2D Geometric Transformations. (Chapter 5 in FVD)
|
|
- Hector Blake
- 5 years ago
- Views:
Transcription
1 2D Geometric Transformations (Chapter 5 in FVD)
2 2D geometric transformation Translation Scaling Rotation Shear Matri notation Compositions Homogeneous coordinates 2
3 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. A point is represented b its Cartesian coordinates: (,). Question: How do we transform a geometric object in the plane? Answer: Let (A,B) be a straight line segment and T a general 2D transformation: T transforms (A,B) into another straight line segment (A,B ), where A =TA and B =TB. 3
4 Translation Translate (a,b): (,) (+a,+b) Translate(2,4) 4
5 Scale Scale (a,b): (,) (a,b) Scale (2,3) Scale (2,3) 5
6 How can we scale an object without moving its origin (lower left corner)? Translate(-,-) Translate(,) Scale(2,3) 6
7 Reflection Scale(-,) Scale(,-) 7
8 Rotation Rotate(θ): (,) ( cos(θ)+ sin(θ), - sin(θ)+ cos(θ)) Rotate(9) Rotate(9) 8
9 How can we rotate an object without moving its origin (lower left corner)? Translate(-,-) Translate(,) Rotate(9) 9
10 Shear Shear (a,b): (,) (+a,+b) Shear(,) Shear(,2)
11 Composition of Transformations Rigid transformation: Translation + Rotation (distance preserving). Similarit transformation: Translation + Rotation + uniform Scale (angle preserving). Affine transformation: Translation + Rotation + Scale + Shear (parallelism preserving). All above transformations are groups where Rigid Similarit Affine.
12 Affine Similarit Rigid 2
13 Matri Notation Let s treat a point (,) as a 2 matri (a column vector): What happens when this vector is multiplied b a 22 matri? a c b d = a c + + b d 3
14 2D Transformations 2D object is represented b points and lines that join them. Transformations can be applied onl to the the points defining the lines. A point (,) is represented b a 2 column vector, and we can represent 2D transformations using 22 matrices: ' ' a = c b d 4
15 Scale Scale(a,b): (,) (a,b) a b a = b If a or b are negative, we get reflection. Inverse: S - (a,b)=s(/a,/b) 5
16 6 Reflection Reflection through the ais: Reflection through the ais: Reflection through =: Reflection through =-:
17 Shear, Rotation Shear(a,b): (,) (+a,+b) b a = + + a b Rotate(θ): (,) (cosθ+sinθ, -sinθ + cosθ) cosθ sinθ Inverse: sinθ cosθ + sinθ = cosθ sinθ + cosθ R - (θ)=r T (θ)=r(-θ) 7
18 Composition of Transformations A sequence of transformations can be collapsed into a single matri: [ ][ ][ ] [ ] A B C Note: order of transformations is important! (otherwise - commutative groups) = D translate rotate rotate translate 8
19 Composition of Transformations (Cont.) D = A B C D - = C - B - A - Proof: D * D- = ABC * C-B-A- = AB* I * B- * A- = A*I*A = I 9
20 Translation Translation(a,b): Problem: Cannot represent translation using 22 matrices. + + a b Solution: Homogeneous Coordinates 2
21 Homogeneous Coordinates Homogeneous Coordinates is a mapping from R n to R n+ : (, ) ( X, Y, W) = ( t, t, t) Note:(t,t,t) all correspond to the same non-homogeneous point (,). E.g. (2,3,) (6,9,3). Inverse mapping: X Y ( X, Y, W ), W W 2
22 22 Translation Translate(a,b): Inverse: T - (a,b)=t(-a,-b) Affine transformation now have the following form: + + = b a b a f d c e b a
23 Geometric Interpretation W (X,Y,) Y (X,Y,W) A 2D point is mapped to a line (ra) in 3D. The non-homogeneous points are obtained b projecting the ras onto the plane Z=. X 23
24 24 Eample: Rotation about an arbitrar point: Actions: Translate the coordinates so that the origin is at (, ). Rotate b θ. Translate back. (, ) θ + = = sin ) cos ( cos sin sin ) cos ( sin cos cos sin sin cos θ θ θ θ θ θ θ θ θ θ θ θ
25 Another eample: Reflection about an Arbitrar Line: p 2 p L=p +t (p 2 -p )=t p 2 +(-t) p Actions: Translate the coordinates so that P is at the origin. Rotate so that L aligns with the - ais. Reflect about the -ais. Rotate back. Translate back. 25
26 Change of Coordinates It is often requires the transformation of object description from one coordinate sstem to another. How do we transform between two Cartesian coordinate sstems? Rule: Transform one coordinate frames towards the other in the opposite direction of the representation change. Representation Transformation 26
27 Change of Coordinates (Cont.) Eample: X P O Y X P O Y 27
28 Eample: Represent the point P=( p, p,) in the (, ) coordinate sstem. where P ' =MP M = R T cosθ = sinθ sinθ cosθ ( p, p ) (, ) θ 28
29 29 Alternative method: Assume =(u,u ) and =(v,v ) in the (,) coordinate sstem. (, ) P =MP ' = v v u u M where (u,u ) (v,v )
30 3 Eample: P is at the ais P=(v,v ): What is the inverse? (u,u ) (v,v ) = = = ' v v v v u u MP P
31 3 Another eample: Reflection about an Arbitrar Line: Define a coordinate sstems (u,v) parallel to P P 2 : p p 2 = u u p p p p 2 2 u = = v v u u v = p p v v u u v u v u p p M
Graphics Example: Type Setting
D Transformations Graphics Eample: Tpe Setting Modern Computerized Tpesetting Each letter is defined in its own coordinate sstem And positioned on the page coordinate sstem It is ver simple, m she thought,
More informationOperations depend on pixel s Coordinates. Context free. Independent of pixel values. I(x,y) I (x,y )
Geometric Transformation Operations depend on piel s Coordinates. Contet free. Independent of piel values. f f (, ) = ' (, ) = ' I(, ) = I' ( f (, ), f ( ) ), (,) (, ) I(,) I (, ) Eample: Translation =
More informationCS 378: Computer Game Technology
CS 378: Computer Game Technolog 3D Engines and Scene Graphs Spring 202 Universit of Teas at Austin CS 378 Game Technolog Don Fussell Representation! We can represent a point, p =,), in the plane! as a
More informationCS 354R: Computer Game Technology
CS 354R: Computer Game Technolog Transformations Fall 207 Universit of Teas at Austin CS 354R Game Technolog S. Abraham Transformations What are the? Wh should we care? Universit of Teas at Austin CS 354R
More informationReading. 4. Affine transformations. Required: Watt, Section 1.1. Further reading:
Reading Required: Watt, Section.. Further reading: 4. Affine transformations Fole, et al, Chapter 5.-5.5. David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, 2 nd Ed., McGraw-Hill,
More informationAffine transformations
Reading Required: Affine transformations Brian Curless CSE 557 Fall 2009 Shirle, Sec. 2.4, 2.7 Shirle, Ch. 5.-5.3 Shirle, Ch. 6 Further reading: Fole, et al, Chapter 5.-5.5. David F. Rogers and J. Alan
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 informationwe must pay attention to the role of the coordinate system w.r.t. which we perform a tform
linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation
More informationTwo conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which?
walters@buffalo.edu CSE 480/580 Lecture 2 Slide 3-D Transformations 3-D space Two conventions for coordinate sstems Left-Hand vs Right-Hand (Thumb is the ais, inde is the ais) Which is which? Most graphics
More informationAffine transformations. Brian Curless CSE 557 Fall 2014
Affine transformations Brian Curless CSE 557 Fall 2014 1 Reading Required: Shirle, Sec. 2.4, 2.7 Shirle, Ch. 5.1-5.3 Shirle, Ch. 6 Further reading: Fole, et al, Chapter 5.1-5.5. David F. Rogers and J.
More informationwe must pay attention to the role of the coordinate system w.r.t. which we perform a tform
linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation
More informationAffine transformations
Reading Optional reading: Affine transformations Brian Curless CSE 557 Autumn 207 Angel and Shreiner: 3., 3.7-3. Marschner and Shirle: 2.3, 2.4.-2.4.4, 6..-6..4, 6.2., 6.3 Further reading: Angel, the rest
More informationCS 335 Graphics and Multimedia. 2D Graphics Primitives and Transformation
C 335 Graphics and Multimedia D Graphics Primitives and Transformation Basic Mathematical Concepts Review Coordinate Reference Frames D Cartesian Reference Frames (a) (b) creen Cartesian reference sstems
More informationComputer Graphics: 2D Transformations. Course Website:
Computer Graphics: D Transformations Course Website: http://www.comp.dit.ie/bmacnamee 5 Contents Wh transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications
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 informationGeometric Image Manipulation. Lecture #4 Wednesday, January 24, 2018
Geometric Image Maniplation Lectre 4 Wednesda, Janar 4, 08 Programming Assignment Image Maniplation: Contet To start with the obvios, an image is a D arra of piels Piel locations represent points on the
More information( ) ( ) ( ) ( ) TNM046: Datorgrafik. Transformations. Linear Algebra. Linear Algebra. Sasan Gooran VT Transposition. Scalar (dot) product:
TNM046: Datorgrafik Transformations Sasan Gooran VT 04 Linear Algebra ( ) ( ) =,, 3 =,, 3 Transposition t = 3 t = 3 Scalar (dot) product: Length (Norm): = t = + + 3 3 = = + + 3 Normaliation: ˆ = Linear
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 informationLinear and affine transformations
Linear and affine transformations Linear Algebra Review Matrices Transformations Affine transformations in Euclidean space 1 The linear transformation given b a matri Let A be an mn matri. The function
More informationMATRIX TRANSFORMATIONS
CHAPTER 5. MATRIX TRANSFORMATIONS INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATRIX TRANSFORMATIONS Matri Transformations Definition Let A and B be sets. A function f : A B
More informationRigid Body Transforms-3D. J.C. Dill transforms3d 27Jan99
ESC 489 3D ransforms 1 igid Bod ransforms-3d J.C. Dill transforms3d 27Jan99 hese notes on (2D and) 3D rigid bod transform are currentl in hand-done notes which are being converted to this file from that
More informationMatrices. VCE Maths Methods - Unit 2 - Matrices
Matrices Introduction to matrices Addition & subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations
More informationPhys 221. Chapter 3. Vectors A. Dzyubenko Brooks/Cole
Phs 221 Chapter 3 Vectors adzubenko@csub.edu http://www.csub.edu/~adzubenko 2014. Dzubenko 2014 rooks/cole 1 Coordinate Sstems Used to describe the position of a point in space Coordinate sstem consists
More informationIntroduction to 3D Game Programming with DirectX 9.0c: A Shader Approach
Introduction to 3D Game Programming with DirectX 90c: A Shader Approach Part I Solutions Note : Please email to frank@moon-labscom if ou find an errors Note : Use onl after ou have tried, and struggled
More informationAffine transformations
Reading Required: Affine transformations Brian Curless CSEP 557 Fall 2016 Angel 3.1, 3.7-3.11 Further reading: Angel, the rest of Chapter 3 Fole, et al, Chapter 5.1-5.5. David F. Rogers and J. Alan Adams,
More information1 HOMOGENEOUS TRANSFORMATIONS
HOMOGENEOUS TRANSFORMATIONS Purpose: The purpose of this chapter is to introduce ou to the Homogeneous Transformation. This simple 4 4 transformation is used in the geometr engines of CAD sstems and in
More informationMathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: Exercise Answers
Mathematical Structures for Computer Graphics Steven J. Janke John Wiley & Sons, 2015 ISBN: 978-1-118-71219-1 Updated /17/15 Exercise Answers Chapter 1 1. Four right-handed systems: ( i, j, k), ( i, j,
More informationLecture 5: 3-D Rotation Matrices.
3.7 Transformation Matri and Stiffness Matri in Three- Dimensional Space. The displacement vector d is a real vector entit. It is independent of the frame used to define it. d = d i + d j + d k = dˆ iˆ+
More informationPhysically Based Rendering ( ) Geometry and Transformations
Phsicall Based Rendering (6.657) Geometr and Transformations 3D Point Specifies a location Origin 3D Point Specifies a location Represented b three coordinates Infinitel small class Point3D { public: Coordinate
More informationOmm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
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 n-vector w is written w 2 2 2 w = w + w2 + + w n Example: the
More informationThe first change comes in how we associate operators with classical observables. In one dimension, we had. p p ˆ
VI. Angular momentum Up to this point, we have been dealing primaril with one dimensional sstems. In practice, of course, most of the sstems we deal with live in three dimensions and 1D quantum mechanics
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 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 in-person grading Anderson@cs.tufts.edu 2/59 Geometric Transform Apply transforms to a hierarchy
More informationCoordinates for Projective Planes
Chapter 8 Coordinates for Projective Planes Math 4520, Fall 2017 8.1 The Affine plane We now have several eamples of fields, the reals, the comple numbers, the quaternions, and the finite fields. Given
More informationDesigning Information Devices and Systems I Discussion 2A
EECS 16A Spring 218 Designing Information Devices and Systems I Discussion 2A 1. Visualizing Matrices as Operations This problem is going to help you visualize matrices as operations. For example, when
More informationCS-184: Computer Graphics. Today
CS-184: Computer Graphics Lecture #3: 2D Transformations Prof. James O Brien Universit of California, Berkele V2006-S-03-1.0 Toda 2D Transformations Primitive Operations Scale, Rotate, Shear, Flip, Translate
More informationReading. Affine transformations. Vector representation. Geometric transformations. x y z. x y. Required: Angel 4.1, Further reading:
Reading Required: Angel 4.1, 4.6-4.10 Further reading: Affine transformations Angel, the rest of Chapter 4 Fole, et al, Chapter 5.1-5.5. David F. Rogers and J. Alan Adams, Mathematical Elements for Computer
More informationSolutions of homework 1. 2 a) Using the stereographic projection from the north pole N = (0, 0, 1) introduce stereographic coordinates
Solutions of homework 1 1 a) Using the stereographic projection from the north pole N (0, 1) introduce stereographic coordinate for the part of the circle S 1 ( + 1) without the north pole. b) Do the same
More informationSome linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013
Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 23 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,
More informationWe could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2
Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1
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 information9. Yes; translate so that the centers align, and then dilate using the ratio of radii to map one circle to another.
9. Yes; translate so that the centers align, and then dilate using the ratio of radii to map one circle to another. 10. Keegan dilated using A as the center of dilation instead of the origin. Point A should
More information5. Nonholonomic constraint Mechanics of Manipulation
5. Nonholonomic constraint Mechanics of Manipulation Matt Mason matt.mason@cs.cmu.edu http://www.cs.cmu.edu/~mason Carnegie Mellon Lecture 5. Mechanics of Manipulation p.1 Lecture 5. Nonholonomic constraint.
More informationCSE4030 Introduction to Computer Graphics
CSE4030 Introduction to Computer Graphics Dongguk University Jeong-Mo Hong Week 5 Living in a 3 dimensional world II Geometric coordinate in 3D How to move your cubes in 3D Objectives Introduce concepts
More informationConsider a slender rod, fixed at one end and stretched, as illustrated in Fig ; the original position of the rod is shown dotted.
4.1 Strain If an object is placed on a table and then the table is moved, each material particle moves in space. The particles undergo a displacement. The particles have moved in space as a rigid bod.
More informationAnd similarly in the other directions, so the overall result is expressed compactly as,
SQEP Tutorial Session 5: T7S0 Relates to Knowledge & Skills.5,.8 Last Update: //3 Force on an element of area; Definition of principal stresses and strains; Definition of Tresca and Mises equivalent stresses;
More informationSET-I SECTION A SECTION B. General Instructions. Time : 3 hours Max. Marks : 100
General Instructions. All questions are compulsor.. This question paper contains 9 questions.. Questions - in Section A are ver short answer tpe questions carring mark each.. Questions 5- in Section B
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 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 one-dimension, two-dimensions, or even three-dimensions can be represented using a magnitude
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 informationMathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes
Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.
More informationSection 1.8/1.9. Linear Transformations
Section 1.8/1.9 Linear Transformations Motivation Let A be a matrix, and consider the matrix equation b = Ax. If we vary x, we can think of this as a function of x. Many functions in real life the linear
More informationGeometry review, part I
Geometr reie, part I Geometr reie I Vectors and points points and ectors Geometric s. coordinate-based (algebraic) approach operations on ectors and points Lines implicit and parametric equations intersections,
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 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 informationUnit 12 Study Notes 1 Systems of Equations
You should learn to: Unit Stud Notes Sstems of Equations. Solve sstems of equations b substitution.. Solve sstems of equations b graphing (calculator). 3. Solve sstems of equations b elimination. 4. Solve
More informationCS 112 Transformations II. Aditi Majumder, CS 112 Slide 1
CS 112 Transformations II Aditi Majumder, CS 112 Slide 1 Composition of Transformations Example: A point P is first translated and then rotated. Translation matrix T, Rotation Matrix R. After Translation:
More informationThree-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional 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 x-coordinate and b is the y-coordinate.
More informationChapter 3. Vectors and Two-Dimensional Motion
Chapter 3 Vectors and Two-Dimensional Motion 1 Vector vs. Scalar Review All physical quantities encountered in this text will be either a scalar or a vector A vector quantity has both magnitude (size)
More informationMathematical Foundations: Intro
Mathematical Foundations: Intro Graphics relies on 3 basic objects: 1. Scalars 2. Vectors 3. Points Mathematically defined in terms of spaces: 1. Vector space 2. Affine space 3. Euclidean space Math required:
More informationAPPLIED MECHANICS I Resultant of Concurrent Forces Consider a body acted upon by co-planar forces as shown in Fig 1.1(a).
PPLIED MECHNICS I 1. Introduction to Mechanics Mechanics is a science that describes and predicts the conditions of rest or motion of bodies under the action of forces. It is divided into three parts 1.
More informationLines and points. Lines and points
omogeneous coordinates in the plane Homogeneous coordinates in the plane A line in the plane a + by + c is represented as (a, b, c). A line is a subset of points in the plane. All vectors (ka, kb, kc)
More information(arrows denote positive direction)
12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional 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 informationSECTION 6.3: VECTORS IN THE PLANE
(Section 6.3: Vectors in the Plane) 6.18 SECTION 6.3: VECTORS IN THE PLANE Assume a, b, c, and d are real numbers. PART A: INTRO A scalar has magnitude but not direction. We think of real numbers as scalars,
More informationA Tutorial on Euler Angles and Quaternions
A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weimann Institute of Science http://www.weimann.ac.il/sci-tea/benari/ Version.0.1 c 01 17 b Moti Ben-Ari. This work
More information4.1 Distance and Length
Chapter Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at vectors
More informationChapter 8. Rigid transformations
Chapter 8. Rigid transformations We are about to start drawing figures in 3D. There are no built-in routines for this purpose in PostScript, and we shall have to start more or less from scratch in extending
More informationThe Cross Product of Two Vectors
The Cross roduct of Two Vectors In proving some statements involving surface integrals, there will be a need to approximate areas of segments of the surface by areas of parallelograms. Therefore it is
More informationMatrices. VCE Maths Methods - Unit 2 - Matrices
Matrices Introduction to matrices Addition subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations
More information9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
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 informationTransformation of kinematical quantities from rotating into static coordinate system
Transformation of kinematical quantities from rotating into static coordinate sstem Dimitar G Stoanov Facult of Engineering and Pedagog in Sliven, Technical Universit of Sofia 59, Bourgasko Shaussee Blvd,
More information11.4 Polar Coordinates
11. Polar Coordinates 917 11. Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of assigning ordered pairs of numbers to points in the plane.
More informationSpan and Linear Independence
Span and Linear Independence It is common to confuse span and linear independence, because although they are different concepts, they are related. To see their relationship, let s revisit the previous
More informationVector Geometry. Chapter 5
Chapter 5 Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at
More informationProperties of Transformations
6. - 6.4 Properties of Transformations P. Danziger Transformations from R n R m. General Transformations A general transformation maps vectors in R n to vectors in R m. We write T : R n R m to indicate
More information15. Eigenvalues, Eigenvectors
5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does
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 informationOptical flow. Subhransu Maji. CMPSCI 670: Computer Vision. October 20, 2016
Optical flow Subhransu Maji CMPSC 670: Computer Vision October 20, 2016 Visual motion Man slides adapted from S. Seitz, R. Szeliski, M. Pollefes CMPSC 670 2 Motion and perceptual organization Sometimes,
More information( ) ( ) ( ) 2 6A: Special Trig Limits! Math 400
2 6A: Special Trig Limits Math 400 This section focuses entirely on the its of 2 specific trigonometric functions. The use of Theorem and the indeterminate cases of Theorem are all considered. a The it
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 informationMathematics Trigonometry: Unit Circle
a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagog Mathematics Trigonometr: Unit Circle Science and Mathematics Education Research Group Supported b UBC Teaching and
More informationDeterminant and Trace
Determinant an Trace Area an mappings from the plane to itself: Recall that in the last set of notes we foun a linear mapping to take the unit square S = {, y } to any parallelogram P with one corner at
More information4 Strain true strain engineering strain plane strain strain transformation formulae
4 Strain The concept of strain is introduced in this Chapter. The approimation to the true strain of the engineering strain is discussed. The practical case of two dimensional plane strain is discussed,
More informationRotation of Axes. By: OpenStaxCollege
Rotation of Axes By: OpenStaxCollege As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions,
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 informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationWhat you will learn today
What you will learn today The Dot Product Equations of Vectors and the Geometry of Space 1/29 Direction angles and Direction cosines Projections Definitions: 1. a : a 1, a 2, a 3, b : b 1, b 2, b 3, a
More informationAdvanced Higher Grade
Prelim Eamination / 5 (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours Read Carefully. Full credit will be given only where the solution contains appropriate woring.. Calculators
More informationMechanics Physics 151
Mechanics Phsics 151 Lecture 8 Rigid Bod Motion (Chapter 4) What We Did Last Time! Discussed scattering problem! Foundation for all experimental phsics! Defined and calculated cross sections! Differential
More information9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
More informationAll parabolas through three non-collinear points
ALL PARABOLAS THROUGH THREE NON-COLLINEAR POINTS 03 All parabolas through three non-collinear points STANLEY R. HUDDY and MICHAEL A. JONES If no two of three non-collinear points share the same -coordinate,
More informationKINEMATIC RELATIONS IN DEFORMATION OF SOLIDS
Chapter 8 KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS Figure 8.1: 195 196 CHAPTER 8. KINEMATIC RELATIONS IN DEFORMATION OF SOLIDS 8.1 Motivation In Chapter 3, the conservation of linear momentum for a
More informationCSE 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 Fall Quarter 2016 Announcements Monday October 3: Discussion Assignment
More informationMATH 532, 736I: MODERN GEOMETRY
MATH 532, 736I: MODERN GEOMETRY Test 2, Spring 2013 Show All Work Name Instructions: This test consists of 5 pages (one is an information page). Put your name at the top of this page and at the top of
More informationExample 25: Determine the moment M AB produced by force F in Figure which tends to rotate the rod about the AB axis.
Eample 25: Determine the moment M AB produced by force F in Figure which tends to rotate the rod about the AB ais. Solution: Because that F is parallel to the z-ais so it has no moment about z-ais. Its
More informationSection 5.8. (i) ( 3 + i)(14 2i) = ( 3)(14 2i) + i(14 2i) = {( 3)14 ( 3)(2i)} + i(14) i(2i) = ( i) + (14i + 2) = i.
1. Section 5.8 (i) ( 3 + i)(14 i) ( 3)(14 i) + i(14 i) {( 3)14 ( 3)(i)} + i(14) i(i) ( 4 + 6i) + (14i + ) 40 + 0i. (ii) + 3i 1 4i ( + 3i)(1 + 4i) (1 4i)(1 + 4i) (( + 3i) + ( + 3i)(4i) 1 + 4 10 + 11i 10
More informationOverview. Distances in R 3. Distance from a point to a plane. Question
Overview Yesterda we introduced equations to describe lines and planes in R 3 : r + tv The vector equation for a line describes arbitrar points r in terms of a specific point and the direction vector v.
More informationME 7502 Lecture 2 Effective Properties of Particulate and Unidirectional Composites
ME 75 Lecture Effective Properties of Particulate and Unidirectional Composites Concepts from Elasticit Theor Statistical Homogeneit, Representative Volume Element, Composite Material Effective Stress-
More information