CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer


 Eugenia Peters
 3 years ago
 Views:
Transcription
1 CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016
2 Announcements Monday October 3: Discussion Assignment 1 Friday October 7: Assignment 1 due 2
3 Today s Topics Vectors and matrices Affine transformations Homogeneous coordinates 3
4 Vectors Give direction and length in 3D Vectors can describe Difference between two 3D points Speed of an object Surface normals (directions perpendicular to surfaces) Normal vector Surface normals Surface 4
5 Vector arithmetic using coordinates a = ax ay az b = bx by bz a + b = ax + bx ay + by az + bz a b = ax bx ay by az bz ax a= ay az sax sa= say saz where s is a scalar 5
6 Vector Magnitude The magnitude (length) of a vector is: v 2 = v x 2 + v y 2 + v z 2 v = v x 2 + v y 2 + v z 2 A vector with length of 1.0 is called unit vector We can also normalize a vector to make it a unit vector Unit vectors are often used as surface normals v v 6
7 Dot Product a b = a i b i a b = a x b x + a y b y + a z b z a b = a b cosθ 7
8 Angle Between Two Vectors a b = a b cosθ cosθ = a b a b b θ = cos 1 a b a b a 8
9 Cross Product a b is a vector perpendicular to both a and b, in the direction defined by the right hand rule a b = a b sinθ a b = area of parallelogram ab a b = 0 if aand b are parallel (or one or both degenerate) 9
10 Cross Product = = 10
11 Cross Product Calculation = = = 11
12 Matrices Rectangular array of numbers Square matrix if m = n In graphics almost always: m = n = 3; m = n = 4 12
13 Matrix Addition 13
14 Multiplication With Scalar 14
15 Matrix Multiplication 15
16 MatrixVector Multiplication 16
17 Identity Matrix 17
18 Matrix Inverse If a square matrix M is nonsingular, there exists a unique inverse M 1 such that 18
19 Today s Topics Vectors and matrices Affine transformations Homogeneous coordinates 19
20 Affine Transformations Most important for graphics: rotation, translation, scaling Wolfram MathWorld: An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). Implemented using matrix multiplications 20
21 Uniform Scale Uniform scale matrix in 2D Analogous in 3D:
22 NonUniform Scale Nonuniform scaling matrix in 2D Analogous in 3D: 22
23 Rotation in 2D Convention: positive angle rotates counterclockwise Rotation matrix 23
24 Rotation in 3D Rotation around coordinate axes 24
25 Rotation in 3D Concatenation of rotations around x, y, z axes are called Euler angles Result depends on matrix order! 25
26 Rotation about an Arbitrary Axis Complicated! Rotate point [x,y,z] about axis [u,v,w] by angle θ: 26
27 How to rotate around a Pivot Point? Rotation around origin: p = R p Rotation around pivot point: p =? 27
28 Rotating point p around a pivot point 1. Translation T 2. Rotation R 3. Translation T 1 p = T 1 R T p 28
29 Concatenating transformations Given a sequence of transformations M 3 M 2 M 1 Note: associativity applies 29
30 Today s Topics Vectors and matrices Affine transformations Homogeneous coordinates 30
31 Translation Translation in 2D t x t y Translation matrix T=? = = = =???? 31
32 Translation Translation in 2D: 3x3 matrix Analogous in 3D: 4x4 matrix 32
33 Homogeneous Coordinates Basic: a trick to unify/simplify computations. Deeper: projective geometry Interesting mathematical properties Good to know, but less immediately practical We will use some aspect of this when we do perspective projection 33
34 Homogeneous Coordinates Add an extra component. 1 for a point, 0 for a vector: p = Combine M and d into single 4x4 matrix: p x p y p z 1 And see what happens when we multiply r v = v x v y v z 0 m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z
35 Homogeneous Point Transform Transform a point: p x p y p z 1 = m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z p x p y p z 1 = m xx p x + m xy p y + m xz p z + d x m yx p x + m yy p y + m yz p z + d y m zx p x + m zy p y + m zz p z + d z Top three rows are the affine transform! Bottom row stays 1 M p x p y p z + r d 35
36 Homogeneous Vector Transform Transform a vector: v x v y v z 0 = m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z v x v y v z 0 = m xx v x + m xy v y + m xz v z + 0 m yx v x + m yy v y + m yz v z + 0 m zx v x + m zy v y + m zz v z M v x v y v z Top three rows are the linear transform Displacement d is properly ignored Bottom row stays 0 36
37 Homogeneous Arithmetic Legal operations always end in 0 or 1! M vector+vector: 0 + M M 0 0 M vectorvector: 0 M M 0 0 scalar*vector: s M M 0 0 M point+vector: 1 + M M 0 1 M pointpoint: 1 M M 1 0 M point+point: 1 + M M 1 2 scalar*point: s M M 1 s weighted average affine combination of points: 1 M M 3 M
38 Homogeneous Transforms Rotation, Scale, and Translation of points and vectors unified in a single matrix transformation: p = M p Matrix has the form: Last row always 0,0,0,1 m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z Transforms can be composed by matrix multiplication Same caveat: order of operations is important Same note: transforms operate righttoleft 38
CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer
CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Spring Quarter 2016 Announcements Project 1 due next Friday at
More informationCOMP 175 COMPUTER GRAPHICS. Lecture 04: Transform 1. COMP 175: Computer Graphics February 9, Erik Anderson 04 Transform 1
Lecture 04: Transform COMP 75: Computer Graphics February 9, 206 /59 Admin Sign up via email/piazza for your inperson grading Anderson@cs.tufts.edu 2/59 Geometric Transform Apply transforms to a hierarchy
More information11.1 ThreeDimensional Coordinate System
11.1 ThreeDimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and zaxes is shown below. The three axes divide the region into
More informationMathematics for 3D Graphics
math 1 Topics Mathematics for 3D Graphics math 1 Points, Vectors, Vertices, Coordinates Dot Products, Cross Products Lines, Planes, Intercepts References Many texts cover the linear algebra used for 3D
More informationA DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any
Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»
More informationGG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS
GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why
More information1 Overview. CS348a: Computer Graphics Handout #8 Geometric Modeling Original Handout #8 Stanford University Thursday, 15 October 1992
CS348a: Computer Graphics Handout #8 Geometric Modeling Original Handout #8 Stanford University Thursday, 15 October 1992 Original Lecture #1: 1 October 1992 Topics: Affine vs. Projective Geometries Scribe:
More informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More informationThreeDimensional Coordinate Systems. ThreeDimensional Coordinate Systems. ThreeDimensional Coordinate Systems. ThreeDimensional Coordinate Systems
To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the xcoordinate and b is the ycoordinate.
More informationLECTURE 5, FRIDAY
LECTURE 5, FRIDAY 20.02.04 FRANZ LEMMERMEYER Before we start with the arithmetic of elliptic curves, let us talk a little bit about multiplicities, tangents, and singular points. 1. Tangents How do we
More information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationCS 4300 Computer Graphics. Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011
CS 4300 Computer Graphics Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011 October 8, 2011 College of Computer and Information Science, Northeastern Universit 1 Toda s Topics Linear Algebra Review
More informationIf the pull is downward (Fig. 1), we want C to point into the page. If the pull is upward (Fig. 2), we want C to point out of the page.
11.5 Cross Product Contemporary Calculus 1 11.5 CROSS PRODUCT This section is the final one about the arithmetic of vectors, and it introduces a second type of vector vector multiplication called the cross
More informationHomogeneous Transformations
Purpose: Homogeneous Transformations The purpose of this chapter is to introduce you to the Homogeneous Transformation. This simple 4 x 4 transformation is used in the geometry engines of CAD systems and
More informationMatrixVector Products and the Matrix Equation Ax = b
MatrixVector Products and the Matrix Equation Ax = b A. Havens Department of Mathematics University of Massachusetts, Amherst January 31, 2018 Outline 1 Matrices Acting on Vectors Linear Combinations
More informationProperties of surfaces II: Second moment of area
Properties of surfaces II: Second moment of area Just as we have discussing first moment of an area and its relation with problems in mechanics, we will now describe second moment and product of area of
More informationLecture Notes for MATH6106. March 25, 2010
Lecture Notes for MATH66 March 25, 2 Contents Vectors 4. Points in Space.......................... 4.2 Distance between Points..................... 4.3 Scalars and Vectors........................ 5.4 Vectors
More informationOmm AlQura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 31 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
More informationExercise 1: Inertia moment of a simple pendulum
Exercise : Inertia moment of a simple pendulum A simple pendulum is represented in Figure. Three reference frames are introduced: R is the fixed/inertial RF, with origin in the rotation center and i along
More informationLecture 8: Coordinate Frames. CITS3003 Graphics & Animation
Lecture 8: Coordinate Frames CITS3003 Graphics & Animation E. Angel and D. Shreiner: Interactive Computer Graphics 6E AddisonWesley 2012 Objectives Learn how to define and change coordinate frames Introduce
More informationClassical Mechanics. Luis Anchordoqui
1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body
More informationAnnouncements Wednesday, September 27
Announcements Wednesday, September 27 The midterm will be returned in recitation on Friday. You can pick it up from me in office hours before then. Keep tabs on your grades on Canvas. WeBWorK 1.7 is due
More informationMAE 323: Lecture 1. Review
This review is divided into two parts. The first part is a minireview of statics and solid mechanics. The second part is a review of matrix/vector fundamentals. The first part is given as an refresher
More informationProblem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1
Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2ad,g,h,j 2.6, 2.9; Chapter 3: 1ad,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate
More informationVectors for Physics. AP Physics C
Vectors for Physics AP Physics C A Vector is a quantity that has a magnitude (size) AND a direction. can be in onedimension, twodimensions, or even threedimensions can be represented using a magnitude
More informationMAC Module 5 Vectors in 2Space and 3Space II
MAC 2103 Module 5 Vectors in 2Space and 3Space II 1 Learning Objectives Upon completing this module, you should be able to: 1. Determine the cross product of a vector in R 3. 2. Determine a scalar triple
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: Any book on linear algebra! [HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationLOWELL WEEKLY JOURNAL
Y » $ 5 Y 7 Y Y Y Q x Q» 75»»/ q } # ]»\   $ { Q» / X x»» 3 q $ 9 ) Y q  5 5 3 3 3 7 Q q   Q _»»/Q Y  9    ) [ X 7» »  )»? / /? Q Y»» # X Q»  ?» Q ) Q \ Q    3? 7» ? #»»» 7  / Q
More informationMSMS Vectors and Matrices
MSMS Vectors and Matrices Basilio Bona DAUIN Politecnico di Torino Semester 1, 20152016 B. Bona (DAUIN) MSMSVectors and matrices Semester 1, 20152016 1 / 39 Introduction Most of the topics introduced
More information6. 3D Kinematics DE2EA 2.1: M4DE. Dr Connor Myant
DE2EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents ThreeDimensional
More informationHomework 1/Solutions. Graded Exercises
MTH 3103 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both
More informationCS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra /34
Linear Algebra /34 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct axes)
More informationMAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012
(Homework 1: Chapter 1: Exercises 17, 9, 11, 19, due Monday June 11th See also the course website for lectures, assignments, etc) Note: today s lecture is primarily about definitions Lots of definitions
More informationProblem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:
Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein
More information03  Basic Linear Algebra and 2D Transformations
03  Basic Linear Algebra and 2D Transformations (invited lecture by Dr. Marcel Campen) Overview In this box, you will find references to Eigen We will briefly overview the basic linear algebra concepts
More informationChapter 2. Vectors and Vector Spaces
2.2. Cartesian Coordinates and Geometrical Properties of Vectors 1 Chapter 2. Vectors and Vector Spaces Section 2.2. Cartesian Coordinates and Geometrical Properties of Vectors Note. There is a natural
More informationBidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups
Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups Darren FunkNeubauer Department of Mathematics and Physics Colorado State University  Pueblo Pueblo, Colorado, USA darren.funkneubauer@colostatepueblo.edu
More informationAnnouncements Monday, September 18
Announcements Monday, September 18 WeBWorK 1.4, 1.5 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 22. Midterms happen during recitation. The exam covers through 1.5. About
More information1. Vectors.
1. Vectors 1.1 Vectors and Matrices Linear algebra is concerned with two basic kinds of quantities: vectors and matrices. 1.1 Vectors and Matrices Scalars and Vectors  Scalar: a numerical value denoted
More informationALGEBRAIC GEOMETRY HOMEWORK 3
ALGEBRAIC GEOMETRY HOMEWORK 3 (1) Consider the curve Y 2 = X 2 (X + 1). (a) Sketch the curve. (b) Determine the singular point P on C. (c) For all lines through P, determine the intersection multiplicity
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
More informationAlgebraic Expressions
Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly
More information12. Stresses and Strains
12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM  Formulation Classification of Problems Scalar Vector 1D T(x) u(x)
More informationCHAPTER 4 Stress Transformation
CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z
More informationTransformations. Chapter D Transformations Translation
Chapter 4 Transformations Transformations between arbitrary vector spaces, especially linear transformations, are usually studied in a linear algebra class. Here, we focus our attention to transformation
More informationChapter 2 Math Fundamentals
Chapter 2 Math Fundamentals Part 1 2.1 Conventions and Definitions 2.2 Matrices 2.3 Fundamentals of Rigid Transforms 1 Outline 2.1 Conventions and Definitions 2.2 Matrices 2.3 Fundamentals of Rigid Transforms
More informationPolynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:
Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a
More informationMatrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.
Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationLECTURE 10, MONDAY MARCH 15, 2004
LECTURE 10, MONDAY MARCH 15, 2004 FRANZ LEMMERMEYER 1. Minimal Polynomials Let α and β be algebraic numbers, and let f and g denote their minimal polynomials. Consider the resultant R(X) of the polynomials
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationContents. 1 Vectors, Lines and Planes 1. 2 Gaussian Elimination Matrices Vector Spaces and Subspaces 124
Matrices Math 220 Copyright 2016 Pinaki Das This document is freely redistributable under the terms of the GNU Free Documentation License For more information, visit http://wwwgnuorg/copyleft/fdlhtml Contents
More informationProblem 1: (3 points) Recall that the dot product of two vectors in R 3 is
Linear Algebra, Spring 206 Homework 3 Name: Problem : (3 points) Recall that the dot product of two vectors in R 3 is a x b y = ax + by + cz, c z and this is essentially the same as the matrix multiplication
More information6. SCALARS, VECTORS, AND TENSORS (FOR ORTHOGONAL COORDINATE SYSTEMS)
(FOR ORTHOGONAL COORDINATE SYSTEMS) I Main Topics A What are scalars, vectors, and tensors? B Order of scalars, vectors, and tensors C Linear transformaoon of scalars and vectors (and tensors) D Matrix
More informationARC 341 Structural Analysis II. Lecture 10: MM1.3 MM1.13
ARC241 Structural Analysis I Lecture 10: MM1.3 MM1.13 MM1.4) Analysis and Design MM1.5) Axial Loading; Normal Stress MM1.6) Shearing Stress MM1.7) Bearing Stress in Connections MM1.9) Method of Problem
More informationNeatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.
» ~ $ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z
More informationReview of Coordinate Systems
Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate
More informationDepartment of Physics, Korea University
Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: Any book on linear algebra! [HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationCSC 470 Introduction to Computer Graphics. Mathematical Foundations Part 2
CSC 47 Introduction to Computer Graphics Mathematical Foundations Part 2 Vector Magnitude and Unit Vectors The magnitude (length, size) of nvector w is written w 2 2 2 w = w + w2 + + w n Example: the
More informationRigid Body Rotation. Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li. Department of Applied Mathematics and Statistics Stony Brook University (SUNY)
Rigid Body Rotation Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li Department of Applied Mathematics and Statistics Stony Brook University (SUNY) Content Introduction Angular Velocity Angular Momentum
More informationChapter 1. Vector Analysis
Chapter 1. Vector Analysis Hayt; 8/31/2009; 11 1.1 Scalars and Vectors Scalar : Vector: A quantity represented by a single real number Distance, time, temperature, voltage, etc Magnitude and direction
More informationHomogeneous Coordinates
Homogeneous Coordinates Basilio Bona DAUINPolitecnico di Torino October 2013 Basilio Bona (DAUINPolitecnico di Torino) Homogeneous Coordinates October 2013 1 / 32 Introduction Homogeneous coordinates
More informationLesson Rigid Body Dynamics
Lesson 8 Rigid Body Dynamics Lesson 8 Outline Problem definition and motivations Dynamics of rigid bodies The equation of unconstrained motion (ODE) User and time control Demos / tools / libs Rigid Body
More informationStress, Strain, Mohr s Circle
Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected
More informationThe geometry of least squares
The geometry of least squares We can think of a vector as a point in space, where the elements of the vector are the coordinates of the point. Consider for example, the following vector s: t = ( 4, 0),
More informationSec. 1.1: Basics of Vectors
Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in ndimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x
More informationMECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso
MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium
More information(A B) 2 + (A B) 2. and factor the result.
Transformational Geometry of the Plane (Master Plan) Day 1. Some Coordinate Geometry. Cartesian (rectangular) coordinates on the plane. What is a line segment? What is a (right) triangle? State and prove
More informationRotational & RigidBody Mechanics. Lectures 3+4
Rotational & RigidBody Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion  Definitions
More informationLast week we presented the following expression for the angles between two vectors u and v in R n ( ) θ = cos 1 u v
Orthogonality (I) Last week we presented the following expression for the angles between two vectors u and v in R n ( ) θ = cos 1 u v u v which brings us to the fact that θ = π/2 u v = 0. Definition (Orthogonality).
More informationLinear Algebra V = T = ( 4 3 ).
Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a dimensional column vector and V is a 5dimensional
More informationCS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra 1/33
Linear Algebra 1/33 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct
More informationLesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality
Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation.
More informationI&C 6N. Computational Linear Algebra
I&C 6N Computational Linear Algebra 1 Lecture 1: Scalars and Vectors What is a scalar? Computer representation of a scalar Scalar Equality Scalar Operations Addition and Multiplication What is a vector?
More informationMath 51, Homework2 Solutions
SSEA Summer 27 Math 5, Homework2 Solutions Write the parametric equation of the plane that contains the following point and line: 3 2, 4 2 + t 3 t R 5 4 By substituting t = and t =, we get two points
More informationThe Sphere OPTIONAL  I Vectors and three dimensional Geometry THE SPHERE
36 THE SPHERE You must have played or seen students playing football, basketball or table tennis. Football, basketball, table tennis ball are all examples of geometrical figures which we call "spheres"
More informationAlgebraic Expressions and Identities
ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 Algebraic Expressions and Identities CHAPTER 9 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or
More informationLOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort
 7 7 Z 8 q ) V x  X > q  < Y Y X V  z     V  V  q \  q q <  V    x   V q > x  x q  x q  x    7 »     6 q x  >   x    x   q q  V  x   ( Y q Y7  >»>  x Y  ] [
More informationMath Exam 2, October 14, 2008
Math 96  Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian
More informationVector and Affine Math
Vector and Affine Math Computer Science Department The Universit of Teas at Austin Vectors A vector is a direction and a magnitude Does NOT include a point of reference Usuall thought of as an arrow in
More informationA geometric interpretation of the homogeneous coordinates is given in the following Figure.
Introduction Homogeneous coordinates are an augmented representation of points and lines in R n spaces, embedding them in R n+1, hence using n + 1 parameters. This representation is useful in dealing with
More informationLinear Algebra Homework and Study Guide
Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of
More information(arrows denote positive direction)
12 Chapter 12 12.1 3dimensional Coordinate System The 3dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate
More informationHilbert s Metric and Gromov Hyperbolicity
Hilbert s Metric and Gromov Hyperbolicity Andrew Altman May 13, 2014 1 1 HILBERT METRIC 2 1 Hilbert Metric The Hilbert metric is a distance function defined on a convex bounded subset of the ndimensional
More information7a3 2. (c) πa 3 (d) πa 3 (e) πa3
1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin
More informationTwo Posts to Fill On School Board
Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationUnit 2: Lines and Planes in 3 Space. Linear Combinations of Vectors
Lesson10.notebook November 28, 2012 Unit 2: Lines and Planes in 3 Space Linear Combinations of Vectors Today's goal: I can write vectors as linear combinations of each other using the appropriate method
More informationVectors and Matrices
Vectors and Matrices Scalars We often employ a single number to represent quantities that we use in our daily lives such as weight, height etc. The magnitude of this number depends on our age and whether
More informationLagrange Multipliers
Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each
More informationLecture 4: Affine Transformations. for Satan himself is transformed into an angel of light. 2 Corinthians 11:14
Lecture 4: Affine Transformations for Satan himself is transformed into an angel of light. 2 Corinthians 11:14 1. Transformations Transformations are the lifeblood of geometry. Euclidean geometry is based
More informationIntro Vectors 2D implicit curves 2D parametric curves. Graphics 2012/2013, 4th quarter. Lecture 2: vectors, curves, and surfaces
Lecture 2, curves, and surfaces Organizational remarks Tutorials: TA sessions for tutorial 1 start today Tutorial 2 will go online after lecture 3 Practicals: Make sure to find a team partner very soon
More informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 2: Rigid Motions and Homogeneous Transformations
MCE/EEC 647/747: Robot Dynamics and Control Lecture 2: Rigid Motions and Homogeneous Transformations Reading: SHV Chapter 2 Mechanical Engineering Hanz Richter, PhD MCE503 p.1/22 Representing Points, Vectors
More informationMath 302 Test 1 Review
Math Test Review. Given two points in R, x, y, z and x, y, z, show the point x + x, y + y, z + z is on the line between these two points and is the same distance from each of them. The line is rt x, y,
More informationCourse 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations
Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.
More informationChapter 2: Numeric, Cell, and Structure Arrays
Chapter 2: Numeric, Cell, and Structure Arrays Topics Covered: Vectors Definition Addition Multiplication Scalar, Dot, Cross Matrices Row, Column, Square Transpose Addition Multiplication ScalarMatrix,
More informationLecture 3: Linear Algebra Review, Part II
Lecture 3: Linear Algebra Review, Part II Brian Borchers January 4, Linear Independence Definition The vectors v, v,..., v n are linearly independent if the system of equations c v + c v +...+ c n v n
More informationHIGHERORDER THEORIES
HIGHERORDER THEORIES THIRDORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 ThirdOrder Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +
More informationUnit IV State of stress in Three Dimensions
Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength
More information