Lecture 1.3 Basic projective geometry. Thomas Opsahl
|
|
- Timothy Harrington
- 7 years ago
- Views:
Transcription
1 Lecture 1.3 Basic projective geometr Thomas Opsahl
2 Motivation For the pinhole camera, the correspondence between observed 3D points in the world and D points in the captured image is given b straight lines through a common point (pinhole) This correspondence can be described b a mathematical model known as the perspective camera model or the pinhole camera model This model can be used to describe the imaging geometr of man modern cameras, hence it plas a central part in computer vision
3 Motivation Before we can stud the perspective camera model in detail, we need to expand our mathematical toolbox We need to be able to mathematicall describe the position and orientation of the camera relative to the world coordinate frame lso we need to get familiar with some basic elements of projective geometr, since this will make it MUCH easier to describe and work with the perspective camera model 3
4 Introduction T B ξ B B We have seen that the pose of a coordinate frame B relative to a coordinate frame, denoted ξ B, can be represented as a homogeneous transformation T B in D r r t 11 1 Bx RB t B ξb TB = = r1 r tbx SE ( )
5 Introduction T B ξ B B We have seen that the pose of a coordinate frame B relative to a coordinate frame, denoted ξ B, can be represented as a homogeneous transformation T B in D and 3D r11 r1 r13 t Bx RB t B r1 r r3 tb ξb TB = = SE ( 3) 0 1 r31 r3 r33 t Bz
6 Introduction nd we have seen how the can transform points from one reference frame to another if we represent points in homogeneous coordinates x x p= = p 1 x x p= p = z z 1 The main reason for representing pose as homogeneous transformations, was the nice algebraic properties that came with the representation
7 Introduction Euclidean geometr ξ B R B, t B Complicated algebra Projective geometr ξ B T B = R B Simple algebra t B 0 1 p= ξ p p= R p+ t B B B B B ( R, t ) ( R R, R t t ) ( R, R t ) ξ = ξ ξ = + B B B C B C C C B C B C B ξ T T B C C C p= ξ p p = T p B B B B B B ξc = ξb ξc TC = TB TC 1 ξb TB In the following we will take a closer look at some basic elements of projective geometr that we will encounter when we stud the geometrical aspects of imaging Homogeneous coordinates, homogeneous transformations
8 The projective plane Points How to describe points in the plane?
9 The projective plane Points x x How to describe points in the plane? Euclidean plane R Choose a D coordinate frame Each point corresponds to a unique pair of Cartesian coordinates x = x, R x = x
10 The projective plane Points How to describe points in the plane? w x Euclidean plane R Choose a D coordinate frame Each point corresponds to a unique pair of Cartesian coordinates 1 x x x x = x, R x = x Projective plane Expand coordinate frame to 3D Each point corresponds to a triple of homogeneous coordinates x x = x,, w R x = w s.t. x,, w = λ x,, w λ R\ 0
11 The projective plane Points w x Observations 1. n point x = x, in the Euclidean plane has a corresponding homogeneous point x = x,, 1 in the projective plane x x. Homogeneous points of the form x,, 0 does not have counterparts in the Euclidean plane The correspond to points at infinit and are called ideal points 1 x
12 The projective plane Points w x Observations 3. When we work with geometrical problems in the plane, we can swap between the Euclidean representation and the projective representation 1 x x x x x x = x = 1 x x w = = x x w w
13 Example 1. These homogeneous vectors are different numerical representations of the same point in the plane x = = 4 = The homogeneous point 1,,3 represents the same point as 1 3, 3 R
14 The projective plane Lines How to describe lines in the plane?
15 The projective plane Lines How to describe lines in the plane? Euclidean plane R 3 parameters a, b, c R l = x, aa + bb + c = 0 l x
16 The projective plane Lines How to describe lines in the plane? w Euclidean plane R Triple a, b, c R 3 \ 0 l = x, aa + bb + c = 0 l x Projective plane Homogeneous vector l = a, b, c T l = x l T x = 0 1 l x
17 The projective plane Lines w Observations 1. Points and lines in the projective plane have the same representation, we sa that points and lines are dual objects in l x. ll lines in the Euclidean plane have a corresponding line in the projective plane 1 l x 3. The line l = 0,0,1 T in the projective plane does not have an Euclidean counterpart This line consists entirel of ideal points, and is know as the line at infinit
18 The projective plane Lines w l x Properties of lines in the projective plane 1. In the projective plane, all lines intersect, parallel lines intersect at infinit Two lines l 1 and l intersect in the point x = l 1 l. The line passing through points x 1 and x is given b l = x 1 x 1 l x
19 Example Determine the line passing through the two points, 4 and 5, 11 Homogeneous representation of points Homogeneous representation of line Equation of the line x 5 = 4 x = l = x1 x = [ x1] x = = 3 = x+ + = 0 = 3x Matrix representation of the cross product u v u v where 0 u3 u def [ u] = u3 0 u 1 u u1 0
20 Example point at infinit 1
21 The projective plane Transformations Some important transformations like the action of a pose ξ on points in the plane happen to be linear in the projective plane and non-linear in the Euclidean plane The most general invertible transformations of the projective plane are known as homographies or projective transformations / linear projective transformations / projectivities / collineations Definition homograph of is a linear transformation on homogeneous 3-vectors represented b a homogeneous, non-singular 3 3 matrix H x w = h 11 h 1 h 13 h 1 h h 3 h 31 h 3 h 33 So H is unique up to scale, i.e. H = λλ λ R\ 0 x w
22 The projective plane Transformations One characteristic of homographies is that the preserve lines, in fact an invertible transformation of that preserves lines is a homograph Examples Central projection from one plane to another is a homograph Hence if we take an image with a perspective camera of a flat surface from an angle, we can remove the perspective distortion with a homograph x xx Perspective distortion Without distortion Images from
23 The projective plane Transformations One characteristic of homographies is that the preserve lines, in fact an invertible transformation of that preserves lines is a homograph Examples Central projection from one plane to another is a homograph Two images, captured b perspective cameras, of the same planar scene is related b a homograph Image 1 Image
24 The projective plane Transformations One characteristic of homographies is that the preserve lines, in fact an invertible transformation of that preserves lines is a homograph Examples Central projection from one plane to another is a homograph Two images, captured b perspective cameras, of the same planar scene is related b a homograph One can show that the product of two homographies also must be a homograph We sa that the homographies constitute a group the projective linear group PP 3 Within this group there are several more specialized subgroups
25 Transformations of the projective plane Transformation of Matrix #DoF Preserves Visualization Translation I t 0 T 1 Euclidean R t 0 T 1 Similarit sr t 0 T 1 ffine a 11 a 1 a 13 a 1 a a Homograph /projective h 11 h 1 h 13 h 1 h h 3 h 31 h 3 h 33 Orientation + all below 3 Lengths + all below 4 ngles + all below 6 Parallelism, line at infinit + all below 8 Straight lines 6
26 The projective space The relationship between the Euclidean space R 3 and the projective space 3 is much like the relationship between R and In the projective space We represent points in homogeneous coordinates x λx λ x = = λ R\ 0 z λz w λw Points at infinit have last homogeneous coordinate equal to zero Planes and points are dual objects Π = x 3 π T x = 0 x x 3 3 x = x = z z 1 x x w 3 3 x = x = z w w z w The plane at infinit are made up of all points at infinit 7
27 Transformations of the projective space Transformation of 3 Matrix #DoF Preserves Translation I t 0 T 1 Euclidean R t 0 T 1 Similarit sr t 0 T 1 3 Orientation + all below 6 Volumes, volume ratios, lengths + all below 7 ngles + all below ffine Parallelism of planes, a1 a a3 a 4 The plane at infinit a a a a + all below Homograph /projective a a a a h11 h1 h13 h14 h1 h h3 h 4 h31 h3 h33 h34 h41 h4 h43 h44 15 Intersection and tangenc of surfaces in contact, straight lines
28 Summar The projective plane Homogeneous coordinates Line at infinit Points & lines are dual dditional reading Szeliski:.1.,.1.3, The projective space 3 Homogeneous coordinates Plane at infinit Points & planes are dual Linear transformations of and 3 Represented b homogeneous matrices Homographies ffine Similarities Euclidean Translations 9
29 Summar The projective plane Homogeneous coordinates Line at infinit Points & lines are dual The projective space 3 Homogeneous coordinates Plane at infinit Points & planes are dual Linear transformations of and 3 Represented b homogeneous matrices Homographies ffine Similarities Euclidean Translations dditional reading Szeliski:.1.,.1.3, MTLB WRNING When we work with linear transformations, we represent them as matrices that act on points b right multiplication T : n n x = M x Matlab seem to prefer left multiplication instead T : n n x = x T T T So if ou use built in matlab functions when ou work with transformations, be careful!!! R M L 30
Lecture 2: Homogeneous Coordinates, Lines and Conics
Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4
More informationProjective Geometry. Projective Geometry
Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,
More informationProjective Geometry: A Short Introduction. Lecture Notes Edmond Boyer
Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Contents 1 Introduction 2 11 Objective 2 12 Historical Background 3 13 Bibliography 4 2 Projective Spaces 5 21 Definitions 5 22 Properties
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 16-18 in TS128 Question 2.1 Given two nonparallel,
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 informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance
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 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 informationB4 Computational Geometry
3CG 2006 / B4 Computational Geometry David Murray david.murray@eng.o.ac.uk www.robots.o.ac.uk/ dwm/courses/3cg Michaelmas 2006 3CG 2006 2 / Overview Computational geometry is concerned with the derivation
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 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 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 informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
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 two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation
More informationConnecting Transformational Geometry and Transformations of Functions
Connecting Transformational Geometr and Transformations of Functions Introductor Statements and Assumptions Isometries are rigid transformations that preserve distance and angles and therefore shapes.
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationRotation Matrices and Homogeneous Transformations
Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n
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 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 informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More information2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
More informationLines and Planes in R 3
.3 Lines and Planes in R 3 P. Daniger Lines in R 3 We wish to represent lines in R 3. Note that a line may be described in two different ways: By specifying two points on the line. By specifying one point
More informationThe elements used in commercial codes can be classified in two basic categories:
CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationx y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are
Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented
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 informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
More information1.5 SOLUTION SETS OF LINEAR SYSTEMS
1-2 CHAPTER 1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS Many of the concepts and computations in linear algebra involve sets of vectors which are visualized geometrically as
More informationRelating Vanishing Points to Catadioptric Camera Calibration
Relating Vanishing Points to Catadioptric Camera Calibration Wenting Duan* a, Hui Zhang b, Nigel M. Allinson a a Laboratory of Vision Engineering, University of Lincoln, Brayford Pool, Lincoln, U.K. LN6
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More information1.2 GRAPHS OF EQUATIONS. Copyright Cengage Learning. All rights reserved.
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationGeometry Review Flash Cards
point is like a star in the night sky. However, unlike stars, geometric points have no size. Think of them as being so small that they take up zero amount of space. point may be represented by a dot on
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics 3D views and projection Adapted from notes by Yong Cao 1 Overview of 3D rendering Modeling: *Define object in local coordinates *Place object in world coordinates (modeling transformation)
More informationComputer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationLecture 5: Singular Value Decomposition SVD (1)
EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
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 informationA 0.9 0.9. Figure A: Maximum circle of compatibility for position A, related to B and C
MEASURING IN WEIGHTED ENVIRONMENTS (Moving from Metric to Order Topology) Claudio Garuti Fulcrum Ingenieria Ltda. claudiogaruti@fulcrum.cl Abstract: This article addresses the problem of measuring closeness
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
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 informationLines & Planes. Packages: linalg, plots. Commands: evalm, spacecurve, plot3d, display, solve, implicitplot, dotprod, seq, implicitplot3d.
Lines & Planes Introduction and Goals: This lab is simply to give you some practice with plotting straight lines and planes and how to do some basic problem solving with them. So the exercises will be
More informationWeek 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test
Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan
More informationQuestions. Strategies August/September Number Theory. What is meant by a number being evenly divisible by another number?
Content Skills Essential August/September Number Theory Identify factors List multiples of whole numbers Classify prime and composite numbers Analyze the rules of divisibility What is meant by a number
More informationPROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin
PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More informationLINEAR EQUATIONS IN TWO VARIABLES
66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that
More informationLinear Equations ! 25 30 35$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development
MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!
More informationAffine Transformations
A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates
More informationRealtime 3D Computer Graphics Virtual Reality
Realtime 3D Computer Graphics Virtual Realit Viewing and projection Classical and General Viewing Transformation Pipeline CPU Pol. DL Pixel Per Vertex Texture Raster Frag FB object ee clip normalized device
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 informationTeacher Page. 1. Reflect a figure with vertices across the x-axis. Find the coordinates of the new image.
Teacher Page Geometr / Da # 10 oordinate Geometr (5 min.) 9-.G.3.1 9-.G.3.2 9-.G.3.3 9-.G.3. Use rigid motions (compositions of reflections, translations and rotations) to determine whether two geometric
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More information1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped
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 information3D Model based Object Class Detection in An Arbitrary View
3D Model based Object Class Detection in An Arbitrary View Pingkun Yan, Saad M. Khan, Mubarak Shah School of Electrical Engineering and Computer Science University of Central Florida http://www.eecs.ucf.edu/
More informationThe Geometry of Perspective Projection
The Geometry o Perspective Projection Pinhole camera and perspective projection - This is the simplest imaging device which, however, captures accurately the geometry o perspective projection. -Rays o
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.
More informationElasticity Theory Basics
G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
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 informationINSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users
INSTRUCTOR WORKBOOK for MATLAB /Simulink Users Developed by: Amir Haddadi, Ph.D., Quanser Peter Martin, M.A.SC., Quanser Quanser educational solutions are powered by: CAPTIVATE. MOTIVATE. GRADUATE. PREFACE
More informationNotes on the representational possibilities of projective quadrics in four dimensions
bacso 2006/6/22 18:13 page 167 #1 4/1 (2006), 167 177 tmcs@inf.unideb.hu http://tmcs.math.klte.hu Notes on the representational possibilities of projective quadrics in four dimensions Sándor Bácsó and
More informationVector Spaces. Chapter 2. 2.1 R 2 through R n
Chapter 2 Vector Spaces One of my favorite dictionaries (the one from Oxford) defines a vector as A quantity having direction as well as magnitude, denoted by a line drawn from its original to its final
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 informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a two-dimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More information4BA6 - Topic 4 Dr. Steven Collins. Chap. 5 3D Viewing and Projections
4BA6 - Topic 4 Dr. Steven Collins Chap. 5 3D Viewing and Projections References Computer graphics: principles & practice, Fole, vandam, Feiner, Hughes, S-LEN 5.644 M23*;-6 (has a good appendix on linear
More information2-View Geometry. Mark Fiala Ryerson University Mark.fiala@ryerson.ca
CRV 2010 Tutorial Day 2-View Geometry Mark Fiala Ryerson University Mark.fiala@ryerson.ca 3-Vectors for image points and lines Mark Fiala 2010 2D Homogeneous Points Add 3 rd number to a 2D point on image
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationChapter 9. Systems of Linear Equations
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
More informationGeometric Optics Converging Lenses and Mirrors Physics Lab IV
Objective Geometric Optics Converging Lenses and Mirrors Physics Lab IV In this set of lab exercises, the basic properties geometric optics concerning converging lenses and mirrors will be explored. The
More informationLecture 1: Systems of Linear Equations
MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables
More informationVectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.
Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonal-diagonal-orthogonal type matrix decompositions Every
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationUseful Mathematical Symbols
32 Useful Mathematical Symbols Symbol What it is How it is read How it is used Sample expression + * ddition sign OR Multiplication sign ND plus or times and x Multiplication sign times Sum of a few disjunction
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 information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional 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 informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the
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 three-space, we write a vector in terms
More informationReview of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
More informationBasic Problem: Map a 3D object to a 2D display surface. Analogy - Taking a snapshot with a camera
3D Viewing Basic Problem: Map a 3D object to a 2D display surface Analogy - Taking a snapshot with a camera Synthetic camera virtual camera we can move to any location & orient in any way then create a
More informationSection 1.7 22 Continued
Section 1.5 23 A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationEpipolar Geometry. Readings: See Sections 10.1 and 15.6 of Forsyth and Ponce. Right Image. Left Image. e(p ) Epipolar Lines. e(q ) q R.
Epipolar Geometry We consider two perspective images of a scene as taken from a stereo pair of cameras (or equivalently, assume the scene is rigid and imaged with a single camera from two different locations).
More informationYear 9 set 1 Mathematics notes, to accompany the 9H book.
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty
More informationMechanics lecture 7 Moment of a force, torque, equilibrium of a body
G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and
More informationSome Tools for Teaching Mathematical Literacy
Some Tools for Teaching Mathematical Literac Julie Learned, Universit of Michigan Januar 200. Reading Mathematical Word Problems 2. Fraer Model of Concept Development 3. Building Mathematical Vocabular
More information