Transformations II. Arbitrary 3D Rotation. What is its inverse? What is its transpose? Can we constructively elucidate this relationship?
|
|
|
- Louise Dorsey
- 5 years ago
- Views:
Transcription
1 Utah School of Computing Fall 25 Transformations II CS46 Computer Graphics From Rich Riesenfeld Fall 25 Arbitrar 3D Rotation What is its inverse? What is its transpose? Can we constructivel elucidate this relationship? 2 Computer Graphics CS46
2 Utah School of Computing Fall 25 Rotate + about ais a:r a (+ a 3 First, Rotate about b : R ( a Now in the (--plane 4 Computer Graphics CS46 2
3 Utah School of Computing Fall 25 Then Rotate about b + : R ( a Rotate in the (--plane 5 Now, + Rotation about -ais: R (+ a Now aligned with -ais 6 Computer Graphics CS46 3
4 Utah School of Computing Fall 25 Then rotate about b - : R (- a Rotate again in the (--plane 7 Now, + Rotation about b : R ( a Now to original position of a 8 Computer Graphics CS46 4
5 Utah School of Computing Fall 25 We Effected + rotation about Arbitrar ais a:r a (+ a 9 We Effected + rotation about Arbitrar ais a:r a (+ a ( R ( R ( R ( ( R ( R R Utah School of Computing Computer Graphics CS46 5
6 Utah School of Computing Fall 25 Rotation about Arbitrar Ais Rotation about a-ais effected b (nonunique composition of 5 elementar rotations We show arbitrar rotation as succession of 5 rotations about principal aes Ra ( ( ( ( ( ( ( ( ( In matri terms, R a (+ = R ( R ( R ( R ( ( R 2 Computer Graphics CS46 6
7 Utah School of Computing Fall 25 Similarl, R a - (+ = R a (-, so Ra ( ( ( ( ( ( ( ( ( R ( 3 t A Recall, [AB] t = B t A t Consequentl, for R t M t R t t. t t R M R M R R t R t A R t M R, because, M t R 4 Computer Graphics CS46 7
8 Utah School of Computing Fall 25 It follows directl that, R t S t M SR t R t S t M t SR 5 R ( R t a a( Ra R t ( ( ( ( ( ( ( ( ( ( Utah School of Computing 6 Computer Graphics CS46 8
9 Utah School of Computing Fall 25 Computer Graphics CS ( ( ( ( ( ( ( ( ( Ra ( R Similarl, R a - (+ = R a (-, so 8 ( ( ( ( ( ( ( ( ( Ra ( R In matri terms, R a (+ = ( R ( R ( R ( R
10 Utah School of Computing Fall 25 Computer Graphics CS46 9 ( ( R t a Ra Constructivel, we have shown, This will be useful later 2 3D Translation in ( d d d T
11 Utah School of Computing Fall 25 Computer Graphics CS46 2 3D Translation in ( d d d T 22 3D Translation in ( d d d T
12 Utah School of Computing Fall 25 Computer Graphics CS D Shear in -direction ( a a a Sh 24 3D Shear in -direction ( b b b Sh
13 Utah School of Computing Fall 25 Computer Graphics CS D Shears: Clamp a Principal Plane, shear in other 2 DoFs ( a a a Sh
14 Utah School of Computing Fall 25 Computer Graphics CS46 4 Spring 23 Utah School of Computing Utah School of Computing 27 3D Shear in -direction ( b b b Sh 28 3D Shear in -direction ( c c c Sh
15 Utah School of Computing Fall 25 Computer Graphics CS D Shear in -direction ( d d d Sh ( d d d Sh
16 Utah School of Computing Fall 25 Computer Graphics CS D Shear in -direction ( c c c Sh 32 3D Shear in -direction ( e e e Sh
17 Utah School of Computing Fall 25 Computer Graphics CS D Shear in ( e e e Sh 34 3D Shear in ( f f f Sh
18 Utah School of Computing Fall 25 What About Elementar Inverses? Scale Shear Rotation Translation 35 Scale Inverse 36 Computer Graphics CS46 8
19 Utah School of Computing Fall 25 Computer Graphics CS Shear Inverse b b a a 38 Shear Inverse b b a a
20 Utah School of Computing Fall 25 Computer Graphics CS Rotation Inverse - - (- (- -(- (- - 4 Rotation Inverse ( ( ( (
21 Utah School of Computing Fall 25 Rotation Inverse - - What is special about this? 4 Rotation Inverse - - What is special about this? Orthonormal transformations: the inverse is the transpose! 42 Computer Graphics CS46 2
22 Utah School of Computing Fall 25 Computer Graphics CS Translation Inverse ( d d d d 44 Translation Inverse d d
23 Utah School of Computing Fall 25 Computer Graphics CS Want the RHR to Work j i k i k j k j i j k i 46 3D Positive Rotations
24 Utah School of Computing Fall 25 Xforms as Change in Coordinate Ss Useful in man situations Use most natural coordination sstem locall Tie things together in a global sstem (object space to world space 47 Eample Computer Graphics CS46 24
25 Utah School of Computing Fall 25 M takes a point Eample is the transformation that i j p ( j in coordinate sstem j and converts it to a point in coordinate sstem i p ( i 49 Eample p ( i M i j p ( j p ( j M j k p ( k M i k M i j M j k 5 Computer Graphics CS46 25
26 Utah School of Computing Fall 25 Eample Eample M T 2 (4,2 M T(2,3 S 23 (/2,/2 M R (45 T(6.7,.8 34 Spring 23 Utah School of Computing 52 Computer Graphics CS46 26
27 Utah School of Computing Fall 25 Eample M R(45 T(6.7, M T 2 (4,2 2 3 M T(2,3 S(/2,/ Recall the Following ( AB B A 54 Computer Graphics CS46 27
28 Utah School of Computing Fall 25 Since M i j M j i M M M T ( 4, 2 2 S(2,2 T( 2, 3 32 R( 45 T( 6.7, Eample M R( 45 T( 6.7, M T ( 4, M S(2,2 T( 2, Computer Graphics CS46 28
29 Utah School of Computing Fall 25 Change of Coordinate Sstem Describe the old coordinate sstem in terms of the new one. 57 Change of Coordinate Sstem Move to the new coordinate sstem and describe the one old. Old is a negative rotation of the new. 58 Computer Graphics CS46 29
30 Utah School of Computing Fall 25 Pre 3. OpenGL Matrices In Pre 3. OpenGL matrices were part of the state Multiple tpes Model-View (GL_MODELVIEW Projection (GL_PROJECTION Teture (GL_TEXTURE Color(GL_COLOR Single set of functions for manipulation Select which to manipulated b glmatrimode(gl_modelview; glmatrimode(gl_projection; Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Wh Deprecation Functions were based on carring out the operations on the CPU as part of the fied function pipeline Current model-view and projection matrices were automaticall applied to all vertices ug CPU We will use the notion of a current transformation matri with the understanding that it ma be applied in the shaders Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Computer Graphics CS46 3
31 Utah School of Computing Fall 25 Current Transformation Matri (CTM In HTML5/Javascript, conceptuall there is a 3 3 homogeneous coordinate matri, the current transformation matri (CTM that is part of the state and is applied to all vertices that pass down the pipeline In WebGL, conceptuall there is a 4 4 homogeneous coordinate matri, the current transformation matri (CTM that is part of the state and is applied to all vertices that pass down the pipeline The CTM is defined in the user program and loaded into a transformation unit vertices p C CTM p =Cp vertices Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 CTM operations The CTM can be altered either b loading a new CTM or b postmutiplication Load an identit matri: C I Load an arbitrar matri: C M Load a translation matri: C T Load a rotation matri: C R Load a scaling matri: C S Postmultipl b an arbitrar matri: C CM Postmultipl b a translation matri: C CT Postmultipl b a rotation matri: C CR Postmultipl b a scaling matri: C CS Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Computer Graphics CS46 3
32 Utah School of Computing Fall 25 Rotation about a Fied Point Start with identit matri: C I Move fied point to origin: C CT Rotate: C CR Move fied point back: C CT - Result: C = TR T which is backwards. This result is a consequence of doing postmultiplications. Let s tr again. Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Reverg the Order We want C = T R T so we must do the operations in the following order C I C CT - C CR C CT Each operation corresponds to one function call in the program. Note that the last operation specified is the first eecuted in the program Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Computer Graphics CS46 32
33 Utah School of Computing Fall 25 CTM in Canvas In 2D, World Space is our canvas. We will use transformation matrices to place objects into the 2D scene. We will use transformations (matrices to move the objects in the scene. Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 CTM in WebGL OpenGL had a model-view and a projection matri in the pipeline which were concatenated together to form the CTM We will emulate this process Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Computer Graphics CS46 33
34 Utah School of Computing Fall 25 USE MODEL->Object/World->Ee- >CLIP->NDC->Window Utah School of Computing Ug the ModelView Matri In WebGL, the model-view matri is used to Position the camera Can be done b rotations and translations but is often easier to use the lookat function in MV.js Build models of objects and put them in world-space The projection matri is used to define the view volume and to select a camera lens Although these matrices are no longer part of the OpenGL state, it is usuall a good strateg to create them in our own applications q = P*MV*p Computer Graphics CS46 34
35 Utah School of Computing Fall 25 Rotation, Translation, Scaling Create an identit matri [note, MV.js initialies to identit]: var m = mat4(; Multipl on right b rotation matri of theta in degrees where (v, v, v define ais of rotation var r = rotate(theta, v, v, v m = mult(m, r; Also have rotatex, rotatey, rotatez Do same with translation and scaling: var s = scale( s, s, s var t = translate(d, d, d; m = mult(s, t; Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Eample Rotation about ais b 3 degrees with a fied point of (., 2., 3. var m = mult(translate(., 2., 3., rotate(3.,.,.,.; m = mult(m, translate(-., -2., -3.; Remember that last matri specified in the program is the first applied Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Computer Graphics CS46 35
36 Utah School of Computing Fall 25 Arbitrar Matrices Can load and multipl b matrices defined in the application program Matrices are stored as one dimensional arra of 6 elements b MV.js but can be treated as 4 4 matrices in row major order OpenGL wants column major data gl.uniformmatri4f has a parameter for automatic transpose but it must be set to false. flatten function converts to column major order which is required b WebGL functions Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Matri Stacks In man situations we want to save transformation matrices for use later Traverg hierarchical data structures (Chapter 9 Pre 3. OpenGL maintained stacks for each tpe of matri Eas to create the same functionalit in JS push and pop are part of Arra object var stack = [ ] stack.push(modelviewmatri; modelviewmatri = stack.pop(; Angel and Shreiner: Interactive Computer Graphics 7E Addison-Wesle 25 Computer Graphics CS46 36
Introduction to Computer Graphics with WebGL
Introduction to Computer Graphics with WebGL Ed Angel Professor Emeritus of Computer Science Founding Director, Arts, Research, Technology and Science Laboratory University of New Mexico WebGL Transformations
CS452/552; EE465/505. Transformations
CS452/552; EE465/55 Transformations 1-29-15 Outline! Transformations Read: Angel, Chapter 4 (study cube.html/cube.js example) Helpful links: Linear Algebra: Khan Academy Lab1 is posted on github, due:
GLOBAL EDITION. Interactive Computer Graphics. A Top-Down Approach with WebGL SEVENTH EDITION. Edward Angel Dave Shreiner
GLOBAL EDITION Interactive Computer Graphics A Top-Down Approach with WebGL SEVENTH EDITION Edward Angel Dave Shreiner This page is intentionall left blank. 4.10 Concatenation of Transformations 219 in
Computer Graphics CS 543 Lecture 5 (Part 2) Implementing Transformations
Computer Graphics CS 543 Lecture 5 (Part 2) Implementing Transformations Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Objectives Learn how to implement transformations
Introduction to Computer Graphics with WebGL
1 Introduction to Computer Graphics with WebGL Ed Angel Transformations General Transformations A transformation maps points to other points and/or vectors to other vectors v=t(u) Q=T(P) 2 Affine Transformations
Transformations. Rotation and Scaling
Transformations In OpenGL, transformation are performed in the opposite order they are called 4 3 2 1 translate(1., 1.,.); rotatez(45.); scale(2., 2.,.); DrawSquare(.,., 1.); 4 3 2 1 scale(2., 2.,.); rotatez(45.);
Introduction to Computer Graphics with WebGL
Introduction to Computer Graphics with WebGL Ed Angel Professor Emeritus of Computer Science Founding Director, Arts, Research, Technology and Science Laboratory University of New Mexico 1 Computer Viewing
Computer Graphics (CS 4731) Lecture 11: Implementing Transformations. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics (CS 47) Lecture : Implementing Transformations Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Objectives Learn how to implement transformations in OpenGL
CSE528 Computer Graphics: Theory, Algorithms, and Applications
CSE528 Computer Graphics: Theor, Algorithms, and Applications Hong Qin State Universit of New York at Ston Brook (Ston Brook Universit) Ston Brook, New York 794--44 Tel: (63)632-845; Fa: (63)632-8334 qin@cs.sunsb.edu
CS 4204 Computer Graphics
CS 424 Computer Graphics 2D Transformations Yong Cao Virginia Tech References: Introduction to Computer Graphics course notes by Doug Bowman Interactive Computer Graphics, Fourth Edition, Ed Angle Transformations
4. Two Dimensional Transformations
4. Two Dimensional Transformations CS362 Introduction to Computer Graphics Helena Wong, 2 In man applications, changes in orientations, sizes, and shapes are accomplished with geometric transformations
Transformations. CS 537 Interactive Computer Graphics Prof. David E. Breen Department of Computer Science
Transformations CS 537 Interactive Computer Graphics Prof. David E. Breen Department of Computer Science 1 Objectives Introduce standard transformations - Rotation - Translation - Scaling - Shear Derive
Objectives. transformation. General Transformations. Affine Transformations. Notation. Pipeline Implementation. Introduce standard transformations
Objectives Transformations CS Interactive Computer Graphics Prof. David E. Breen Department of Computer Science Introduce standard transformations - Rotation - Translation - Scaling - Shear Derive homogeneous
CS Computer Graphics: Transformations & The Synthetic Camera
CS 543 - Computer Graphics: Transformations The Snthetic Camera b Robert W. Lindeman gogo@wpi.edu (with help from Emmanuel Agu ;-) Introduction to Transformations A transformation changes an objects Size
Computer Graphics (CS 4731) Lecture 11: Implementing Transformations. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics (CS 47) Lecture : Implementing Transformations Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Objectives Learn how to implement transformations in OpenGL
Transformations II. Week 2, Wed Jan 17
Universit of British Columbia CPSC 34 Computer Graphics Jan-Apr 27 Tamara Munzner Transformations II Week 2, Wed Jan 7 http://www.ugrad.cs.ubc.ca/~cs34/vjan27 Readings for Jan 5-22 FCG Chap 6 Transformation
Viewing and Projection
Viewing and Projection Sheelagh Carpendale Camera metaphor. choose camera position 2. set up and organie objects 3. choose a lens 4. take the picture View Volumes what gets into the scene perspective view
Modeling Transformations
Modeling Transformations Thomas Funkhouser Princeton Universit CS 426, Fall 2 Modeling Transformations Specif transformations for objects Allos definitions of objects in on coordinate sstems Allos use
What does OpenGL do?
Theor behind Geometrical Transform What does OpenGL do? So the user specifies a lot of information Ee Center Up Near, far, UP EE Left, right top, bottom, etc. f b CENTER left right top bottom What does
Lecture 4: Viewing. Topics:
Lecture 4: Viewing Topics: 1. Classical viewing 2. Positioning the camera 3. Perspective and orthogonal projections 4. Perspective and orthogonal projections in OpenGL 5. Perspective and orthogonal projection
Modeling Transformations
Transformations Transformations Specif transformations for objects o Allos definitions of objects in on coordinate sstems o Allos use of object definition multiple times in a scene Adam Finkelstein Princeton
1. We ll look at: Types of geometrical transformation. Vector and matrix representations
Tob Howard COMP272 Computer Graphics and Image Processing 3: Transformations Tob.Howard@manchester.ac.uk Introduction We ll look at: Tpes of geometrical transformation Vector and matri representations
Modeling Transformations
שיעור 3 גרפיקה ממוחשבת תשס"ח ב ליאור שפירא Modeling Transformations Heavil based on: Thomas Funkhouser Princeton Universit CS 426, Fall 2 Modeling Transformations Specif transformations for objects Allows
(x, y) (ρ, θ) ρ θ. Polar Coordinates. Cartesian Coordinates
Coordinate Sstems Point Representation in two dimensions Cartesian Coordinates: (; ) Polar Coordinates: (; ) (, ) ρ θ (ρ, θ) Cartesian Coordinates Polar Coordinates p = CPS1, 9: Computer Graphics D Geometric
Transformations III. Week 2, Fri Jan 19
Universit of British Columbia CPSC 34 Computer Graphics Jan-Apr 2007 Tamara Munzner Transformations III Week 2, Fri Jan 9 http://www.ugrad.cs.ubc.ca/~cs34/vjan2007 Readings for Jan 5-22 FCG Chap 6 Transformation
Computer Graphics. P04 Transformations. Aleksandra Pizurica Ghent University
Computer Graphics P4 Transformations Aleksandra Pizurica Ghent Universit Telecommunications and Information Processing Image Processing and Interpretation Group Transformations in computer graphics Goal:
CS559: Computer Graphics
CS559: Computer Graphics Lecture 8: 3D Transforms Li Zhang Spring 28 Most Slides from Stephen Chenne Finish Color space Toda 3D Transforms and Coordinate sstem Reading: Shirle ch 6 RGB and HSV Green(,,)
2D Transformations. Why Transformations. Translation 4/17/2009
4/7/9 D Tansfomations Wh Tansfomations Coodinate sstem tansfomations Placing objects in the wold Move/animate the camea fo navigation Dawing hieachical chaactes Animation Tanslation + d 5,4 + d,3 d 4,
CSE328 Fundamentals of Computer Graphics: Theory, Algorithms, and Applications
CSE328 Fundamentals of Computer Graphics: Theor, Algorithms, and Applications Hong in State Universit of New York at Ston Brook (Ston Brook Universit) Ston Brook, New York 794-44 Tel: (63)632-845; Fa:
What and Why Transformations?
2D transformations What and Wh Transformations? What? : The geometrical changes of an object from a current state to modified state. Changing an object s position (translation), orientation (rotation)
Computer Graphics. Geometric Transformations
Computer Graphics Geometric Transformations Contents coordinate sstems scalar values, points, vectors, matrices right-handed and left-handed coordinate sstems mathematical foundations transformations mathematical
CSC 470 Computer Graphics
CSC 470 Computer Graphics Transformations of Objects CSC 470 Computer Graphics, Dr.N. Georgieva, CSI/CUNY 1 Transformations of objects - 2D CSC 470 Computer Graphics, Dr.N. Georgieva, CSI/CUNY 2 Using
Computer Viewing. CS 537 Interactive Computer Graphics Prof. David E. Breen Department of Computer Science
Computer Viewing CS 537 Interactive Computer Graphics Prof. David E. Breen Department of Computer Science 1 Objectives Introduce the mathematics of projection Introduce OpenGL viewing functions Look at
CS770/870 Spring 2017 Transformations
CS770/870 Spring 2017 Transformations Coordinate sstems 2D Transformations Homogeneous coordinates Matrices, vectors, points Coordinate Sstems Coordinate sstems used in graphics Screen coordinates: the
Notes. University of British Columbia
Notes Drop-bo is no. 14 You can hand in our assignments Assignment 0 due Fri. 4pm Assignment 1 is out Office hours toda 16:00 17:00, in lab or in reading room Uniersit of Uniersit of Chapter 4 - Reminder
High Dimensional Rendering in OpenGL
High Dimensional Rendering in OpenGL Josh McCo December, 2003 Description of Project Adding high dimensional rendering capabilit to the OpenGL graphics programming environment is the goal of this project
More on Transformations. COS 426, Spring 2019 Princeton University
More on Transformations COS 426, Spring 2019 Princeton Universit Agenda Grab-bag of topics related to transformations: General rotations! Euler angles! Rodrigues s rotation formula Maintaining camera transformations!
Computer Graphics. Geometric Transformations
Contents coordinate sstems scalar values, points, vectors, matrices right-handed and left-handed coordinate sstems mathematical foundations transformations mathematical descriptions of geometric changes,
Image Warping. Some slides from Steve Seitz
Image Warping http://www.jeffre-martin.com Some slides from Steve Seitz 5-463: Computational Photograph Aleei Efros, CMU, Spring 2 Image Transformations image filtering: change range of image g() = T(f())
Modeling Transformations
Modeling Transformations Michael Kazhdan (601.457/657) HB Ch. 5 FvDFH Ch. 5 Overview Ra-Tracing so far Modeling transformations Ra Tracing Image RaTrace(Camera camera, Scene scene, int width, int heigh,
Modeling Transformations Revisited
Modeling Transformations Revisited Basic 3D Transformations Translation Scale Shear Rotation 3D Transformations Same idea as 2D transformations o Homogeneous coordinates: (,,z,w) o 44 transformation matrices
CS 428: Fall Introduction to. Viewing and projective transformations. Andrew Nealen, Rutgers, /23/2009 1
CS 428: Fall 29 Introduction to Computer Graphics Viewing and projective transformations Andrew Nealen, Rutgers, 29 9/23/29 Modeling and viewing transformations Canonical viewing volume Viewport transformation
Homogeneous Coordinates
COMS W4172 3D Math 2 Steven Feiner Department of Computer Science Columbia Universit New York, NY 127 www.cs.columbia.edu/graphics/courses/csw4172 Februar 1, 218 1 Homogeneous Coordinates w X W Y X W Y
Image Warping : Computational Photography Alexei Efros, CMU, Fall Some slides from Steve Seitz
Image Warping http://www.jeffre-martin.com Some slides from Steve Seitz 5-463: Computational Photograph Aleei Efros, CMU, Fall 2 Image Transformations image filtering: change range of image g() T(f())
Order of Transformations
Order of Transformations Because the same transformation is applied to many vertices, the cost of forming a matrix M=ABCD is not significant compared to the cost of computing Mp for many vertices p Note
Modeling Transformations
Modeling Transformations Michael Kazhdan (601.457/657) HB Ch. 5 FvDFH Ch. 5 Announcement Assignment 2 has been posted: Due: 10/24 ASAP: Download the code and make sure it compiles» On windows: just build
6. Modelview Transformations
6. Modelview Transformations Transformation Basics Transformations map coordinates from one frame of reference to another through matri multiplications Basic transformation operations include: - translation
Scene Graphs & Modeling Transformations COS 426
Scene Graphs & Modeling Transformations COS 426 3D Object Representations Points Range image Point cloud Surfaces Polgonal mesh Subdivision Parametric Implicit Solids Voels BSP tree CSG Sweep High-level
Image Warping. Some slides from Steve Seitz
Image Warping http://www.jeffre-martin.com Some slides from Steve Seitz 5-463: Computational Photograph Aleei Efros, CMU, Fall 26 Image Warping image filtering: change range of image g() T(f()) f T f image
Image Warping (Szeliski Sec 2.1.2)
Image Warping (Szeliski Sec 2..2) http://www.jeffre-martin.com CS94: Image Manipulation & Computational Photograph Aleei Efros, UC Berkele, Fall 7 Some slides from Steve Seitz Image Transformations image
Fachhochschule Regensburg, Germany, February 15, 2017
s Operations Fachhochschule Regensburg, Germany, February 15, 2017 s Motivating Example s Operations To take a photograph of a scene: Set up your tripod and point camera at the scene (Viewing ) Position
Transforms 1 Christian Miller CS Fall 2011
Transforms 1 Christian Miller CS 354 - Fall 2011 Transformations What happens if you multiply a square matrix and a vector together? You get a different vector with the same number of coordinates These
2D Transformations. 7 February 2017 Week 5-2D Transformations 1
2D Transformations 7 Februar 27 Week 5-2D Transformations Matri math Is there a difference between possible representations? a c b e d f ae bf ce df a c b d e f ae cf be df a b c d e f ae bf ce df 7 Februar
CS F-07 Objects in 2D 1
CS420-2010F-07 Objects in 2D 1 07-0: Representing Polgons We want to represent a simple polgon Triangle, rectangle, square, etc Assume for the moment our game onl uses these simple shapes No curves for
General Purpose Computation (CAD/CAM/CAE) on the GPU (a.k.a. Topics in Manufacturing)
ME 29-R: General Purpose Computation (CAD/CAM/CAE) on the GPU (a.k.a. Topics in Manufacturing) Sara McMains Spring 29 lecture 2 Toda s GPU eample: moldabilit feedback Two-part mold [The Complete Sculptor
Uses of Transformations. 2D transformations Homogeneous coordinates. Transformations. Transformations. Transformations. Transformations and matrices
Uses of Transformations 2D transformations Homogeneous coordinates odeling: position and resie parts of a comple model; Viewing: define and position the virtual camera Animation: define how objects move/change
Computer Graphics. Transformation
(SBE 36) Dr. Aman Eldeib Spring 2 SBE 36 i a fundamental corner tone of computer graphic and i a central to OpenGL a well a mot other graphic tem.(2d and 3D ) Given an object, tranformation i to change
Image Warping CSE399b, Spring 07 Computer Vision
Image Warping CSE399b, Spring 7 Computer Vision http://maps.a9.com http://www.cs.ubc.ca/~mbrown/autostitch/autostitch.html http://www.cs.ubc.ca/~mbrown/autostitch/autostitch.html Autostiching on A9.com
3-Dimensional Viewing
CHAPTER 6 3-Dimensional Vieing Vieing and projection Objects in orld coordinates are projected on to the vie plane, hich is defined perpendicular to the vieing direction along the v -ais. The to main tpes
Classical and Computer Viewing. Adapted From: Ed Angel Professor of Emeritus of Computer Science University of New Mexico
Classical and Computer Viewing Adapted From: Ed Angel Professor of Emeritus of Computer Science University of New Mexico Planar Geometric Projections Standard projections project onto a plane Projectors
Recall: Function Calls to Create Transform Matrices
Reall: Fntion Calls to Create Transform Matries Previosl made fntion alls to generate 44 matries for identit, translate, sale, rotate transforms Pt transform matri into CTM Eample mat4 m = Identit(; CTM
To Do. Demo (Projection Tutorial) Motivation. What we ve seen so far. Outline. Foundations of Computer Graphics (Fall 2012) CS 184, Lecture 5: Viewing
Foundations of Computer Graphics (Fall 0) CS 84, Lecture 5: Viewing http://inst.eecs.berkele.edu/~cs84 To Do Questions/concerns about assignment? Remember it is due Sep. Ask me or TAs re problems Motivation
Announcement. Project 1 has been posted online and in dropbox. Due: 11:59:59 pm, Friday, October 14
Announcement Project 1 has been posted online and in dropbox Due: 11:59:59 pm, Friday, October 14 Project 1: Interactive Viewing of Two Teapots How to create a teapot? Before OpenGL 3., glutsolidteapot
Image warping. image filtering: change range of image. image warping: change domain of image g(x) = f(h(x)) h(y)=0.5y+0.5. h([x,y])=[x,y/2] f h
![Image warping. image filtering: change range of image. image warping: change domain of image g(x) = f(h(x)) h(y)=0.5y+0.5. h([x,y])=[x,y/2] f h](/thumbs/71/65976347.jpg "Image warping. image filtering: change range of image. image warping: change domain of image g(x) = f(h(x)) h(y)=0.5y+0.5. h([x,y])=[x,y/2] f h")
Image warping Image warping image filtering: change range of image g() () = h(f()) h(f()) f h g h()=0.5+0.5 image warping: change domain of image g() = f(h()) f h g h([,])=[,/2] Parametric (global) warping
Chap 7, 2009 Spring Yeong Gil Shin
Three-Dimensional i Viewingi Chap 7, 29 Spring Yeong Gil Shin Viewing i Pipeline H d fi i d? How to define a window? How to project onto the window? Rendering "Create a picture (in a snthetic camera) Specification
Two Dimensional Viewing
Two Dimensional Viewing Dr. S.M. Malaek Assistant: M. Younesi Two Dimensional Viewing Basic Interactive Programming Basic Interactive Programming User controls contents, structure, and appearance of objects
Computer Graphics (CS 543) Lecture 6a: Hierarchical 3D Models. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)
Computer Graphics (CS 543) Lecture 6a: Hierarchical 3D Models Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Instance Transformation Start with unique object (a symbol)
Determining the 2d transformation that brings one image into alignment (registers it) with another. And
Last two lectures: Representing an image as a weighted combination of other images. Toda: A different kind of coordinate sstem change. Solving the biggest problem in using eigenfaces? Toda Recognition
3D Geometry and Camera Calibration
3D Geometr and Camera Calibration 3D Coordinate Sstems Right-handed vs. left-handed 2D Coordinate Sstems ais up vs. ais down Origin at center vs. corner Will often write (u, v) for image coordinates v
GEOMETRIC TRANSFORMATIONS AND VIEWING
GEOMETRIC TRANSFORMATIONS AND VIEWING 2D and 3D 1/44 2D TRANSFORMATIONS HOMOGENIZED Transformation Scaling Rotation Translation Matrix s x s y cosθ sinθ sinθ cosθ 1 dx 1 dy These 3 transformations are
What is Perspective?
Fall 25 M ss =M screen * M ersective * M view What is Persective? A mechanism for ortraing 3D in 2D True Persective corresons to rojection onto a lane True Persective corresons to an ieal camera image
CS452/552; EE465/505. Models & Viewing
CS452/552; EE465/505 Models & Viewing 2-03 15 Outline! Building Polygonal Models Vertex lists; gl.drawarrays( ) Edge lists: gl.drawelements( )! Viewing Classical Viewing Read: Viewing in Web3D Angel, Section
One or more objects A viewer with a projection surface Projectors that go from the object(s) to the projection surface
Classical Viewing Viewing requires three basic elements One or more objects A viewer with a projection surface Projectors that go from the object(s) to the projection surface Classical views are based
CS 450: COMPUTER GRAPHICS 2D TRANSFORMATIONS SPRING 2016 DR. MICHAEL J. REALE
CS 45: COMUTER GRAHICS 2D TRANSFORMATIONS SRING 26 DR. MICHAEL J. REALE INTRODUCTION Now that we hae some linear algebra under our resectie belts, we can start ug it in grahics! So far, for each rimitie,
CS 4731/543: Computer Graphics Lecture 5 (Part I): Projection. Emmanuel Agu
CS 4731/543: Computer Graphics Lecture 5 (Part I): Projection Emmanuel Agu 3D Viewing and View Volume Recall: 3D viewing set up Projection Transformation View volume can have different shapes (different
Computer Viewing. CITS3003 Graphics & Animation. E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley
Computer Viewing CITS3003 Graphics & Animation E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012 1 Objectives Introduce the mathematics of projection Introduce OpenGL viewing
Viewing/Projection IV. Week 4, Fri Jan 29
Universit of British Columbia CPSC 314 Computer Graphics Jan-Apr 2010 Tamara Munner Viewing/Projection IV Week 4, Fri Jan 29 http://www.ugrad.cs.ubc.ca/~cs314/vjan2010 News etra TA office hours in lab
1/29/13. Computer Graphics. Transformations. Simple Transformations
/29/3 Computer Graphics Transformations Simple Transformations /29/3 Contet 3D Coordinate Sstems Right hand (or counterclockwise) coordinate sstem Left hand coordinate sstem Not used in this class and
Graphics pipeline and transformations. Composition of transformations
Graphics pipeline and transformations Composition of transformations Order matters! ( rotation * translation translation * rotation) Composition of transformations = matrix multiplication: if T is a rotation
Viewing. Cliff Lindsay, Ph.D. WPI
Viewing Cliff Lindsa, Ph.D. WPI Building Virtual Camera Pipeline l Used To View Virtual Scene l First Half of Rendering Pipeline Related To Camera l Takes Geometr From ApplicaHon To RasteriaHon Stages
3D Viewing. Introduction to Computer Graphics Torsten Möller. Machiraju/Zhang/Möller
3D Viewing Introduction to Computer Graphics Torsten Möller Machiraju/Zhang/Möller Reading Chapter 4 of Angel Chapter 13 of Hughes, van Dam, Chapter 7 of Shirley+Marschner Machiraju/Zhang/Möller 2 Objectives
CS 475 / CS 675 Computer Graphics. Lecture 7 : The Modeling-Viewing Pipeline
CS 475 / CS 675 Computer Graphics Lecture 7 : The Modeling-Viewing Pipeline Taonom Planar Projections Parallel Perspectie Orthographic Aonometric Oblique Front Top Side Trimetric Dimetric Isometric Caalier
3D Viewing. CMPT 361 Introduction to Computer Graphics Torsten Möller. Machiraju/Zhang/Möller
3D Viewing CMPT 361 Introduction to Computer Graphics Torsten Möller Reading Chapter 4 of Angel Chapter 6 of Foley, van Dam, 2 Objectives What kind of camera we use? (pinhole) What projections make sense
3D Viewing Episode 2
3D Viewing Episode 2 1 Positioning and Orienting the Camera Recall that our projection calculations, whether orthographic or frustum/perspective, were made with the camera at (0, 0, 0) looking down the
Affine Transformations in 3D
Affine Transformations in 3D 1 Affine Transformations in 3D 1 Affine Transformations in 3D General form 2 Translation Elementary 3D Affine Transformations 3 Scaling Around the Origin 4 Along x-axis Shear
CS 432 Interactive Computer Graphics
CS 432 Interactive Computer Graphics Lecture 4 3D Viewing Matt Burlick - Drexel University - CS 432 1 Reading Angel Chapters 3-4 Red Book Chapter 5, Appendix E Matt Burlick - Drexel University - CS 432
Computer Graphics. 2D transformations. Transforma3ons in computer graphics. Overview. Basic classes of geometric transforma3ons
Transforma3ons in computer graphics omputer Graphics Transforma3ons leksandra Piurica Goal: introduce methodolog to hange coordinate sstem Move and deform objects Principle: transforma3ons are applied
3D Viewing Episode 2
3D Viewing Episode 2 1 Positioning and Orienting the Camera Recall that our projection calculations, whether orthographic or frustum/perspective, were made with the camera at (0, 0, 0) looking down the
Motivation. What we ve seen so far. Demo (Projection Tutorial) Outline. Projections. Foundations of Computer Graphics
Foundations of Computer Graphics Online Lecture 5: Viewing Orthographic Projection Ravi Ramamoorthi Motivation We have seen transforms (between coord sstems) But all that is in 3D We still need to make
Realtime 3D Computer Graphics & Virtual Reality. Viewing
Realtime 3D Computer Graphics & Virtual Realit Viewing Transformation Pol. Per Verte Pipeline CPU DL Piel Teture Raster Frag FB v e r t e object ee clip normalied device Modelview Matri Projection Matri
3D Viewing and Projec5on. Taking Pictures with a Real Camera. Steps: Graphics does the same thing for rendering an image for 3D geometric objects
3D Vieing and Projec5on Taking Pictures ith a Real Camera Steps: Iden5 interes5ng objects Rotate and translate the camera to desired viepoint Adjust camera seings such as ocal length Choose desired resolu5on
3D graphics rendering pipeline (1) 3D graphics rendering pipeline (3) 3D graphics rendering pipeline (2) 8/29/11
3D graphics rendering pipeline (1) Geometr Rasteriation 3D Coordinates & Transformations Prof. Aaron Lanterman (Based on slides b Prof. Hsien-Hsin Sean Lee) School of Electrical and Computer Engineering
Computer Graphics. Chapter 7 2D Geometric Transformations
Computer Graphics Chapter 7 2D Geometric Transformations Chapter 7 Two-Dimensional Geometric Transformations Part III. OpenGL Functions for Two-Dimensional Geometric Transformations OpenGL Geometric Transformation
The 3-D Graphics Rendering Pipeline
The 3-D Graphics Rendering Pipeline Modeling Trival Rejection Illumination Viewing Clipping Projection Almost ever discussion of 3-D graphics begins here Seldom are an two versions drawn the same wa Seldom
Last Time. Correct Transparent Shadow. Does Ray Tracing Simulate Physics? Does Ray Tracing Simulate Physics? Refraction and the Lifeguard Problem
Graphics Pipeline: Projective Last Time Shadows cast ra to light stop after first intersection Reflection & Refraction compute direction of recursive ra Recursive Ra Tracing maimum number of bounces OR
News. Projections and Picking. Transforming View Volumes. Projections recap. Basic Perspective Projection. Basic Perspective Projection
Universit of British Columbia CPSC 44 Computer Graphics Projections and Picking Wed 4 Sep 3 project solution demo recap: projections projections 3 picking News Project solution eecutable available idea
CS770/870 Spring 2017 Transformations
CS770/870 Spring 2017 Transformations Coordinate sstems 2D Transformations Homogeneous coordinates Matrices, vectors, points 01/29/2017 1 Coordinate Sstems Coordinate sstems used in graphics Screen coordinates:
Viewing/Projections III. Week 4, Wed Jan 31
Universit of British Columbia CPSC 34 Computer Graphics Jan-Apr 27 Tamara Munner Viewing/Projections III Week 4, Wed Jan 3 http://www.ugrad.cs.ubc.ca/~cs34/vjan27 News etra TA coverage in lab to answer
The Graphics Pipeline and OpenGL I: Transformations!
! The Graphics Pipeline and OpenGL I: Transformations! Gordon Wetzstein! Stanford University! EE 267 Virtual Reality! Lecture 2! stanford.edu/class/ee267/!! Albrecht Dürer, Underweysung der Messung mit
3D Coordinates & Transformations
3D Coordinates & Transformations Prof. Aaron Lanterman (Based on slides b Prof. Hsien-Hsin Sean Lee) School of Electrical and Computer Engineering Georgia Institute of Technolog 3D graphics rendering pipeline