# Geometric Transformation CS 211A

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Geometric Transformation CS 211A

2 What is transformation? Moving points (x,y) moves to (x+t, y+t) Can be in any dimension 2D Image warps 3D 3D Graphics and Vision Can also be considered as a movement to the coordinate axes

3 Homogeneous Coordinates Note: (2x/y, 2), (3x/y,3), and (x/y,1) represent the same 1D point P P (x,y) y=1 Q (2x,2y) Any point on the same vector has the same homogeneous coordinates P (x/y,1) 1D points on the line is represented by 2D array, called homogeneous coordinates

4 Generalize to Higher Dimensions Z P (x, y, z) 2D points represented by homogeneous coordinates P (x/z, y/z, 1) Similarly, 3D points are represented by homogeneous coordinates If (x,y,z,w) is the homogeneous coordinate of a 3D point, where w = 1, then the 3D point is given by (x/w,y/w,z/w,1)

5 Practically [x y z w], w 1 Then, [x/w, y/w, z/w, 1] Try to put it the w=1 hyperplane Why? Can represent pts at infinity 2D [, ] 3D homogeneous [2, 3, 0] Point at infinity in the direction of [2, 3] Distinguish between points and vectors [2, 3, 1] vs [2, 3, 0]

6 Linear Transformation L(ap+bq) = al(p) + bl(q) Lines/planes transform to lines/planes If transformation of vertices are known, transformation of linear combination of vertices can be achieved p and q are points or vectors in (n+1)x1 homogeneous coordinates For 2D, 3x1 homogeneous coordinates For 3D, 4x1 homogeneous coordinates L is a (n+1)x(n+1) square matrix For 2D, 3x3 matrix For 3D, 4x4 matrix

7 Linear Transformations Euclidian Length and angles are preserved Affine Ratios of lengths and angles are preserved Projective Can move points at infinity in range and finite points to infinity

8 Euclidian Transformations Lengths and angles are preserved Translation Rotation

9 2D Translation Line between two points is transformed to a line between the transformed points

10 3D Translation x x P = y z 1 P = y z 1 P = TP x = x + t x y = y + t y z = z + t z T = t x t y t z Denoted by T(t x, t y, t z )

11 Inverse Translation x x P = y z 1 P = y z 1 P = T -1 P x = x - t x y = y - t y z = z - t z T -1 = t x t y t z T -1 = T(-t x, -t y, -t z )

12 x = ρ cos φ y = ρ sin φ x = ρ cos (θ + φ) y = ρ sin (θ + φ) 2D Rotation θ φ (x,y ) ρ (x,y) x = ρ cosθ cosφ - ρ sinθ sinφ = x cosθ y sinθ y = ρ sinθ cosφ + ρ cosθ sinφ = x sinθ + y cosθ x y 1 = R x y 1 R = cosθ sinθ 0 sinθ cosθ

13 Example

14 Rotation in 3D about z axis x = x cosθ y sinθ y = x sinθ + y cosθ z = z R 0 cosθ sinθ 0 x = 0 sinθ cosθ R z = P = R z P cosθ sinθ 0 0 sinθ cosθ Denoted by R(θ) R -1 = R(-θ) = R T (θ) Where R = R x or R y or R z

15 Affine Transformations Ratio of lengths and angles are preserved Scale Lengths are not preserved Shear Angles are not preserved

16 Scaling P = x y z 1 x = s x x y = s y y z = s z z P = x y z 1 P = SP s x S = 0 s y s z Denoted by S(s x, s y, s z ) S -1 = S(1/s x, 1/s y, 1/s z )

17 Shear Translation of one coordinate of a point is proportional to the value of the other coordinate of the same point. Point : (x,y) After y-shear : (x+ay,y) After x-shear : (x,y+bx) Changes the shape of the object.

18 Using matrix for Shear Example: Z-shear (Z coordinate does not change) 1 0 a 0 x x+az x 0 1 b y z = y+bz z = y z

19 General Affine Transformation The last row is fixed Has 12 degrees of A = freedom Does not change degrees of polynomial Parallel and intersecting lines/planes to the same α 11 α 12 α 13 α 14 α 21 α 22 α 23 α 24 α 31 α 32 α 33 α

20 Fixed Points and Lines Some points and lines can be fixed under a transformation Scaling Point Origin Rotation Line Axis of rotation

21 Scaling About a point (0,2) (2,2) (0,4) (4,4) (0,0) (2,0) (0,0) (4,0) Origin is fixed with transformation -> Scaling about origin

22 Concatenation of Transformations How do we use multiple transformation? Apply F Get to a known situation Apply the required L Apply F -1

23 Scaling About a point (0,2) (2,2) (-1,3) (3,3) (1,1) (1,1) (0,0) (2,0) (-1,-1) (3,-1) Scaling about center -> Center is fixed with transformation

24 Done by concatenation (0,2) (2,2) (1,1) (-1,1) (1,1) (0,0) (2,0) (1,1) (-1,-1) (1,-1) Translate so that center coincides with origin - T(-1,-1).

25 Done by concatenation (0,2) (2,2) (-2,2) (2,2) (1,1) (0,0) (2,0) (1,1) (-2,-2) (2,-2) Scale the points about the center S(2,2)

26 Done by concatenation (0,2) (2,2) (-1,3) (3,3) (1,1) (1,1) (0,0) (2,0) (-1,-1) (3,-1) Translate it back by reverse parameters T(1,1) Total Transformation: T(1,1) S(2,2) T(-1,-1) P

27 Rotation about a fixed point z-axis rotation of θ about its center P f Translate by P f : T(-P f ) Rotate about z-axis : R z (θ) Translate back by P f : T(P f ) Total Transformation M = T(P f )R z (θ)t(-p f )

28 Rotation About an Arbitrary Axis Axis given by two points P 1 (starting point) and P 2 (ending point) P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) Anticlockwise angle of rotation is θ Rotate all points to around P 1 P 2 by θ Z P1 P2

29 Rotation about an Arbitrary Axis Make P 1 P 2 coincide with Z-axis Translate P 1 to origin: T(-x 1,-y 1,-z 1 ) Coincides one point of the axis with origin P1 P2 Z

30 Rotation about an Arbitrary Axis Make P 1 P 2 coincide with Z-axis Translate P 1 to origin: T(-x 1,-y 1,-z 1 ) Coincides one point of the axis with origin Rotate shifted axis to coincide with Z axis Z P1 P2

31 Rotation about an Arbitrary Axis Make P 1 P 2 coincide with Z-axis Translate P 1 to origin: T(-x 1,-y 1,-z 1 ) Coincides one point of the axis with origin Rotate shifted axis to coincide with Z axis R 1 : Rotate about to lie on Z plane Z P1 P2

32 Rotation about an Arbitrary Axis Make P 1 P 2 coincide with Z-axis Translate P 1 to origin: T(-x 1,-y 1,-z 1 ) Coincides one point of the axis with origin Rotate shifted axis to coincide with Z axis R 1 : Rotate about to lie on Z plane R 2 : Rotate about to lie on Z axis Z P1 P2

33 Rotation about an Arbitrary Axis Make P 1 P 2 coincide with Z-axis Translate P 1 to origin: T(-x 1,-y 1,-z 1 ) Coincides one point of the axis with origin Rotate shifted axis to coincide with Z axis R 1 : Rotate about to lie on Z plane R 2 : Rotate about to lie on Z axis Z P2 P1

34 Rotation about an Arbitrary Axis Make the axis P 1 P 2 coincide with the Z-axis Translation to move P 1 to the origin: T(-x 1,-y 1,-z 1 ) Coincides one point of the axis with origin Rotation to coincide the shifted axis with Z axis R 1 : Rotation around such that the axis lies on the Z plane. R 2 : Rotation around such that the axis coincides with the Z axis R 3 : Rotate the scene around the Z axis by an angle θ Inverse transformations of R 2, R 1 and T 1 to bring back the axis to the original position M = T R 1 R 2 R 3 R 2 R 1 T

35 After translation Translation P 2 P 2 Axis V = P 2 P 1 = (x 1 -x 2,y 1 -y 2,z 1 -z 2 ) P 1 u P 1 u = V V = (a, b, c) Z

36 Rotation about axis Rotate u about so that it coincides with Z plane u = (0, b, c) u = (a, b, c) α Z α u = (a, 0, d) R 1 = Project u on Z plane : u (0, b, c) α is the angle made by u with Z axis Cos α = c/ b 2 +c 2 = c/d Sin α = b/d c/d -b/d 0 0 b/d c/d

37 Rotation about axis Rotate u about so that it coincides with Z axis Cos β = d/ a 2 +d 2 = d/ a 2 +b 2 +c 2 = d Sin β = a Z β u = (a, 0, d) R 2 = d 0 -a a 0 d

38 Rotation about Z axis Rotate by θ about Z axis R 3 = cosθ sinθ 0 0 sinθ cosθ

39 Rotation about Arbitrary Axis M = T -1 R 1-1 R 2-1 R 3 (θ) R 2 (β) R 1 (α) T = T -1 R x -1 R y -1 R z (θ) R y (β) R x (α) T = T -1 R x (-α)r y (-β)r z (θ) R y (β) R x (α) T

40 Coordinate Systems Represent a point as a linear combination of three vectors and the origin Linearly independent vectors basis Orthogonal vectors are linearly independent v 2 w w = α 1 v 1 + α 2 v 2 + α 3 v 3 + R α 2 v 1 v 2 v 3 R α 1 = w α 2 v 3 α 1 α 3 v 1 α 3 1

41 Coordinate Systems Represent a point as a linear combination of three vectors and the origin Linearly independent vectors basis Orthogonal vectors are linearly independent v 2 w w = α 1 v 1 + α 2 v 2 + α 3 v 3 + R α α 1 α 2 = w v 3 α 1 α 3 v α 3 1

42 Change of Coordinates First coordinate v 1, v 2, v 3, R 1 Second coordinate u 1, u 2, u 3, R 2 u 2 v 2 u 1 P α 1 α 2 = C 1 R 2 R 1 v 1 α 3 1 v 3 u 3

43 Change of Coordinates First coordinate v 1, v 2, v 3, R 1 Second coordinate u 1, u 2, u 3, R 2 u 2 v 2 v 2 u 1 P u 1 = γ 11 v 1 + γ 21 v 2 + γ 31 v 3 R 2 u 2 = γ 12 v 1 + γ 22 v 2 + γ 32 v 3 R 1 v 1 u 3 = γ 13 v 1 + γ 23 v 2 + γ 33 v 3 v 3 R 2 = γ 14 v 1 + γ 24 v 2 + γ 34 v 3 + R 1 u 3 u 3

44 Change of Coordinates u 1 = γ 11 v 1 + γ 21 v 2 + γ 31 v 3 u 2 = γ 12 v 1 + γ 22 v 2 + γ 32 v 3 u 3 = γ 13 v 1 + γ 23 v 2 + γ 33 v 3 R 2 = γ 14 v 1 + γ 24 v 2 + γ 34 v 3 + R 1 γ 11 γ 12 γ 13 γ 14 u 1 u 2 u 3 R 2 = v 1 v 2 v 3 R 1 M M = γ 21 γ 22 γ 23 γ 24 γ 31 γ 32 γ 33 γ

45 Change of Coordinates If R 1 = R 2, then γ 14 = γ 24 = γ 34 = 0 M = γ 11 γ 12 γ 0 13 γ 21 γ 22 γ 0 23 γ 31 γ 32 γ

46 Change of Coordinates P = u 1 u 2 u 3 R 2 C 2 = = v 1 v 2 v 3 R 1 v 1 v 2 v 3 R 1 M C 2 C 1 Hence, C 1 = MC 2

47 What is this matrix? ou need translate Coincide origins ou need to rotate To make the axis match This is a rotation and translation concatenated M = γ 11 γ 12 γ γ γ 21 γ 22 γ γ γ 31 γ 32 γ γ

48 What is this concatenation? r 11 r 12 r t x R = r 21 r 22 r 0 23 r 31 r 32 r T = t y t z r 11 r 12 r 13 t x M = r 21 r 22 r 23 t y r 31 r 32 r 33 t z = [R RT]

49 How to simplify rotation about arbitrary axis? Translation goes in the last column The rotation matrix defined by finding a new coordinate with the arbitrary axis as a, or Z axis

50 How to find coordinate axes? N = U x V V W = N x U U

51 Faster Way Faster way to find R 2 R 1 u x, u y, u z are unit vectors in the,, Z direction Set up a coordinate system where u = u z u z = u u u = u z u y = u x u x u y u x u x u u x = u y x u z x u x1 u u x2 x3 0 Z R 1-1 R 2-1 = R -1 R = R 2 R 1 = u y1 u y2 u 0 y3 u z1 u z2 u 0 z

52 Properties of Concatenation Not commutative R T R T RTP TRP

53 Properties of Concatenation Associative Does not matter how to multiply ((AB)C)P = (A(BC))P What is the interpretation of these two? Till now we were doing (A(BC))P Right to left Transforming points Coordinate axes same across A, B and C GLOBAL COORDINATES

54 Properties of Concatenation What is the geometric interpretation of (AB)C Left to right Transforms the axes (not the points) LOCAL COORDINATES Results are the same as long as the matrix is (ABC)

55 Local/Global Coordinate Systems GCS: Right to Left: point is first translated and then rotated Local Local Local Local LCS: Left to Right: coordinate first rotated and then translated

56 Local / Global Coordinate Systems GCS: Right to Left: point is first scaled and then rotated LCS: Left to Right: coordinate first rotated and then scaled

57 Projective Transformation Most general form of linear transformation Note that we are interested in points such that w =1 Takes finite points to infinity and vice versa

58 Example Take a circle in (x,y) x 2 +y 2 =1 Show that it goes to parabola Take two parallel lines Show that they go to intersecting lines Intersection lines can become parallel Degree of the polynomials are preserved Since linear

59 Images are two-dimensional patterns of brightness values. Figure from US Navy Manual of Basic Optics and Optical Instruments, prepared by Bureau of Naval Personnel. Reprinted by Dover Publications, Inc., They are formed by the projection of 3D objects.

60 Distant objects appear smaller

61 Parallel lines meet vanishing point

62 Vanishing points H VPL VPR VP 1 VP 2 To different directions correspond different vanishing points VP 3

### Lecture 3: Coordinate Systems and Transformations

Lecture 3: Coordinate Systems and Transformations Topics: 1. Coordinate systems and frames 2. Change of frames 3. Affine transformations 4. Rotation, translation, scaling, and shear 5. Rotation about an

### Lecture 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

### TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

### Affine Transformations. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell

Affine Transformations University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Logistics Required reading: Watt, Section 1.1. Further reading: Foley, et al, Chapter 5.1-5.5. David

### Linear transformations Affine transformations Transformations in 3D. Graphics 2011/2012, 4th quarter. Lecture 5: linear and affine transformations

Lecture 5 Linear and affine transformations Vector transformation: basic idea Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Multiplication of an n n matrix

### Rotation 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

### The calibration problem was discussed in details during lecture 3.

1 2 The calibration problem was discussed in details during lecture 3. 3 Once the camera is calibrated (intrinsics are known) and the transformation from the world reference system to the camera reference

### 2D 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.

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

### Geometric Transformations

Geometric Transformations Moving objects relative to a stationary coordinate system Common transformations: Translation Rotation Scaling Implemented using vectors and matrices Quick Review of Matrix Algebra

### Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

### Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Welcome everybody. We continue the discussion on 2D

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### 2D Geometric Transformations. COMP 770 Fall 2011

2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector multiplication Matrix-matrix multiplication

### The Vector or Cross Product

The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero

### Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns

Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

### Review B: Coordinate Systems

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of hysics 8.02 Review B: Coordinate Systems B.1 Cartesian Coordinates... B-2 B.1.1 Infinitesimal Line Element... B-4 B.1.2 Infinitesimal Area Element...

### Projective 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

### geometric transforms

geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition

### Addition and Subtraction of Vectors

ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### Geometric 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

### Geometric 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

### Geometric Image Transformations

Geometric Image Transformations Part One 2D Transformations Spatial Coordinates (x,y) are mapped to new coords (u,v) pixels of source image -> pixels of destination image Types of 2D Transformations Affine

### The 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

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

### 521493S 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,

### Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### Complex Numbers. w = f(z) z. Examples

omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If

### THREE 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,

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### Geometric Transformations

CS3 INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and 3D CS3 INTRODUCTION TO COMPUTER GRAPHICS Grading Plan to be out Wednesdas one week after the due date CS3 INTRODUCTION TO COMPUTER

### 4. How many integers between 2004 and 4002 are perfect squares?

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

### Parametric Equations and the Parabola (Extension 1)

Parametric Equations and the Parabola (Extension 1) Parametric Equations Parametric equations are a set of equations in terms of a parameter that represent a relation. Each value of the parameter, when

### a 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

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)

0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3-space. This time the outcome will be a vector in 3-space. Definition

### Figure 1: u2 (x,y) D u 1 (x,y) yρ(x, y, z)dv. xρ(x, y, z)dv. M yz = E

Contiune on.7 Triple Integrals Figure 1: [ ] u2 (x,y) f(x, y, z)dv = f(x, y, z)dz da u 1 (x,y) Applications of Triple Integrals Let be a solid region with a density function ρ(x, y, z). Volume: V () =

### Lecture 6 : Aircraft orientation in 3 dimensions

Lecture 6 : Aircraft orientation in 3 dimensions Or describing simultaneous roll, pitch and yaw 1.0 Flight Dynamics Model For flight dynamics & control, the reference frame is aligned with the aircraft

### 3-D Geometric Transformations

3-D Geometric Transformations 3-D Viewing Transformation Projection Transformation 3-D Geometric Transformations Move objects in a 3-D scene Extension of 2-D Affine Transformations Three important ones:

### 5. Orthogonal matrices

L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### Cross 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.

### 6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:

6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an inner-product space. This is the usual n- dimensional

### How to Solve a Cubic Equation Part 1 The Shape of the Discriminant

How to Solve a Cubic Equation Part The Shape of the Discriminant James F. Blinn Microsoft Research blinn@microsoft.com Originally published in IEEE Computer Graphics and Applications May/Jun 006, pages

### Assignment 2: Transformation and Viewing

Assignment : Transformation and Viewing 5-46 Graphics I Spring Frank Pfenning Sample Solution Based on the homework by Kevin Milans kgm@andrew.cmu.edu Three-Dimensional Homogeneous Coordinates (5 pts)

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE. Contents

LINEAR MAPS, THE TOTAL DERIVATIVE AND THE CHAIN RULE ROBERT LIPSHITZ Abstract We will discuss the notion of linear maps and introduce the total derivative of a function f : R n R m as a linear map We will

### B4 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

### Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

### Given a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes:

3 3.1 2D Given a point cloud, polygon, or sampled parametric curve, we can use transformations for several purposes: 1. Change coordinate frames (world, window, viewport, device, etc). 2. Compose objects

### 9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes

The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is so-called because when the scalar product of

### Geometry for Computer Graphics

Computer Graphics and Visualisation Geometry for Computer Graphics Student Notes Developed by F Lin K Wyrwas J Irwin C Lilley W T Hewitt T L J Howard Computer Graphics Unit Manchester Computing Centre

### A 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

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus

Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations

### CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

### Dot 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

### Orthogonal 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

### 6 Systems of Linear Equations

6 Systems of Linear Equations A system of equations of the form or is called a linear system of equations. x + y = 7 x y = 5p 6q + r = p + q 5r = 7 6p q + r = Definition. An equation involving certain

### 1 Spherical Kinematics

ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

### IMAGE FORMATION. Antonino Furnari

IPLab - Image Processing Laboratory Dipartimento di Matematica e Informatica Università degli Studi di Catania http://iplab.dmi.unict.it IMAGE FORMATION Antonino Furnari furnari@dmi.unict.it http://dmi.unict.it/~furnari

### Linear Algebra: Vectors

A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector

### 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

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

### θ = 45 θ = 135 θ = 225 θ = 675 θ = 45 θ = 135 θ = 225 θ = 675 Trigonometry (A): Trigonometry Ratios You will learn:

Trigonometr (A): Trigonometr Ratios You will learn: () Concept of Basic Angles () how to form simple trigonometr ratios in all 4 quadrants () how to find the eact values of trigonometr ratios for special

### Linear 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

### Orthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then

Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal

### Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj

Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that

### Summary: Transformations. Lecture 14 Parameter Estimation Readings T&V Sec 5.1-5.3. Parameter Estimation: Fitting Geometric Models

Summary: Transformations Lecture 14 Parameter Estimation eadings T&V Sec 5.1-5.3 Euclidean similarity affine projective Parameter Estimation We will talk about estimating parameters of 1) Geometric models

### BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 2009 Page 1 of?? ENG rectangle. Rectangle

Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and

### Fundamentals of Computer Animation

Fundamentals of Computer Animation Quaternions as Orientations () page 1 Visualizing a Unit Quaternion Rotation in 4D Space ( ) = w + x + y z q = Norm q + q = q q [ w, v], v = ( x, y, z) w scalar q =,

### 13 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

### Section 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

### MATRIX 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 +

### Theorem 5. The composition of any two symmetries in a point is a translation. More precisely, S B S A = T 2

Isometries. Congruence mappings as isometries. The notion of isometry is a general notion commonly accepted in mathematics. It means a mapping which preserves distances. The word metric is a synonym to

### 3D Viewing. Chapter 7. Projections. 3D clipping. OpenGL viewing functions and clipping planes

3D Viewing Chapter 7 Projections 3D clipping OpenGL viewing functions and clipping planes 1 Projections Parallel Perspective Coordinates are transformed along parallel lines Relative sizes are preserved

### Deriving character tables: Where do all the numbers come from?

3.4. Characters and Character Tables 3.4.1. Deriving character tables: Where do all the numbers come from? A general and rigorous method for deriving character tables is based on five theorems which in

### Rotations and Quaternions

Uniersity of British Columbia CPSC 414 Computer Graphics Rotations and Quaternions Week 9, Wed 29 Oct 2003 Tamara Munzner 1 Uniersity of British Columbia CPSC 414 Computer Graphics Clipping recap Tamara

### www.sakshieducation.com

LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### Affine 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

### Differentiation of vectors

Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

### Section 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

### Epipolar Geometry Prof. D. Stricker

Outline 1. Short introduction: points and lines Epipolar Geometry Prof. D. Stricker 2. Two views geometry: Epipolar geometry Relation point/line in two views The geometry of two cameras Definition of the

### Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method. Version 2 CE IIT, Kharagpur

Module Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version CE IIT, haragpur esson 7 The Direct Stiffness Method: Beams Version CE IIT, haragpur Instructional Objectives