC O M P U T E R G R A P H I C S. Computer Graphics. Three-Dimensional Graphics I. Guoying Zhao 1 / 52
Computer Graphics Three-Dimensional Graphics I Guoying Zhao 1 / 52 Geometry Guoying Zhao 2 / 52 Objectives Introduce the elements of geometry Scalars Vectors Points Develop mathematical operations among
More informationHomework 5: Transformations in geometry
Math 21b: Linear Algebra Spring 2018 Homework 5: Transformations in geometry This homework is due on Wednesday, February 7, respectively on Thursday February 8, 2018. 1 a) Find the reflection matrix at
More informationComputer Vision Projective Geometry and Calibration. Pinhole cameras
Computer Vision Projective Geometry and Calibration Professor Hager http://www.cs.jhu.edu/~hager Jason Corso http://www.cs.jhu.edu/~jcorso. Pinhole cameras Abstract camera model - box with a small hole
More informationGeometric Hand-Eye Calibration for an Endoscopic Neurosurgery System
2008 IEEE International Conference on Robotics and Automation Pasadena, CA, USA, May 19-23, 2008 Geometric Hand-Eye Calibration for an Endoscopic Neurosurgery System Jorge Rivera-Rovelo Silena Herold-Garcia
More informationJorg s Graphics Lecture Notes Coordinate Spaces 1
Jorg s Graphics Lecture Notes Coordinate Spaces Coordinate Spaces Computer Graphics: Objects are rendered in the Euclidean Plane. However, the computational space is better viewed as one of Affine Space
More informationTransformations. Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico
Transformations Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico Angel: Interactive Computer Graphics 4E Addison-Wesley 25 1 Objectives
More informationIntroduction to Transformations. In Geometry
+ Introduction to Transformations In Geometry + What is a transformation? A transformation is a copy of a geometric figure, where the copy holds certain properties. Example: copy/paste a picture on your
More informationAutonomous Navigation for Flying Robots
Computer Vision Group Prof. Daniel Cremers Autonomous Navigation for Flying Robots Lecture 3.1: 3D Geometry Jürgen Sturm Technische Universität München Points in 3D 3D point Augmented vector Homogeneous
More informationLecture 6 Sections 4.3, 4.6, 4.7. Wed, Sep 9, 2009
Lecture 6 Sections 4.3, 4.6, 4.7 Hampden-Sydney College Wed, Sep 9, 2009 Outline 1 2 3 4 re are three mutually orthogonal axes: the x-axis, the y-axis, and the z-axis. In the standard viewing position,
More informationDIMENSIONAL SYNTHESIS OF SPATIAL RR ROBOTS
DIMENSIONAL SYNTHESIS OF SPATIAL RR ROBOTS ALBA PEREZ Robotics and Automation Laboratory University of California, Irvine Irvine, CA 9697 email: maperez@uci.edu AND J. MICHAEL MCCARTHY Department of Mechanical
More informationModule 1 Session 1 HS. Critical Areas for Traditional Geometry Page 1 of 6
Critical Areas for Traditional Geometry Page 1 of 6 There are six critical areas (units) for Traditional Geometry: Critical Area 1: Congruence, Proof, and Constructions In previous grades, students were
More informationGeometry. Chapter 5. Types of Curves and Surfaces
Chapter 5. Geometry Geometry refers to the physical items represented by the model (such as points, curves, and surfaces), independent of their spatial or topological relationships. The ACIS free form
More informationFor each question, indicate whether the statement is true or false by circling T or F, respectively.
True/False For each question, indicate whether the statement is true or false by circling T or F, respectively. 1. (T/F) Rasterization occurs before vertex transformation in the graphics pipeline. 2. (T/F)
More informationG 6,3 GEOMETRIC ALGEBRA
9 th International Conference on Clifford Algebras and their Applications in Mathematical Physics K. Gürlebeck (ed.) Weimar, Germany, 15 0 July 011 G 6,3 GEOMETRIC ALGEBRA Julio Zamora-Esquivel Intel,VPG,Guadalajara
More informationEECE 478. Learning Objectives. Learning Objectives. Linear Algebra and 3D Geometry. Linear algebra in 3D. Coordinate systems
EECE 478 Linear Algebra and 3D Geometry Learning Objectives Linear algebra in 3D Define scalars, points, vectors, lines, planes Manipulate to test geometric properties Coordinate systems Use homogeneous
More informationINTRODUCTION TO CAD/CAM SYSTEMS IM LECTURE HOURS PER WEEK PRESENTIAL
COURSE CODE INTENSITY MODALITY CHARACTERISTIC PRE-REQUISITE CO-REQUISITE CREDITS ACTUALIZATION DATE INTRODUCTION TO CAD/CAM SYSTEMS IM0242 3 LECTURE HOURS PER WEEK 48 HOURS CLASSROOM ON 16 WEEKS, 96 HOURS
More informationGaigen: a Geometric Algebra Implementation Generator
Gaigen: a Geometric Algebra Implementation Generator Daniël Fontijne, Tim Bouma, Leo Dorst University of Amsterdam July 28, 2002 Abstract This paper describes an approach to implementing geometric algebra.
More informationCS 445 / 645 Introduction to Computer Graphics. Lecture 21 Representing Rotations
CS 445 / 645 Introduction to Computer Graphics Lecture 21 Representing Rotations Parameterizing Rotations Straightforward in 2D A scalar, θ, represents rotation in plane More complicated in 3D Three scalars
More informationPLAY WITH GEOMETRY ANIMATED AND INTERACTIVE, FREE, INSTANT ACCESS, ONLINE GEOMETRIC ALGEBRA JAVA APPLETS WITH CINDERELLA
Fukui University International Congress 2002, International Symposium on Advanced Mechanical Engineering, Workshop on Mechanical Engineering between Fukui-Pukyong National Universities, 11-13 September
More informationComputer Vision I - Appearance-based Matching and Projective Geometry
Computer Vision I - Appearance-based Matching and Projective Geometry Carsten Rother 05/11/2015 Computer Vision I: Image Formation Process Roadmap for next four lectures Computer Vision I: Image Formation
More informationIndex. 3D reconstruction, point algorithm, point algorithm, point algorithm, point algorithm, 253
Index 3D reconstruction, 123 5+1-point algorithm, 274 5-point algorithm, 260 7-point algorithm, 255 8-point algorithm, 253 affine point, 43 affine transformation, 55 affine transformation group, 55 affine
More informationAspects of Geometry. Finite models of the projective plane and coordinates
Review Sheet There will be an exam on Thursday, February 14. The exam will cover topics up through material from projective geometry through Day 3 of the DIY Hyperbolic geometry packet. Below are some
More informationCMSC427: Computer Graphics Lecture Notes Last update: November 21, 2014
CMSC427: Computer Graphics Lecture Notes Last update: November 21, 2014 TA: Josh Bradley 1 Linear Algebra Review 1.1 Vector Multiplication Suppose we have a vector a = [ x a y a ] T z a. Then for some
More informationVISUALIZING QUATERNIONS
THE MORGAN KAUFMANN SERIES IN INTERACTIVE 3D TECHNOLOGY VISUALIZING QUATERNIONS ANDREW J. HANSON «WW m.-:ki -. " ;. *' AMSTERDAM BOSTON HEIDELBERG ^ M Ä V l LONDON NEW YORK OXFORD
More informationOrientation & Quaternions
Orientation & Quaternions Orientation Position and Orientation The position of an object can be represented as a translation of the object from the origin The orientation of an object can be represented
More informationOverview. By end of the week:
Overview By end of the week: - Know the basics of git - Make sure we can all compile and run a C++/ OpenGL program - Understand the OpenGL rendering pipeline - Understand how matrices are used for geometric
More informationDual Numbers: Simple Math, Easy C++ Coding, and Lots of Tricks. Gino van den Bergen
Dual Numbers: Simple Math, Easy C++ Coding, and Lots of Tricks Gino van den Bergen gino@dtecta.com Introduction Dual numbers extend the real numbers, similar to complex numbers. Complex numbers adjoin
More informationRobot Vision: Projective Geometry
Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals Understand homogeneous coordinates Understand points, line, plane parameters and interpret them geometrically
More informationarxiv:cs.cg/ v1 9 Oct 2003
Circle and sphere blending with conformal geometric algebra Chris Doran 1 Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK. arxiv:cs.cg/0310017 v1 9 Oct 2003 Abstract Blending
More informationLecture 4: Transformations and Matrices. CSE Computer Graphics (Fall 2010)
Lecture 4: Transformations and Matrices CSE 40166 Computer Graphics (Fall 2010) Overall Objective Define object in object frame Move object to world/scene frame Bring object into camera/eye frame Instancing!
More informationImage warping , , Computational Photography Fall 2017, Lecture 10
Image warping http://graphics.cs.cmu.edu/courses/15-463 15-463, 15-663, 15-862 Computational Photography Fall 2017, Lecture 10 Course announcements Second make-up lecture on Friday, October 6 th, noon-1:30
More informationChapter 2 - Basic Mathematics for 3D Computer Graphics
Chapter 2 - Basic Mathematics for 3D Computer Graphics Three-Dimensional Geometric Transformations Affine Transformations and Homogeneous Coordinates Combining Transformations Translation t + t Add a vector
More informationFreeform Curves on Spheres of Arbitrary Dimension
Freeform Curves on Spheres of Arbitrary Dimension Scott Schaefer and Ron Goldman Rice University 6100 Main St. Houston, TX 77005 sschaefe@rice.edu and rng@rice.edu Abstract Recursive evaluation procedures
More information3D Transformations World Window to Viewport Transformation Week 2, Lecture 4
CS 430/536 Computer Graphics I 3D Transformations World Window to Viewport Transformation Week 2, Lecture 4 David Breen, William Regli and Maxim Peysakhov Geometric and Intelligent Computing Laboratory
More informationHomework 5: Transformations in geometry
Math b: Linear Algebra Spring 08 Homework 5: Transformations in geometry This homework is due on Wednesday, February 7, respectively on Thursday February 8, 08. a) Find the reflection matrix at the line
More informationAdvances in Applied Clifford Algebras Boosted Surfaces: Synthesis of 3D Meshes using Point Pair Generators in the Conformal Model
Advances in Applied Clifford Algebras Boosted Surfaces: Synthesis of 3D Meshes using Point Pair Generators in the Conformal Model --Manuscript Draft-- Manuscript Number: Full Title: Boosted Surfaces: Synthesis
More informationPlanes Intersecting Cones: Static Hypertext Version
Page 1 of 12 Planes Intersecting Cones: Static Hypertext Version On this page, we develop some of the details of the plane-slicing-cone picture discussed in the introduction. The relationship between the
More informationCredit: UTokyo Online Education, GFK Series 2016 TSUBOI, Takashi
Credit: UTokyo Online Education, GFK Series 2016 TSUBOI, Takashi License: You may use this material in a page unit for educational purposes. Except where otherwise noted, this material is licensed under
More information