geometric transforms


 Justina Copeland
 1 years ago
 Views:
Transcription
1 geometric transforms 1
2 linear algebra review 2
3 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3
4 matrix operations addition scalar multiplication T = M + N t ij + m ij n ij [ ] = [ ] T = am [ t ij ] = [ am ij ] 4
5 matrix operations matrixmatrix multiplication rowcolumn multiplication not commutative associative =[ ] T = MN = [ t ij ] m ik n kj k t 11 t 12 t 21 t 22 = m 11 m 12 n 11 m 21 m 22 n 22 n 21 n 12 5
6 matrix operations matrixvector multiplication rowcolumn multiplication u 1 u 2 = u = Mv m 11 m 12 m 21 m 22 v 2 v 1 6
7 matrix operations transpose: flip along diagonal T = M T [ t ij ] = [ t ji ] inverse (not computed explicitly in this course) T = M 1 MT = M M 1 = M 1 M = I 7
8 special matrices identity: invariant for multiplication ={ 1 I = [ i ij ] =[ I 1 ] 1 i = j i j : M : M = MI = IM 8
9 special matrices zero: invariant for addition O = [ i ij ] = =[ I ] : M : M = M + O = O + M 9
10 matrix operation properties linearity of multiplication and addition a(a + B) = aa + ab M(aA + bb) = ama + bmb associativity of multiplication A(BC) = (AB)C 1
11 matrix operation properties transpose and inverse of multiplication (AB) T = B T A T (AB) 1 = B 1 A 1 11
12 2d transformation 12
13 geometric transformations functions that maps points to points p Ü p = X(p) different transformations have restrictions on the form of X 13
14 translation T t (p) = p + t Tt 1 (p) = T t (p) = p t 14
15 linear transformations fundamental property: can be represented in matrix form: properties: maps origin to origin maps lines to lines parellel lines remain parallel length ratios are preserved closed under composition X(ap + bq) = ax(p) + bx(q) X(p) = Mp 15
16 uniform scale S s =[ ][ ] p s p x = [ sp x ] s p y sp y Ss 1 = S 1/s 16
17 nonuniform scale =[ ][ p ] p x = [ s x p x ] s y p y s y p y S s s x Ss 1 = S 1/s 17
18 rotation =[ ][ ] p sin? p x = [ p x p y ] sin? cos? p x sin? + p y cos? R? cos? p y cos? sin? R 1? = R? 18
19 shear =[ 1 s ][ ] S p = [ ] h x p x p x + s x p y s s y 1 p y s y p x + p y 19
20 reflection R p =[ ][ ] = [ ] R p = ) l x 1 1 p x p y p x p y =[ ][ ] R p = [ ] l 1 p x p x o 1 p y p y l y 2
21 affine transforms combine translation with linear transformation rigid body transformation are a subset of this X M,t (p) = Mp + t properties does not map origin to origin maps lines to lines parallel lines remain parallel length ratios are preserved closed under composition 21
22 transforming points and vectors points and vectors are different entities vectors: encode direction and length (difference of points) points: encode position (origin plus a vector) points: transform as reported above X(p) = Mp + t vectors: transform like points, but no translation is applied X(v) = Mv directions (normalized vectors): normalize after transform X( d^ ) = M d^ / M d^ 22
23 transforming points and vectors proof that vectors transform as such v X(p) X(v) = p q = Mp + t = X(p) X(q) = = (Mp + t) (Mq + t) = = M(p q) = = Mv 23
24 homogeneous coordinates represent points/vectors with an additional coordinate set it to 1 for points (or multiply all per arbitrary ) p x p = p y 1 wp x wp y w w w set it to for vectors v = v x v y 24
25 homogeneous coordinates translation: represent as 3x3 matrix T t t x p x 1 p = 1 t y p y = 1 1 p x p y t x t y shorthand notation for points and vectors T t =[ ][ ] p I t p = [ p + t] =[ ][ ] v I t v = [ v] 1 T t 25
26 homogeneous coordinates linear transform: 3x3 matrix by adding one row,column m 11 m 12 p x Mp = m = 21 m 22 p 1 y shorthand notation for points and vectors 1 m 11 p x m 21 p x =[ ][ ] Mp M p = [ Mp] =[ Mv M ][ v ] = [ Mv ] 1 + m 12 p y + m 22 p y 1 26
27 affine transformations combine linear and translation in one matrix =[ Xp M =[ Xv M t 1 ][ p] 1 = Mp + t t][ v] 1 = Mv 27
28 combining transforms apply one transformation after the another express by function composition p = X 2 ( X 1 (p)) = ( X 2 Q X 1 (p)) for affine transformations, compute by matrix multiplication ( X 2 Q X 1 (p)) = X 2 ( X 1 (p)) = M 2 ( M 1 )p = ( M 2 M 1 )p 28
29 combining transforms translation [ I t 2 ][ I t 1 ] = [ I t 1 + t 2 ] linear transformations [ ][ ] = [ M 2 M 1 ] M 2 M 1 29
30 combining transforms affine transformations [ t 2 ][ t 1 ] = [ M 2 M 1 M 2 t 1 + t 2 ] M 2 M 1 3
31 composition is not commutative original rotation translation original transation rotation 31
32 complex transformations represent as combination of simpler ones intuitive geometric interpretation rotation around arbitrary axis at a of angle? translate axis center to origin rotate (about origin) translate back R a,? = T a R? T a 32
33 complex transformations 33
34 complex transformations 34
35 transforms and coordinate systems change of coordinate system can be written as affine matrix for f f, we have p = Ü that in matrix form becomes f O p xf x p xf y p xf z =[ ][ ] p M t p = [ f ][ ] x f y f z f o p
36 transforms and coordinate systems for f Ü f, we invert the previous equation since p =[ M 1 t][ p] 1 1 M 1 M is orthonormal, M 1 = p = x T y T z T M T x t y t z t 1 p x p y p z 1 36
37 3d transformations 37
38 3d transformations adopt homogeneous formulation in 3d point have 4 coordinates use 4x4 matrices for transformations most concepts generalize very easily rotation much more complex 38
39 translation T t = t x t y t z 1 39
40 scale S s = s x s y s z 1 4
41 rotation around z R z? = cos? sin? sin? cos?
42 rotation around y R y? = cos? sin? 1 sin? cos? 1 42
43 rotation around x R x? = 1 cos? sin? sin? cos? 1 43
44 rotation around arbitrary axis in 2D, rotation are around a point: R a,? change coordinate frame (translation) rotation around the origin change coordinate frame back simple geometric construction in 3D, rotation are around an axis: R a,? change coordinate frame (align with axis ) rotation around change coordinate frame back complex geometric construction z z = T a R? T a = F 1 a R? F a a 44
45 representing rotations Euler angles: 3 rotations around major axis remember to choose order simple but has quirks when combining rotations will use this for simplicity axis and angle combinations of rotations can be represented this way with more formalism become elegant and consistent quaternions 45
46 transforming normals points and vectors works tangents, i.e. differences of points, work too normals works differently defined as orthogonal to the transformed surface i.e. orthogonal to all tangents 46
47 transforming normals t n = t T n = (Mt) T (Xn) = t t T M T Xn = t T M T ( M T ) 1 n = by definition: after tranform: for all we have: which gives: normals are transformed by the inverse transpose 47
48 transformation hierarchies 48
49 transformation hierarchies often need to transform an object wrt another e.g. the computer on the table when the table moves, the computer moves naturally build a hierarchy of transformation to transform the table, apply its transform to transform the computer, apply the table and the computer transform 49
50 transformation hierarchies represented as a tree data structure transformation nodes object nodes / leaves walk the tree when drawing very convenient representation for objects all objects can be defined in their simplest form e.g. every sphere can be represented by a transformation applied to the unit sphere 5
51 transformation hierarchies 51
52 implementing transform hierarchies transformation function for each node get the parent matrix multiply the parent and current matrices pass the combined matrix when calling children stack of transforms push/pop when walking down/up used by graphics libraries (OpenGL) more flexible generalized mechanism for all attributes 52
53 raytracing and transformations transform the object simple for triangles, since they transforms to triangles but most objects require complex intersection tests spheres do not transforms to spheres, but ellipsoids transform the ray much more elegant works on any surface allow for much simpler intersection tests only worry about unit sphere, all others are transformed 53
54 raytracing and transformations transforming rays transform origin/direction as point/vector note that direction is not normalized now i.e. ray parameter is not the distance intersect a transformed object transform the ray by matrix inverse intersect surface transform hit point and normal by matrix works like changes in coodinate system 54
2D Geometric Transformations. COMP 770 Fall 2011
2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrixvector multiplication Matrixmatrix multiplication
More informationAlgebra and Linear Algebra
Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...
More informationMotivation. Goals. General Idea. Outline. (Nonuniform) Scale. Foundations of Computer Graphics
Foundations of Computer Graphics Basic 2D Transforms Motivation Many different coordinate systems in graphics World, model, body, arms, To relate them, we must transform between them Also, for modeling
More informationLinear transformations Affine transformations Transformations in 3D. Graphics 2011/2012, 4th quarter. Lecture 5: linear and affine transformations
Lecture 5 Linear and affine transformations Vector transformation: basic idea Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Multiplication of an n n matrix
More information3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)
3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationMATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationMatrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,
LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationLecture 3: Coordinate Systems and Transformations
Lecture 3: Coordinate Systems and Transformations Topics: 1. Coordinate systems and frames 2. Change of frames 3. Affine transformations 4. Rotation, translation, scaling, and shear 5. Rotation about an
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationCoordinates and Transformations
Coordinates and Transformations MIT ECCS 6.837 Wojciech Matusik many slides follow Steven Gortler s book 1 Hierarchical modeling Many coordinate systems: Camera Static scene car driver arm hand... Image
More information1 Introduction. 2 The G 4 Algebra. 2.1 Definition of G 4 Algebra
1 Introduction Traditionally translations and rotations have been analysed/computed with either matrices, quaternions, biquaternions, dual quaternions or double quaternions depending of ones preference
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationRotation Matrices and Homogeneous Transformations
Rotation Matrices and Homogeneous Transformations A coordinate frame in an ndimensional space is defined by n mutually orthogonal unit vectors. In particular, for a twodimensional (2D) space, i.e., n
More informationLecture 2 Matrix Operations
Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrixvector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or
More informationChapter 4: Binary Operations and Relations
c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,
More information1 Introduction to Matrices
1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns
More informationContent. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11
Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2D Welcome everybody. We continue the discussion on 2D
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationWe have just introduced a first kind of specifying change of orientation. Let s call it AxisAngle.
2.1.5 Rotations in 3D around the origin; Axis of rotation In threedimensional space, it will not be sufficient just to indicate a center of rotation, as we did for plane kinematics. Any change of orientation
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationSection 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj
Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationVector algebra Christian Miller CS Fall 2011
Vector algebra Christian Miller CS 354  Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority
More informationBasics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20
Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationA linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form
Section 1.3 Matrix Products A linear combination is a sum of scalars times quantities. Such expressions arise quite frequently and have the form (scalar #1)(quantity #1) + (scalar #2)(quantity #2) +...
More information2D Geometrical Transformations. Foley & Van Dam, Chapter 5
2D Geometrical Transformations Fole & Van Dam, Chapter 5 2D Geometrical Transformations Translation Scaling Rotation Shear Matri notation Compositions Homogeneous coordinates 2D Geometrical Transformations
More information2D Geometric Transformations
2D Geometric Transformations (Chapter 5 in FVD) 2D Geometric Transformations Question: How do we represent a geometric object in the plane? Answer: For now, assume that objects consist of points and lines.
More informationAffine Transformations
Affine Transformations Reading Foley et al., Chapter 5.6 and Chapter 6 Supplemental David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, Second edition 2D geometry Pipeline 3D
More informationMatrices, transposes, and inverses
Matrices, transposes, and inverses Math 40, Introduction to Linear Algebra Wednesday, February, 202 Matrixvector multiplication: two views st perspective: A x is linear combination of columns of A 2 4
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationMetrics on SO(3) and Inverse Kinematics
Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationGrassmann Algebra in Game Development. Eric Lengyel, PhD Terathon Software
Grassmann Algebra in Game Development Eric Lengyel, PhD Terathon Software Math used in 3D programming Dot / cross products, scalar triple product Planes as 4D vectors Homogeneous coordinates Plücker coordinates
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 3 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 3 1 / 12 Vector product and volumes Theorem. For three 3D vectors u, v, and w,
More informationGiven a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes:
3 3.1 2D Given a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes: 1. Change coordinate frames (world, window, viewport, device, etc). 2. Compose objects
More informationLinear Algebra: Matrices
B Linear Algebra: Matrices B 1 Appendix B: LINEAR ALGEBRA: MATRICES TABLE OF CONTENTS Page B.1. Matrices B 3 B.1.1. Concept................... B 3 B.1.2. Real and Complex Matrices............ B 3 B.1.3.
More informationCalculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants
Calculus and linear algebra for biomedical engineering Week 4: Inverse matrices and determinants Hartmut Führ fuehr@matha.rwthaachen.de Lehrstuhl A für Mathematik, RWTH Aachen October 30, 2008 Overview
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationAffine Transformations. University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell
Affine Transformations University of Texas at Austin CS384G  Computer Graphics Fall 2010 Don Fussell Logistics Required reading: Watt, Section 1.1. Further reading: Foley, et al, Chapter 5.15.5. David
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More information3D geometry basics (for robotics) lecture notes
lecture notes Marc Toussaint Machine Learning & Robotics lab, FU Berlin Arnimallee 7, 495 Berlin, Germany October 30, 20 This document introduces to some basic geometry, focussing on 3D transformations,
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationThe calibration problem was discussed in details during lecture 3.
1 2 The calibration problem was discussed in details during lecture 3. 3 Once the camera is calibrated (intrinsics are known) and the transformation from the world reference system to the camera reference
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationModern Geometry Homework.
Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z
More information3 Orthogonal Vectors and Matrices
3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first
More informationDefinition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that
0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationRotation in the Space
Rotation in the Space (Com S 477/577 Notes) YanBin Jia Sep 6, 2016 The position of a point after some rotation about the origin can simply be obtained by multiplying its coordinates with a matrix One
More informationAdvanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz
Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics 2D and 3D Transformations Doug Bowman Adapted from notes by Yong Cao Virginia Tech 1 Transformations What are they? changing something to something else via rules mathematics:
More informationThe Solution of Linear Simultaneous Equations
Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve
More informationQuaternion Math. Application Note. Abstract
Quaternion Math Application Note Abstract This application note provides an overview of the quaternion attitude representation used by VectorNav products and how to convert it into other common attitude
More information5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES
5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES Definition 5.3. Orthogonal transformations and orthogonal matrices A linear transformation T from R n to R n is called orthogonal if it preserves
More informationMatrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: b k
MATRIX ALGEBRA FOR STATISTICS: PART 1 Matrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: y = x 1 b 1 + x 2 + x 3
More informationTWODIMENSIONAL TRANSFORMATION
CHAPTER 2 TWODIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 08 4 01 1 0 11
More informationGeometric Transformations
CS3 INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and 3D CS3 INTRODUCTION TO COMPUTER GRAPHICS Grading Plan to be out Wednesdas one week after the due date CS3 INTRODUCTION TO COMPUTER
More information1 Spherical Kinematics
ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationGeometric Modelling Summer 2016
Geometric Modelling Summer 2016 Prof. Dr. Hans Hagen http://gfx.unikl.de/~gm Prof. Dr. Hans Hagen Geometric Modelling Summer 2016 1 Ane Spaces, Elliptic Prof. Dr. Hans Hagen Geometric Modelling Summer
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More information2D Geometrical Transformations
2D Geometrical Transformations Translation Moves points to new locations by adding translation amounts to the coordinates of the points T P(,y) P (,y ) = + d, =y + dy or = y P =P + T + d dy Totranslate
More information1.5 Elementary Matrices and a Method for Finding the Inverse
.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:
More information0.1 Linear Transformations
.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called
More informationGeometric Transformation CS 211A
Geometric Transformation CS 211A What is transformation? Moving points (x,y) moves to (x+t, y+t) Can be in any dimension 2D Image warps 3D 3D Graphics and Vision Can also be considered as a movement to
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More informationCS3220 Lecture Notes: QR factorization and orthogonal transformations
CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss
More informationMatrix Calculations: Kernels & Images, Matrix Multiplication
Matrix Calculations: Kernels & Images, Matrix Multiplication A. Kissinger (and H. Geuvers) Institute for Computing and Information Sciences Intelligent Systems Version: spring 2016 A. Kissinger Version:
More informationHelpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:
Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants
More informationTopic 1: Matrices and Systems of Linear Equations.
Topic 1: Matrices and Systems of Linear Equations Let us start with a review of some linear algebra concepts we have already learned, such as matrices, determinants, etc Also, we shall review the method
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationQuaternions and Rotations
CSCI 520 Computer Animation and Simulation Quaternions and Rotations Jernej Barbic University of Southern California 1 Rotations Very important in computer animation and robotics Joint angles, rigid body
More information(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationGeometric Camera Parameters
Geometric Camera Parameters What assumptions have we made so far? All equations we have derived for far are written in the camera reference frames. These equations are valid only when: () all distances
More informationMatrixvector multiplication in terms of dotproducts
Matrixvector multiplication in terms of dotproducts Let M be an R C matrix. DotProduct Definition of matrixvector multiplication: M u is the Rvector v such that v[r] is the dotproduct of row r of
More information6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:
6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an innerproduct space. This is the usual n dimensional
More information1 Quiz on Linear Equations with Answers. This is Theorem II.3.1. There are two statements of this theorem in the text.
1 Quiz on Linear Equations with Answers (a) State the defining equations for linear transformations. (i) L(u + v) = L(u) + L(v), vectors u and v. (ii) L(Av) = AL(v), vectors v and numbers A. or combine
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More information4. Matrix inverses. left and right inverse. linear independence. nonsingular matrices. matrices with linearly independent columns
L. Vandenberghe EE133A (Spring 2016) 4. Matrix inverses left and right inverse linear independence nonsingular matrices matrices with linearly independent columns matrices with linearly independent rows
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationSolving 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
More informationSolutions to Assignment 4
Solutions to Assignment 4 Math 217, Fall 2002 2.7.10 Consider the following geometric 2D transformations: D, a dilation (in which xcoordinates and y coordinates are scaled by the same factor); R, a rotation;
More information521493S Computer Graphics. Exercise 2 & course schedule change
521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 1618 in TS128 Question 2.1 Given two nonparallel,
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More information