Fall CSCI 420: Computer Graphics. 2.2 Transformations. Hao Li.
|
|
- Calvin Potter
- 6 years ago
- Views:
Transcription
1 Fall 2017 CSCI 420: Computer Graphics 2.2 Transformations Hao Li 1
2 OpenGL Transformations Matrices Model-view matrix (4x4 matrix) Projection matrix (4x4 matrix) vertices in 3D Model-view vertices in canonical 3D world coordinate system Projection vertices in 2D 2
3 4x4 Model-view Matrix (this lecture) Translate, rotate, scale objects Position the camera vertices in 3D Model-view vertices in canonical 3D world coordinate system Projection vertices in 2D 3
4 4x4 Model-view Matrix (next lecture) Projection from 3D to 2D vertices in 3D Model-view vertices in canonical 3D world coordinate system Projection vertices in 2D 4
5 OpenGL Transformation Matrices Model-view Projection Manipulated separately in OpenGL (must set matrix mode) : glmatrixmode (GL_MODELVIEW); glmatrixmode (GL_PROJECTION; 5
6 Setting the Current Model-view Matrix Load or post-multiply glmatrixmode (GL_MODELVIEW); glloadidentity(); // very common usage float m[16] = { }; glloadmatrixf(m); // rare, advanced glmultmatrixf(m); // rare, advanced Use library functions gltranslatef(dx, dy, dz); glrotatef(angle, vx, vy, vz); glscalef(sx, sy, sz); 6
7 Translated, rotated, scaled object world 7
8 The rendering coordinate system Initially (after glloadidentity()) : rendering coordinate system = world coordinate system world 8
9 The rendering coordinate system gltranslatef(x, y, z); world [x, y,z] rendering coordinate system 9
10 The rendering coordinate system glrotatef(angle,ax, ay, az); world rendering coordinate system 10
11 The rendering coordinate system glscalef(sx, sy, sz); world rendering coordinate system 11
12 OpenGL code glmatrixmode (GL_MODELVIEW); glloadidentity(); gltranslatef(x, y, z); glrotatef(angle, ax, ay, az); glscalef(sx, sy, sz); renderbunny(); world rendering coordinate system 12
13 Rendering more objects How to obtain this frame? world rendering coordinate system 13
14 Solution 1 How to obtain this frame? Find gltranslate( ), glrotatef( ), glscalef( ) world 14
15 Solution 2: gl{push,pop}matrix glmatrixmode (GL_MODELVIEW); glloadidentity(); // render first bunny glpushmatrix(); // store current matrix gltranslate3f( ); glrotatef( ); renderbunny(); glpopmatrix(); // pop matrix // render second bunny glpushmatrix(); // store current matrix gltranslate3f( ); glrotatef( ); renderbunny(); glpopmatrix(); // pop matrix 15
16 Recall: Linear Algebra 16
17 Scalars Scalars,, from a scalar field () 1 Operations,,,,, Expected laws apply Examples: rationals or reals with addition and multiplication 17
18 Vectors Vectors u, v, w from a vector space u + v u v Vector addition, subtraction Zero vector 0 Scalar multiplication v 18
19 Euclidean Space Vector space over real numbers Three-dimensional in computer graphics Dot product: = u > v = u 1 v 1 + u 2 v 2 + u 3 v 3 0 > 0 =0, u, v are orthogonal if u > v =0 kvk 2 = v > v kvk v defines, the length of 19
20 Lines and Line Segments Parametric form of line: p( ) =p 0 + d Line segment between q and r : for p( ) =(1 )q + r 0 apple apple 1 20
21 Convex Hull Convex hull defined by p = 1 p n p n n =1 for and 0 apple 1 apple 1,i=1...n 21
22 Projection Dot product projects one vector onto another vector u > v = u 1 v 1 + u 2 v 2 + u 3 v 3 = kukkvk cos( ) v (u) =(u > v)v/kvk 2 22
23 Cross Product ka bk = a b sin( ) Cross product is perpendicular to both Right-hand rule a and b 23
24 Plane Plane defined by point and vectors u and v u and v should not be parallel Parametric form: p 0 t(, )=p 0 + u + v n = u v/ku vk is the normal n > (p p 0 )=0if and only if p lies in plane 24
25 Coordinate Systems v 1, v 2, v 3 Let be three linearly independent vectors in a 3-dimensional vector space Can write any vector w as w = 1 v v v 3 for some scalars 1, 2, 3 25
26 Frames p 0 Frame = origin + coordinate system Any point p = p v v v 3 26
27 In Practice, Frames are Often Orthogonal 27
28 Change of Coordinate System u 1, u 2, u 3 v 1, v 2, v 3 Bases { } and { } u i Express basis vectors in terms of v j u 1 = 11 v v v 3 u 2 = 21 v v v 3 u 3 = 31 v v v 3 Represent in matrix form: 2 4 u > 1 u > = M 4 v 1 > v 2 > 3 5 M u > 3 v > 3 28
29 Representing 3D transformations (and model-view matrices) 29
30 Linear Transformations 3 x 3 matrices represent linear transformations a = Mb Can represent rotation, scaling, and reflection Cannot represent translation M 30
31 Homogeneous Coordinates In order to represent rotations, scales AND translations [ 1, 2, 3 ] > Augment by adding a fourth component (1): p =[ 1, 2, 3, 1] > Homogeneous property:, for any scalar p =[ 1, 2, 3, 1] > =[ 1, 2, 3 ] > =0 31
32 Homogeneous Coordinates Homogeneous coordinates are transformed by 4x4 matrices p q q = Ap world 4-vector 4x4 matrix 4-vector 32
33 Affine Transformations (4x4 matrices) Translation Rotation Scaling Any composition of the above Later: projective (perspective) transformations -Also expressible as 4 x 4 matrices! 33
34 Translation q = p + d d =[ x, y, z, 0] > where, p =[x, y, z, 1] >, q =[x 0,y 0,z 0, 1] >, q = Tp Express in matrix form and solve for T 34
35 Scaling x 0 = y 0 = x x y y z 0 = z z Express as q = Sp and solve for S 35
36 Rotation in 2 Dimensions Rotation by about the origin x 0 = x cos( ) y sin( ) y 0 = x sin( )+y cos( ) Express in matrix form: Note that the determinant is 1 36
37 Rotation in 3 Dimensions Orthogonal matrices: RR > = R > R = I det(r) =1 Affine transformation: 37
38 Affine Matrices are Composed by Matrix Multiplication A = A 1 A 2 A 3 Applied from right to left Ap =(A 1 A 2 A 3 )p = A 1 (A 2 (A 3 p)) When calling gltranslate3f, glrotatef, or glscalef, OpenGL forms the corresponding 4x4 matrix, and multiplies the current modelview matrix with it. 38
39 Summary OpenGL Transformation Matrices Vector Spaces Frames Homogeneous Coordinates Transformation Matrices 39
40 Next Time: Viewing & Projection 40
41 Thanks! 41
Transformations. CSCI 420 Computer Graphics Lecture 4
CSCI 420 Computer Graphics Lecture 4 Transformations Jernej Barbic University of Southern California Vector Spaces Euclidean Spaces Frames Homogeneous Coordinates Transformation Matrices [Angel, Ch. 4]
More informationTransformations. CSCI 420 Computer Graphics Lecture 5
CSCI 420 Computer Graphics Lecture 5 Transformations Jernej Barbic University of Southern California Vector Spaces Euclidean Spaces Frames Homogeneous Coordinates Transformation Matrices [Angel, Ch. 3]
More informationTransformations. OpenGL Transformations. 4x4 Model-view Matrix (this lecture) OpenGL Transformation Matrices. 4x4 Projection Matrix (next lecture)
CSCI 420 Computer Graphics Lecture 5 OpenGL Transformations Transformations Vector Spaces Euclidean Spaces Frames Homogeneous Coordinates Transformation Matrices Jernej Barbic [Angel, Ch. 3] University
More informationTransformations Computer Graphics I Lecture 4
15-462 Computer Graphics I Lecture 4 Transformations Vector Spaces Affine and Euclidean Spaces Frames Homogeneous Coordinates Transformation Matrices January 23, 2003 [Angel, Ch. 4] Frank Pfenning Carnegie
More informationTransformations Computer Graphics I Lecture 4
15-462 Computer Graphics I Lecture 4 Transformations Vector Spaces Affine and Euclidean Spaces Frames Homogeneous Coordinates Transformation Matrices January 24, 2002 [Angel, Ch. 4] Frank Pfenning Carnegie
More informationLecture 4: Transformations and Matrices. CSE Computer Graphics (Fall 2010)
Lecture 4: Transformations and Matrices CSE 40166 Computer Graphics (Fall 2010) Overall Objective Define object in object frame Move object to world/scene frame Bring object into camera/eye frame Instancing!
More informationCS 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
More informationComputer 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
More informationCSC 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
More informationModeling Transform. Chapter 4 Geometric Transformations. Overview. Instancing. Specify transformation for objects 李同益
Modeling Transform Chapter 4 Geometric Transformations 李同益 Specify transformation for objects Allow definitions of objects in own coordinate systems Allow use of object definition multiple times in a scene
More informationLecture 5b. Transformation
Lecture 5b Transformation Refresher Transformation matrices [4 x 4]: the fourth coordinate is homogenous coordinate. Rotation Transformation: Axis of rotation must through origin (0,0,0). If not, translation
More informationGL_MODELVIEW transformation
lecture 3 view transformations model transformations GL_MODELVIEW transformation view transformations: How do we map from world coordinates to camera/view/eye coordinates? model transformations: How do
More informationCS D Transformation. Junqiao Zhao 赵君峤
CS10101001 3D Transformation Junqiao Zhao 赵君峤 Department of Computer Science and Technology College of Electronics and Information Engineering Tongji University Review Translation Linear transformation
More informationOrder 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
More informationEECE 478. Learning Objectives. Learning Objectives. Linear Algebra and 3D Geometry. Linear algebra in 3D. Coordinate systems
EECE 478 Linear Algebra and 3D Geometry Learning Objectives Linear algebra in 3D Define scalars, points, vectors, lines, planes Manipulate to test geometric properties Coordinate systems Use homogeneous
More informationCS 428: Fall Introduction to. Transformations in OpenGL + hierarchical modeling. Andrew Nealen, Rutgers, /21/2009 1
CS 428: Fall 2009 Introduction to Computer Graphics Transformations in OpenGL + hierarchical modeling 9/21/2009 1 Review of affine transformations Use projective geometry staple of CG Euclidean (x,z) (x,y,z)
More informationComputer Graphics Geometric Transformations
Computer Graphics 2016 6. Geometric Transformations Hongxin Zhang State Key Lab of CAD&CG, Zhejiang University 2016-10-31 Contents Transformations Homogeneous Co-ordinates Matrix Representations of Transformations
More informationCSCI E-74. Simulation and Gaming
CSCI E-74 Virtual and Augmented Reality for Simulation and Gaming Fall term 2017 Gianluca De Novi, PhD Lesson 3 General Introduction to OpenGL APIs and TRS Perspective Simulation Perspective simulation
More informationLecture 6 Sections 4.3, 4.6, 4.7. Wed, Sep 9, 2009
Lecture 6 Sections 4.3, 4.6, 4.7 Hampden-Sydney College Wed, Sep 9, 2009 Outline 1 2 3 4 re are three mutually orthogonal axes: the x-axis, the y-axis, and the z-axis. In the standard viewing position,
More informationSpring 2013, CS 112 Programming Assignment 2 Submission Due: April 26, 2013
Spring 2013, CS 112 Programming Assignment 2 Submission Due: April 26, 2013 PROJECT GOAL: Write a restricted OpenGL library. The goal of the project is to compute all the transformation matrices with your
More information3.1 Viewing and Projection
Fall 2017 CSCI 420: Computer Graphics 3.1 Viewing and Projection Hao Li http://cs420.hao-li.com 1 Recall: Affine Transformations Given a point [xyz] > form homogeneous coordinates [xyz1] > The transformed
More informationChapter 3: Modeling Transformation
Chapter 3: Modeling Transformation Graphics Programming, 8th Sep. Graphics and Media Lab. Seoul National University 2011 Fall OpenGL Steps Every step in the graphics pipeline is related to the transformation.
More information1 Transformations. Chapter 1. Transformations. Department of Computer Science and Engineering 1-1
Transformations 1-1 Transformations are used within the entire viewing pipeline: Projection from world to view coordinate system View modifications: Panning Zooming Rotation 1-2 Transformations can also
More informationViewing and Projection
CSCI 480 Computer Graphics Lecture 5 Viewing and Projection Shear Transformation Camera Positioning Simple Parallel Projections Simple Perspective Projections [Geri s Game, Pixar, 1997] January 26, 2011
More informationBasic Elements. Geometry is the study of the relationships among objects in an n-dimensional space
Basic Elements Geometry is the study of the relationships among objects in an n-dimensional space In computer graphics, we are interested in objects that exist in three dimensions We want a minimum set
More informationReading. Hierarchical Modeling. Symbols and instances. Required: Angel, sections , 9.8. Optional:
Reading Required: Angel, sections 9.1 9.6, 9.8 Optional: Hierarchical Modeling OpenGL rogramming Guide, the Red Book, chapter 3 cse457-07-hierarchical 1 cse457-07-hierarchical 2 Symbols and instances Most
More information3D Transformation. In 3D, we have x, y, and z. We will continue use column vectors:. Homogenous systems:. x y z. x y z. glvertex3f(x, y,z);
3D Transformation In 3D, we have x, y, and z. We will continue use column vectors:. Homogenous systems:. 3D Transformation glvertex3f(x, y,z); x y z x y z A Right Handle Coordinate System x y z; y z x;
More informationC OMPUTER G RAPHICS Thursday
C OMPUTER G RAPHICS 2017.04.27 Thursday Professor s original PPT http://calab.hanyang.ac.kr/ Courses Computer Graphics practice3.pdf TA s current PPT not uploaded yet GRAPHICS PIPELINE What is Graphics
More information3D Graphics Pipeline II Clipping. Instructor Stephen J. Guy
3D Graphics Pipeline II Clipping Instructor Stephen J. Guy 3D Rendering Pipeline (for direct illumination) 3D Geometric Primitives 3D Model Primitives Modeling Transformation 3D World Coordinates Lighting
More informationComputer Graphics. Chapter 5 Geometric Transformations. Somsak Walairacht, Computer Engineering, KMITL
Chapter 5 Geometric Transformations Somsak Walairacht, Computer Engineering, KMITL 1 Outline Basic Two-Dimensional Geometric Transformations Matrix Representations and Homogeneous Coordinates Inverse Transformations
More informationC O M P U T E R G R A P H I C S. Computer Graphics. Three-Dimensional Graphics I. Guoying Zhao 1 / 52
Computer Graphics Three-Dimensional Graphics I Guoying Zhao 1 / 52 Geometry Guoying Zhao 2 / 52 Objectives Introduce the elements of geometry Scalars Vectors Points Develop mathematical operations among
More informationCIS 636 Interactive Computer Graphics CIS 736 Computer Graphics Spring 2011
CIS 636 Interactive Computer Graphics CIS 736 Computer Graphics Spring 2011 Lab 1a of 7 OpenGL Setup and Basics Fri 28 Jan 2011 Part 1a (#1 5) due: Thu 03 Feb 2011 (before midnight) The purpose of this
More informationViewing and Projection
CSCI 480 Computer Graphics Lecture 5 Viewing and Projection January 25, 2012 Jernej Barbic University of Southern California Shear Transformation Camera Positioning Simple Parallel Projections Simple Perspective
More informationChapter 3 - Basic Mathematics for 3D Computer Graphics
Chapter 3 - Basic Mathematics for 3D Computer Graphics Three-Dimensional Geometric Transformations Affine Transformations and Homogeneous Coordinates OpenGL Matrix Logic Translation Add a vector t Geometrical
More informationHierarchical Modeling. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Hierarchical Modeling University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell Reading Angel, sections 9.1-9.6 [reader pp. 169-185] OpenGL Programming Guide, chapter 3 Focus especially
More informationIntroduction 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
More informationComputer Graphics. Chapter 10 Three-Dimensional Viewing
Computer Graphics Chapter 10 Three-Dimensional Viewing Chapter 10 Three-Dimensional Viewing Part I. Overview of 3D Viewing Concept 3D Viewing Pipeline vs. OpenGL Pipeline 3D Viewing-Coordinate Parameters
More informationCS 591B Lecture 9: The OpenGL Rendering Pipeline
CS 591B Lecture 9: The OpenGL Rendering Pipeline 3D Polygon Rendering Many applications use rendering of 3D polygons with direct illumination Spring 2007 Rui Wang 3D Polygon Rendering Many applications
More informationRendering Pipeline and Coordinates Transforms
Rendering Pipeline and Coordinates Transforms Alessandro Martinelli alessandro.martinelli@unipv.it 16 October 2013 Rendering Pipeline (3): Coordinates Transforms Rendering Architecture First Rendering
More informationModeling with Transformations
Modeling with Transformations Prerequisites This module requires some understanding of 3D geometry, particularly a sense of how objects can be moved around in 3-space. The student should also have some
More informationTransforms 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
More informationCS380: Computer Graphics 2D Imaging and Transformation. Sung-Eui Yoon ( 윤성의 ) Course URL:
CS380: Computer Graphics 2D Imaging and Transformation Sung-Eui Yoon ( 윤성의 ) Course URL: http://sglab.kaist.ac.kr/~sungeui/cg Class Objectives Write down simple 2D transformation matrixes Understand the
More informationModeling Objects by Polygonal Approximations. Linear and Affine Transformations (Maps)
Modeling Objects by Polygonal Approximations Define volumetric objects in terms of surfaces patches that surround the volume Each surface patch is approximated set of polygons Each polygon is specified
More information7. 3D Viewing. Projection: why is projection necessary? CS Dept, Univ of Kentucky
7. 3D Viewing Projection: why is projection necessary? 1 7. 3D Viewing Projection: why is projection necessary? Because the display surface is 2D 2 7.1 Projections Perspective projection 3 7.1 Projections
More information3D 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
More informationCS4202: Test. 1. Write the letter corresponding to the library name next to the statement or statements that describe library.
CS4202: Test Name: 1. Write the letter corresponding to the library name next to the statement or statements that describe library. (4 points) A. GLUT contains routines that use lower level OpenGL commands
More informationDH2323 DGI16 Lab2 Transformations
DH2323 DGI16 Lab2 Transformations In Lab1, you used SDL (Simple DirectMedia Layer) which is a cross-platform development library designed to provide low level access to audio and input devices. This lab
More informationTranslation. 3D Transformations. Rotation about z axis. Scaling. CS 4620 Lecture 8. 3 Cornell CS4620 Fall 2009!Lecture 8
Translation 3D Transformations CS 4620 Lecture 8 1 2 Scaling Rotation about z axis 3 4 Rotation about x axis Rotation about y axis 5 6 Transformations in OpenGL Stack-based manipulation of model-view transformation,
More informationTransformations. Mike Bailey. Oregon State University. Oregon State University. Computer Graphics. transformations.
1 Transformations Mike Bailey mjb@cs.oregonstate.edu transformations.pptx Geometry vs. Topology 2 Original Object 4 1 2 4 Geometry: Where things are (e.g., coordinates) 3 1 3 2 Geometry = changed Topology
More informationAffine 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
More informationCS 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
More informationLast week. Machiraju/Zhang/Möller
Last week Machiraju/Zhang/Möller 1 Overview of a graphics system Output device Input devices Image formed and stored in frame buffer Machiraju/Zhang/Möller 2 Introduction to CG Torsten Möller 3 Ray tracing:
More informationLecture 5: Viewing. CSE Computer Graphics (Fall 2010)
Lecture 5: Viewing CSE 40166 Computer Graphics (Fall 2010) Review: from 3D world to 2D pixels 1. Transformations are represented by matrix multiplication. o Modeling o Viewing o Projection 2. Clipping
More informationVector Algebra Transformations. Lecture 4
Vector Algebra Transformations Lecture 4 Cornell CS4620 Fall 2008 Lecture 4 2008 Steve Marschner 1 Geometry A part of mathematics concerned with questions of size, shape, and relative positions of figures
More informationComputer Graphics. Coordinate Systems and Change of Frames. Based on slides by Dianna Xu, Bryn Mawr College
Computer Graphics Coordinate Systems and Change of Frames Based on slides by Dianna Xu, Bryn Mawr College Linear Independence A set of vectors independent if is linearly If a set of vectors is linearly
More informationCT5510: Computer Graphics. Transformation BOCHANG MOON
CT5510: Computer Graphics Transformation BOCHANG MOON 2D Translation Transformations such as rotation and scale can be represented using a matrix M.., How about translation? No way to express this using
More informationCS452/552; EE465/505. Geometry Transformations
CS452/552; EE465/505 Geometry Transformations 1-26-15 Outline! Geometry: scalars, points & vectors! Transformations Read: Angel, Chapter 4 (study cube.html/cube.js example) Appendix B: Spaces (vector,
More informationFachhochschule 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
More information蔡侑庭 (Yu-Ting Tsai) National Chiao Tung University, Taiwan. Prof. Wen-Chieh Lin s CG Slides OpenGL 2.1 Specification
蔡侑庭 (Yu-Ting Tsai) Department of Computer Science National Chiao Tung University, Taiwan Prof. Wen-Chieh Lin s CG Slides OpenGL 2.1 Specification OpenGL Programming Guide, Chap. 3 & Appendix F 2 OpenGL
More informationProf. Feng Liu. Fall /19/2016
Prof. Feng Liu Fall 26 http://www.cs.pdx.edu/~fliu/courses/cs447/ /9/26 Last time More 2D Transformations Homogeneous Coordinates 3D Transformations The Viewing Pipeline 2 Today Perspective projection
More informationColumn and row space of a matrix
Column and row space of a matrix Recall that we can consider matrices as concatenation of rows or columns. c c 2 c 3 A = r r 2 r 3 a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 The space spanned by columns of
More informationTransforms. COMP 575/770 Spring 2013
Transforms COMP 575/770 Spring 2013 Transforming Geometry Given any set of points S Could be a 2D shape, a 3D object A transform is a function T that modifies all points in S: T S S T v v S Different transforms
More informationComputer Graphics I. Midterm Examination. October 23, This is a closed-book exam; only one double-sided sheet of notes is permitted.
15-462 Computer Graphics I Midterm Examination October 23, 2003 Name: Andrew User ID: This is a closed-book exam; only one double-sided sheet of notes is permitted. Write your answer legibly in the space
More informationOverview. By end of the week:
Overview By end of the week: - Know the basics of git - Make sure we can all compile and run a C++/ OpenGL program - Understand the OpenGL rendering pipeline - Understand how matrices are used for geometric
More information3.2 Hierarchical Modeling
Fall 2018 CSCI 420: Computer Graphics 3.2 Hierarchical Modeling Hao Li http://cs420.hao-li.com 1 Importance of shadows Source: UNC 2 Importance of shadows Source: UNC 3 Importance of shadows Source: UNC
More informationComputer Graphics. Bing-Yu Chen National Taiwan University
Computer Graphics Bing-Yu Chen National Taiwan University Introduction to OpenGL General OpenGL Introduction An Example OpenGL Program Drawing with OpenGL Transformations Animation and Depth Buffering
More informationGeometry. CS 537 Interactive Computer Graphics Prof. David E. Breen Department of Computer Science
Geometry CS 537 Interactive Computer Graphics Prof. David E. Breen Department of Computer Science E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012. 1 Objectives Introduce
More informationComputer Graphics Hands-on
Computer Graphics Hands-on Two-Dimensional OpenGL Transformations Objectives To get hands-on experience manipulating the OpenGL current transformation (MODELVIEW) matrix to achieve desired effects To gain
More informationComputer 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
More informationAffine Transformations Computer Graphics Scott D. Anderson
Affine Transformations Computer Graphics Scott D. Anderson 1 Linear Combinations To understand the poer of an affine transformation, it s helpful to understand the idea of a linear combination. If e have
More informationScene Graphs. CS4620/5620: Lecture 7. Announcements. HW 1 out. PA 1 will be out on Wed
CS4620/5620: Lecture 7 Scene Graphs 1 Announcements HW 1 out PA 1 will be out on Wed Next week practicum will have an office hour type session on Open GL 2 Example Can represent drawing with flat list
More informationIntroduction to OpenGL Transformations, Viewing and Lighting. Ali Bigdelou
Introduction to OpenGL Transformations, Viewing and Lighting Ali Bigdelou Modeling From Points to Polygonal Objects Vertices (points) are positioned in the virtual 3D scene Connect points to form polygons
More informationOpenGL Transformations
OpenGL Transformations R. J. Renka Department of Computer Science & Engineering University of North Texas 02/18/2014 Introduction The most essential aspect of OpenGL is the vertex pipeline described in
More informationComputer Graphics: Viewing in 3-D. Course Website:
Computer Graphics: Viewing in 3-D Course Website: http://www.comp.dit.ie/bmacnamee 2 Contents Transformations in 3-D How do transformations in 3-D work? 3-D homogeneous coordinates and matrix based transformations
More informationThe Viewing Pipeline adaptation of Paul Bunn & Kerryn Hugo s notes
The Viewing Pipeline adaptation of Paul Bunn & Kerryn Hugo s notes What is it? The viewing pipeline is the procession of operations that are applied to the OpenGL matrices, in order to create a 2D representation
More informationWhere We Are: Static Models
Where We Are: Static Models Our models are static sets of polygons We can only move them via transformations translate, scale, rotate, or any other 4x4 matrix This does not allow us to simulate very realistic
More informationDescribe the Orthographic and Perspective projections. How do we combine together transform matrices?
Aims and objectives By the end of the lecture you will be able to Work with multiple transform matrices Describe the viewing process in OpenGL Design and build a camera control APIs Describe the Orthographic
More informationGEOMETRIC 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
More informationPerspective projection and Transformations
Perspective projection and Transformations The pinhole camera The pinhole camera P = (X,,) p = (x,y) O λ = 0 Q λ = O λ = 1 Q λ = P =-1 Q λ X = 0 + λ X 0, 0 + λ 0, 0 + λ 0 = (λx, λ, λ) The pinhole camera
More informationComputer 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
More informationTransformation Pipeline
Transformation Pipeline Local (Object) Space Modeling World Space Clip Space Projection Eye Space Viewing Perspective divide NDC space Normalized l d Device Coordinatesd Viewport mapping Screen space Coordinate
More informationComputer graphics MN1
Computer graphics MN1 http://www.opengl.org Todays lecture What is OpenGL? How do I use it? Rendering pipeline Points, vertices, lines,, polygons Matrices and transformations Lighting and shading Code
More informationUsing GLU/GLUT Objects
Using GLU/GLUT Objects GLU/GLUT provides very simple object primitives glutwirecone glutwirecube glucylinder glutwireteapot GLU/GLUT Objects Each glu/glut object has its default size, position, and orientation
More informationComputer Graphics 7: Viewing in 3-D
Computer Graphics 7: Viewing in 3-D In today s lecture we are going to have a look at: Transformations in 3-D How do transformations in 3-D work? Contents 3-D homogeneous coordinates and matrix based transformations
More informationCS452/552; 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:
More informationCSE Intro to Computer Graphics. ANSWER KEY: Midterm Examination. November 18, Instructor: Sam Buss, UC San Diego
CSE 167 - Intro to Computer Graphics ANSWER KEY: Midterm Examination November 18, 2003 Instructor: Sam Buss, UC San Diego Write your name or initials on every page before beginning the exam. You have 75
More informationThe Importance of Matrices in the DirectX API. by adding support in the programming language for frequently used calculations.
Hermann Chong Dr. King Linear Algebra Applications 28 November 2001 The Importance of Matrices in the DirectX API In the world of 3D gaming, there are two APIs (Application Program Interface) that reign
More informationToday. Today. Introduction. Matrices. Matrices. Computergrafik. Transformations & matrices Introduction Matrices
Computergrafik Matthias Zwicker Universität Bern Herbst 2008 Today Transformations & matrices Introduction Matrices Homogeneous Affine transformations Concatenating transformations Change of Common coordinate
More informationS U N G - E U I YO O N, K A I S T R E N D E R I N G F R E E LY A VA I L A B L E O N T H E I N T E R N E T
S U N G - E U I YO O N, K A I S T R E N D E R I N G F R E E LY A VA I L A B L E O N T H E I N T E R N E T Copyright 2018 Sung-eui Yoon, KAIST freely available on the internet http://sglab.kaist.ac.kr/~sungeui/render
More informationTransformation Algebra
Transformation Algebra R. J. Renka Department of Computer Science & Engineering University of North Texas 0/07/205 Linear Transformations A point with Cartesian coordinates (x, y, z) is taken to be a column
More informationComputer Viewing Computer Graphics I, Fall 2008
Computer Viewing 1 Objectives Introduce mathematics of projection Introduce OpenGL viewing functions Look at alternate viewing APIs 2 Computer Viewing Three aspects of viewing process All implemented in
More informationCoordinate Transformations & Homogeneous Coordinates
CS-C31 (Introduction to) Computer Graphics Jaakko Lehtinen Coordinate Transformations & Homogeneous Coordinates Lots of slides from Frédo Durand CS-C31 Fall 216 Lehtinen Outline Intro to Transformations
More informationCS230 : Computer Graphics Lecture 6: Viewing Transformations. Tamar Shinar Computer Science & Engineering UC Riverside
CS230 : Computer Graphics Lecture 6: Viewing Transformations Tamar Shinar Computer Science & Engineering UC Riverside Rendering approaches 1. image-oriented foreach pixel... 2. object-oriented foreach
More informationCOMP Computer Graphics and Image Processing. 5: Viewing 1: The camera. In part 1 of our study of Viewing, we ll look at ˆʹ U ˆ ʹ F ˆʹ S
COMP27112 Û ˆF Ŝ Computer Graphics and Image Processing ˆʹ U ˆ ʹ F C E 5: iewing 1: The camera ˆʹ S Toby.Howard@manchester.ac.uk 1 Introduction In part 1 of our study of iewing, we ll look at iewing in
More informationModeling Objects. Modeling. Symbol-Instance Table. Instance Transformation. Each appearance of the object in the model is an instance
Modeling Objects Modeling Hierarchical Transformations Hierarchical Models Scene Graphs A prototype has a default size, position, and orientation You need to perform modeling transformations to position
More informationLuiz Fernando Martha André Pereira
Computer Graphics for Engineering Numerical simulation in technical sciences Color / OpenGL Luiz Fernando Martha André Pereira Graz, Austria June 2014 To Remember Computer Graphics Data Processing Data
More informationComputer Graphics Lighting
Lighting 1 Mike Bailey mjb@cs.oregonstate.edu This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International License Lighting.pptx Why Do We Care About Lighting?
More informationComputer Graphics Lighting. Why Do We Care About Lighting?
Lighting 1 Mike Bailey mjb@cs.oregonstate.edu This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International License Lighting.pptx Why Do We Care About Lighting?
More informationTwo possible ways to specify transformations. Each part of the object is transformed independently relative to the origin
Transformations Two possible ways to specify transformations Each part of the object is transformed independently relative to the origin - Not convenient! z y Translate the base by (5,0,0); Translate the
More informationTransformations IV. Week 3, Wed Jan 20
University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2010 Tamara Munzner Transformations IV Week 3, Wed Jan 20 http://www.ugrad.cs.ubc.ca/~cs314/vjan2010 Assignments 2 Correction: Assignments
More information