Projections. Let us start with projections in 2D, because there are easier to visualize.
|
|
- Rolf George
- 6 years ago
- Views:
Transcription
1 Projetions Let us start ith projetions in D, beause there are easier to visualie. Projetion parallel to the -ais: Ever point in the -plane ith oordinates (, ) ill be transformed into the point ith oordinates (, ). he matri of the transformation is Projetion along a line ith slope m: he point (,) ill be transformed into the point (- /m, ). he point (, ) ill be transformed into ( p, ). Using similar triangles on the net figure, i.e the triangle (,), (-/m, ) and (,) and the triangle (,), ( p, ) and (,), e find that - p (/m) herefore the matri of the transformation is m or equivalentl m m
2 Perspetive projetion: Here e have a OP given b (,, ) Given an point (,, ) on the plane, e dra the line joining it ith the OP. he projetion point is the point here this line meets the -ais, at ( p, ). Using again similar triangles in the previous figure, i.e the triangle (, ), ( p,), (, ) and the triangle (,), ( p, ), (, ), e an rite p p Solving for p e find p. In other ords, using homogeneous oordinates, e obtain the projeted point to have the folloing oordinates: (,, ), or using the propert of homogeneous oordinates (,, ). his generates the matri of the transformation: hanged to: or if the enter is given b (,, ) then the matri ill be REMARK: he parallel projetion is a partiular ase of the perspetive projetion the a e have defined in the last matri.
3 If the enter of projetion is the diretion of the oblique line (, m, ), then b substitution e find the matri of the beginning of the page. B generaliation (using the same tehniques to obtain the values of the projetion matri), e an rite that the projetion onto the -plane, using the enter of projetion (,,, ) is the folloing one: 3-D Projetions Parallel Projetion. his is a projetion folloing a diretion. In the lassial vieing e have the folloing projetions: Orthographi projetion: he diretion of projetion is perpendiular to the projetion plane. In a multivie orthographi projetion, the projetion plane is parallel to one of the prinipal faes of the objet. Aonometri projetion: he DOP is perpendiular to the plane of projetion, but the objet ma have an orientation ith respet to the projetion plane. Isometri projetion is a tpe of aonometri projetion in hih the projetion plane is plaed smmetriall ith respet to three prinipal faes of the objet that meet at a orner. Dimetri projetion is another aonometri projetion, but in this one, the projetion plane is plaed smmetriall ith respet to to prinipal faes of the objet. Oblique projetions: he diretion of projetion is not orthogonal to the projetion plane. Among them the most famous are the avalier and the abinet projetions. In both of them one prinipal fae of the objet is parallel to the projetion plane, but the diretion of projetion is suh that in the ase of abinet projetion: he prinipal fae perpendiular to the projetion plane has half the sale that the fae parallel to the projetion plane.
4 avalier projetion: he prinipal fae perpendiular to the projetion plane has the same sale that the fae parallel to the projetion plane. PerspetiveProjetion. he enter of projetion is a point at a finite distane. In lassial vieing for the perspetive projetion, e have one-, to-, and three-point perspetives. he one-point perspetive projetion has a prinipal fae of the objet parallel to the projetion plane. he to-point one has a line of the objet parallel to the projetion plane. he three-point does not have an line parallel to the projetion plane. he one-point perspetive projetion has a vanishing point (point here the parallel lines no parallel to the projetion plane onverge) he to-point perspetive projetion has to vanishing points (that generate the line of the horion, b the vanishing points of the parallel lines not parallel to the projetion plane) he three-point perspetive projetion has three vanishing points. All these projetions are partiular ases of the projetion transformation that e ill generate. Projetion eample: Let s assume that the enter of projetion is P [,,, ]. e have the bo A [,,, ] ; B [5,,, ] ; [5,, -5, ] ; D [,, -5, ] ; E [,,, ] ; F [5,,, ] ; G [5,, -5, ] ; H [,, -5, ] ; e ant to find the oordinates of the projeted bo. First e find the projetion onto the -plane, using the OP given b P. But before omputing the projetion matri, e reall ompute the prearping matri.
5 P Note that e keep the pseudo-distane (for the -buffer) to distinguish the points that are farther aa. hen the onl thing left to obtain the projetion is to multipl b the orthogonal projetion matri, hih is equivalent to hanging the -oordinate value to ero. First e ould immediatel rite hat happens in the -plane, sine the OP is there, and so are the points A, B,, D. As a matter of fat, e an obtain P. A P. [,,, ] [,,, ]. Note that the pseudo-depth is. P. B P. [5,,, ] [5,,, ]. Note that the pseudo-depth is again ero. P. P. [5,, -5, ] [8,, -.6, ]. Note that the pseudo-depth is.6. P. D P. [,, -5, ] [6,, -.6, ]. Pseudo-depth is.6. P. E P. [,,, ] [,,, ]. Pseudo-depth is again ero. P. F P. [5,,, ] [5,,, ]. Pseudo-depth is again ero. P. G P. [5,, -5, ] [8, 4, -.6, ]. Pseudo-depth is.6. P. H P. [,, -5, ] [6, 4, -.6, ]. Pseudo-depth is.6.
6 If e represent these ne points, ithout the pseudo-depth, e obtain the folloing figure: Note that the vanishing point V has oordinates [,,, ]. Oblique projetion: It is a parallel projetion hose diretion of projetion is not perpendiular to the vieing plane. e need to find the matri of the transformation.
7 learl the diretion of projetion is given b [ p -, p,, ], but in order to determine the values of p, p, e ould need the diretion of projetion. So e need another approah. e ould determine p, p, if e kne the angle α, and the angle φ. he angle α is determined b the line going from the point [,,, ] to [ p, p,, ] and the line going from the same point [ p, p,, ] to the point [,,, ] ( See the previous figure). he angle φ is determined b the line going from [,,, ] to the point [ p, p,, ] and the -ais. If these to angles ere given, e ould determine the diretion of projetion. Let us see: Let L be the distane from the point ( p, p, ) to the point (,, ). he e an rite, p L os φ p L sin φ. But L tan - α herefore p tan - α os φ p tan - α sin φ and onsequentl the projetion matri is: ' osφ tanα sinφ tanα
8 omparing this result ith the general projetion matri e enounter before, e an establish that the enter of projetion for the oblique projetion is given b,, tan sin, tan os α φ α φ OP If tan α, then e have a abinet projetion. If it is equal to, then e have the avalier projetion. r some eamples to see the relationship beteen the distanes on the plane parallel to the projetion plane, and the distanes on a plane perpendiular to the projetion plane. Projetion Normaliation. Ever projetion an be transformed into the produt of to transformations. First e appl a distortion, and then an orthogonal projetion. he first transformation, hih makes the distortion of the objet, is sometimes referred as prearping. Let s ork on D, and then e an generalie. Imagine e have the projetion matri,, that an be deomposed into the produt of these to: if, or (note that and an not both be ero simultaneousl) Let us all the matri on the right and the projetion matri. Note that the other matri orresponds to an orthogonal projetion.
9 hen e appl e obtain a pseudo-distane to the -ais as a bprodut. his pseudodistane preserves the relative distane order of the points ith respet to the -ais. Eample: Assume (,, ) (,, ) hen the matri ill be given b. onsider the points ( - - α, -α, ) α α α α. So the ne points in artesian oordinates are α α α,, and the distanes to the -ais are given b. α α he distane to the -ais of the original points as α, and the ne distane is α. α You ma hek that the ne distane preserves the order of the points. he ones originall farther aa, go to a ne distane that is also further aa. For instane, if α.5, the ne distane is 3, if α as.75, the ne distane is 7 3, if α as, the ne distane is.5, if α as 3, then the ne distane is 4 3, et But before e even appl the distortion of the objet, e perform another transformation, to hange the given vieing volume to the ube ith enter at the origin and determined b the planes ±, ±, ±. his is alled the anonial vie volume. Note that e ould have used a unit ube, ith a orner at the origin, and the results ould be ver similar. Let s start ith an orthogonal projetion. e have to suppl the vieing volume, given b the folloing parallelepiped: ( L, R, B,, N, F ) - L stands for left, R stands for right, B stands for bottom, stands for top, N stands for near, and F stands for far. Other notations an be found, ith the folloing meaning: L min, R ma, B min, ma, N min, F ma.
10 In order to make the hange to the anonial vie, e have to make a translation first, hose matri is given b N F B L R hen e make a saling, given b the matri S: N F B L R S Multipling both matries e obtain the transformation matri N F N F N F B B B L R L R L R P Oblique and perspetive projetion. In this ase a good tehnique is to define the orld-vieing indo in the folloing a. First e define a vieing indo in the plane, i.e. e delare L, R, B and. hen e delare the to planes N, F. he vieing 3D indo is the region of the spae beteen these to planes, that projets into the retangle ( L, R, B, ). In this ase the prearping hanges the orld-vieing indo into a parallelepiped, and e are in the previous ase. Eample in D: onsider the enter of projetion (,, ), and the indo (in the -ais) determined b (,,) and (,,). For the orld-vieing volume e fi - (near) and - (far). his determines that the orld-vieing volume is the quadrilateral defined b the folloing points: (,-,), (3, -, ), (6, -,) and (4, -,). (See net figure)
11 he matri for the prearping is given b, hih transforms the quadrilateral into a retangle, hose verties are given b (, -.5,), (, ,), (, ,) and (,-.5,). No e an hange this retangle into the anonial vie, before e appl the orthographi projetion.
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
More informationGeometric Model of Camera
Geometric Model of Camera Dr. Gerhard Roth COMP 42A Winter 25 Version 2 Similar Triangles 2 Geometric Model of Camera Perspective projection P(X,Y,Z) p(,) f X Z f Y Z 3 Parallel lines aren t 4 Figure b
More information1. Inversions. A geometric construction relating points O, A and B looks as follows.
1. Inversions. 1.1. Definitions of inversion. Inversion is a kind of symmetry about a irle. It is defined as follows. he inversion of degree R 2 entered at a point maps a point to the point on the ray
More information3D 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
More informationImproved Real-Time Shadow Mapping for CAD Models
Improved Real-Time Shado Mapping for CAD Models Vitor Barata R. B. Barroso TeGraf, Computer Siene Dept., PUC-Rio vbarata@tegraf.pu-rio.br Waldemar Celes TeGraf, Computer Siene Dept., PUC-Rio eles@tegraf.pu-rio.br
More information2D transformations and homogeneous coordinates
2D transformations and homogeneous coordinates Dr Nicolas Holzschuch Universit of Cape Ton e-mail: holzschu@cs.uct.ac.za Map of the lecture Transformations in 2D: vector/matri notation eample: translation,
More informationChap 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
More informationSimultaneous image orientation in GRASS
Simultaneous image orientation in GRASS Alessandro BERGAMINI, Alfonso VITTI, Paolo ATELLI Dipartimento di Ingegneria Civile e Ambientale, Università degli Studi di Trento, via Mesiano 77, 38 Trento, tel.
More informationLecture 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
More information5.8.3 Oblique Projections
278 Chapter 5 Viewing y (, y, ) ( p, y p, p ) Figure 537 Oblique projection P = 2 left right 0 0 left+right left right 0 2 top bottom 0 top+bottom top bottom far+near far near 0 0 far near 2 0 0 0 1 Because
More informationRed-Black Trees 10/19/2009. Red-Black Trees. Example. Red-Black Properties. Black Height. Example
lgorithms Red-lak Trees 13-2 Red-lak Trees Red-lak Trees ll binar searh tree operations take O(h) time, here h is the height of the tree Therefore, it is important to `balane the tree so that its height
More informationSelf-Location of a Mobile Robot with uncertainty by cooperation of an heading sensor and a CCD TV camera
Self-oation of a Mobile Robot ith unertainty by ooperation of an heading sensor and a CCD TV amera E. Stella, G. Ciirelli, A. Distante Istituto Elaborazione Segnali ed Immagini - C.N.R. Via Amendola, 66/5-706
More informationECE Digital Image Processing and Introduction to Computer Vision. Outline
ECE592-064 Digital Image Processing and Introduction to Computer Vision Depart. of ECE, NC State University Instructor: Tianfu (Matt) Wu Spring 2017 1. Recap Outline 2. Modeling Projection and Projection
More informationMATH STUDENT BOOK. 12th Grade Unit 6
MATH STUDENT BOOK 12th Grade Unit 6 Unit 6 TRIGONOMETRIC APPLICATIONS MATH 1206 TRIGONOMETRIC APPLICATIONS INTRODUCTION 3 1. TRIGONOMETRY OF OBLIQUE TRIANGLES 5 LAW OF SINES 5 AMBIGUITY AND AREA OF A TRIANGLE
More informationA radiometric analysis of projected sinusoidal illumination for opaque surfaces
University of Virginia tehnial report CS-21-7 aompanying A Coaxial Optial Sanner for Synhronous Aquisition of 3D Geometry and Surfae Refletane A radiometri analysis of projeted sinusoidal illumination
More informationTransforms II. Overview. Homogeneous Coordinates 3-D Transforms Viewing Projections. Homogeneous Coordinates. x y z w
Transforms II Overvie Homogeneous Coordinates 3- Transforms Vieing Projections 2 Homogeneous Coordinates Allos translations to be included into matri transform. Allos us to distinguish beteen a vector
More informationMixture models and clustering
1 Lecture topics: Miture models and clustering, k-means Distance and clustering Miture models and clustering We have so far used miture models as fleible ays of constructing probability models for prediction
More informationImage 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())
More informationReview (Law of sines and cosine) cosines)
Date:03/7,8/01 Review 6.1-6. Objetive: Apply the onept to use the law of the sines and osines to solve oblique triangles Apply the onept to find areas using the law of sines and osines Agenda: Bell ringer
More informationChap 7, 2008 Spring Yeong Gil Shin
Three-Dimensional i Viewingi Chap 7, 28 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 synthetic camera)
More informationInteractive Order-Independent Transparency
Interative Order-Independent Transpareny Cass Everitt NVIDIA OpenGL Appliations Engineering ass@nvidia.om (a) (b) Figure. These images illustrate orret (a) and inorret (b) rendering of transparent surfaes.
More informationTransformation Packet
Name Transformation Packet UE: TEST: 1 . Transformation Vocabular Transformation Related Terms Sketch Reflection (flip across a line) Line of reflection Pre-image and image Rigid Rotation (turn about a
More informationAnnouncements. Introduction to Cameras. The Key to Axis Angle Rotation. Axis-Angle Form (review) Axis Angle (4 steps) Mechanics of Axis Angle
Ross Beerige Bruce Draper Introuction to Cameras September th 25 Announcements PA ue eek from Tuesa Q: hat i I mean b robust I/O? Hanle arious numbers of erte/face features Check for count matches Goo
More informationThree Lorentz Transformations. Considering Two Rotations
Ad. Stdies Theor. Phs., Vol. 6,, no., 9 Three orentz Transformations Considering To otations Mkl Chandra Das* Singhania Uniersit, ajasthan, India mkldas.@gmail.om ampada Misra Department of eletronis,
More informationOrientation of the coordinate system
Orientation of the oordinate system Right-handed oordinate system: -axis by a positive, around the -axis. The -axis is mapped to the i.e., antilokwise, rotation of The -axis is mapped to the -axis by a
More informationCSCI-4972/6963 Advanced Computer Graphics
Luo Jr. CSCI-497/696 Advaned Computer Graphis http://www.s.rpi.edu/~utler/lasses/advanedgraphis/s8/ Professor Barb Cutler utler@s.rpi.edu MRC 9A Piar Animation Studios, 986 Topis for the Semester Mesh
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 informationGLOBAL 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
More informationChapter 8: Right Triangle Trigonometry
Haberman MTH 11 Setion I: The Trigonometri Funtions Chapter 8: Right Triangle Trigonometry As we studied in Part 1 of Chapter 3, if we put the same angle in the enter of two irles of different radii, we
More informationModeling 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,
More information15.4 Constrained Maxima and Minima
15.4 Constrained Maxima and Minima Question 1: Ho do ou find the relative extrema of a surface hen the values of the variables are constrained? Question : Ho do ou model an optimization problem ith several
More informationGray Codes for Reflectable Languages
Gray Codes for Refletable Languages Yue Li Joe Sawada Marh 8, 2008 Abstrat We lassify a type of language alled a refletable language. We then develop a generi algorithm that an be used to list all strings
More informationImage 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
More informationEvening s Goals. Mathematical Transformations. Discuss the mathematical transformations that are utilized for computer graphics
Evening s Goals Discuss the mathematical transformations that are utilized for computer graphics projection viewing modeling Describe aspect ratio and its importance Provide a motivation for homogenous
More informationCSE528 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
More informationCSE328 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:
More informationRecall: 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
More informationMachine Vision. Laboratory Exercise Name: Student ID: S
Mahine Vision 521466S Laoratory Eerise 2011 Name: Student D: General nformation To pass these laoratory works, you should answer all questions (Q.y) with an understandale handwriting either in English
More informationRotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors
Eurographis Symposium on Geometry Proessing (003) L. Kobbelt, P. Shröder, H. Hoppe (Editors) Rotation Invariant Spherial Harmoni Representation of 3D Shape Desriptors Mihael Kazhdan, Thomas Funkhouser,
More informationComputer Graphics. Ch 6. 3D Viewing
Computer Graphics Ch 6. 3D Viewing 3D Viewing Basic do you see this image as flat? 12 lines 3D Coordinate System 3D homogeneous coordinates: p = [x y z w] T Our textbook and OpenGL use a RIGHT-HANDED system
More informationComputer Graphics. Jeng-Sheng Yeh 葉正聖 Ming Chuan University (modified from Bing-Yu Chen s slides)
Computer Graphics Jeng-Sheng Yeh 葉正聖 Ming Chuan Universit (modified from Bing-Yu Chen s slides) Viewing in 3D 3D Viewing Process Specification of an Arbitrar 3D View Orthographic Parallel Projection Perspective
More information-z c = c T - c T B B-1 A 1 - c T B B-1 b. x B B -1 A 0 B -1 b. (a) (b) Figure 1. Simplex Tableau in Matrix Form
3. he Revised Simple Method he LP min, s.t. A = b ( ),, an be represented by Figure (a) below. At any Simple step, with known and -, the Simple tableau an be represented by Figure (b) below. he minimum
More informationObjectives To identify isometries To find translation images of figures
-8 9-1 Translations ontent tandards G.O. epresent transformations in the plane... describe transformations as functions that take points in the plane as inputs and give other points as outputs... Also
More informationAn Automatic Laser Scanning System for Accurate 3D Reconstruction of Indoor Scenes
An Automati Laser Sanning System for Aurate 3D Reonstrution of Indoor Senes Danrong Li, Honglong Zhang, and Zhan Song Shenzhen Institutes of Advaned Tehnology Chinese Aademy of Sienes Shenzhen, Guangdong
More informationColouring contact graphs of squares and rectilinear polygons de Berg, M.T.; Markovic, A.; Woeginger, G.
Colouring ontat graphs of squares and retilinear polygons de Berg, M.T.; Markovi, A.; Woeginger, G. Published in: nd European Workshop on Computational Geometry (EuroCG 06), 0 Marh - April, Lugano, Switzerland
More informationMatrix Transformations. Affine Transformations
Matri ransformations Basic Graphics ransforms ranslation Scaling Rotation Reflection Shear All Can be Epressed As Linear Functions of the Original Coordinates : A + B + C D + E + F ' A ' D 1 B E C F 1
More informationSpecial Relativistic (Flight-)Simulator
Speial Relativisti (Flight-)Simulator Anton Tsoulos and Wolfgang Knopki Abstrat With speial relativisti visualisation it is possible to experiene and simulate relativisti effets whih our when traveling
More informationTo name coordinates of special figures by using their properties
6-8 Appling Coordinate Geometr Content tandard Prepares for G.GP.4 Use coordinates to prove simple geometric theorems algebraicall. bjective o name coordinates of special figures b using their properties
More informationSupplementary Material: Geometric Calibration of Micro-Lens-Based Light-Field Cameras using Line Features
Supplementary Material: Geometri Calibration of Miro-Lens-Based Light-Field Cameras using Line Features Yunsu Bok, Hae-Gon Jeon and In So Kweon KAIST, Korea As the supplementary material, we provide detailed
More information9 3 Rotations 9 4 Symmetry
h 9: Transformations 9 1 Translations 9 Reflections 9 3 Rotations 9 Smmetr 9 1 Translations: Focused Learning Target: I will be able to Identif Isometries. Find translation images of figures. Vocabular:
More informationLecture 02 Image Formation
Institute of Informatis Institute of Neuroinformatis Leture 2 Image Formation Davide Saramuzza Outline of this leture Image Formation Other amera parameters Digital amera Perspetive amera model Lens distortion
More informationDrawing lines. Naïve line drawing algorithm. drawpixel(x, round(y)); double dy = y1 - y0; double dx = x1 - x0; double m = dy / dx; double y = y0;
Naïve line drawing algorithm // Connet to grid points(x0,y0) and // (x1,y1) by a line. void drawline(int x0, int y0, int x1, int y1) { int x; double dy = y1 - y0; double dx = x1 - x0; double m = dy / dx;
More informationLAMC Junior Circle April 15, Constructing Triangles.
LAMC Junior Cirle April 15, 2012 Olga Radko radko@math.ula.edu Oleg Gleizer oleg1140@gmail.om Construting Triangles. Copyright: for home use only. This handout is a part of the book in preparation. Using
More informationYear 11 GCSE Revision - Re-visit work
Week beginning 6 th 13 th 20 th HALF TERM 27th Topis for revision Fators, multiples and primes Indies Frations, Perentages, Deimals Rounding 6 th Marh Ratio Year 11 GCSE Revision - Re-visit work Understand
More informationToday s class. Geometric objects and transformations. Informationsteknologi. Wednesday, November 7, 2007 Computer Graphics - Class 5 1
Toda s class Geometric objects and transformations Wednesda, November 7, 27 Computer Graphics - Class 5 Vector operations Review of vector operations needed for working in computer graphics adding two
More informationThe Happy Ending Problem
The Happy Ending Problem Neeldhara Misra STATUTORY WARNING This doument is a draft version 1 Introdution The Happy Ending problem first manifested itself on a typial wintery evening in 1933 These evenings
More informationRealtime 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
More information[ ] [ ] Orthogonal Transformation of Cartesian Coordinates in 2D & 3D. φ = cos 1 1/ φ = tan 1 [ 2 /1]
Orthogonal Transformation of Cartesian Coordinates in 2D & 3D A vector is specified b its coordinates, so it is defined relative to a reference frame. The same vector will have different coordinates in
More informationWhat 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)
More informationMeasurement of the stereoscopic rangefinder beam angular velocity using the digital image processing method
Measurement of the stereosopi rangefinder beam angular veloity using the digital image proessing method ROMAN VÍTEK Department of weapons and ammunition University of defense Kouniova 65, 62 Brno CZECH
More informationLecture 1: Turtle Graphics. the turtle and the crane and the swallow observe the time of their coming; Jeremiah 8:7
Lecture 1: Turtle Graphics the turtle and the crane and the sallo observe the time of their coming; Jeremiah 8:7 1. Turtle Graphics Motion generates geometry. The turtle is a handy paradigm for investigating
More informationComputer Graphics. Bing-Yu Chen National Taiwan University The University of Tokyo
Computer Graphics Bing-Yu Chen National Taiwan Universit The Universit of Toko Viewing in 3D 3D Viewing Process Classical Viewing and Projections 3D Snthetic Camera Model Parallel Projection Perspective
More informationChapter 1. Turtle Graphics. 1.1 Turtle Graphics. The turtle and the crane and the swallow observe the time of their coming Jeremiah 8:7
Goldman/An Integrated Introduction to Computer Graphics and Geometric Modeling K10188_C001 Revise Proof page 3 26.3.2009 7:54am Compositor Name: VAmoudavally Chapter 1 Turtle Graphics The turtle and the
More informationImage 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())
More informationPre-Critical incidence
Seismi methods: Refration I Refration reading: Sharma p58-86 Pre-Critial inidene Refletion and refration Snell s Law: sin i sin R sin r P P P P P P where p is the ray parameter and is onstant along eah
More informationPlane-based Calibration of a Camera with Varying Focal Length: the Centre Line Constraint
Planebased Calibration of a Camera with Varying Foal Length: the Centre Line Constraint Pierre GURDOS and René PAYRISSAT IRITUPS TCI 118 route de Narbonne 31062 Toulouse Cedex 4 FRANCE PierreGurdjos@iritfr
More informationModeling 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
More informationSection 1.4 Limits involving infinity
Section. Limits involving infinit (/3/08) Overview: In later chapters we will need notation and terminolog to describe the behavior of functions in cases where the variable or the value of the function
More informationSolving Problems Using Quadratic Models. LEARN ABOUT the Math. concert, where n is the number of tickets sold.
6.6 Solving Problems Using Quadratic Models YOU WILL NEED grid paper ruler graphing calculator GOAL Solve problems that can be modelled by quadratic relations using a variety of tools and strategies. LEARN
More informationFind the length, x, in the diagram, rounded to the nearest tenth of a centimetre.
The tangent ratio relates two sides of a right triangle and an angle. If ou know an angle and the length of one of the legs of the triangle, ou can find the length of the other leg. Eample Find a Side
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mathematics SKE, Strand J UNIT J Further Transformations: Tet STRND J: TRNSFORMTIONS, VETORS and MTRIES J Further Transformations Tet ontents Section J.1 Translations * J. ombined Transformations Mathematics
More informationThree-Dimensional Viewing Hearn & Baker Chapter 7
Three-Dimensional Viewing Hearn & Baker Chapter 7 Overview 3D viewing involves some tasks that are not present in 2D viewing: Projection, Visibility checks, Lighting effects, etc. Overview First, set up
More informationViewing Transformations I Comp 535
Viewing Transformations I Comp 535 Motivation Want to see our virtual 3-D worl on a 2-D screen 2 Graphics Pipeline Moel Space Moel Transformations Worl Space Viewing Transformation Ee/Camera Space Projection
More information3D Computer Vision II. Reminder Projective Geometry, Transformations. Nassir Navab. October 27, 2009
3D Computer Vision II Reminder Projective Geometr, Transformations Nassir Navab based on a course given at UNC b Marc Pollefes & the book Multiple View Geometr b Hartle & Zisserman October 27, 29 2D Transformations
More information3D Model Based Pose Estimation For Omnidirectional Stereovision
3D Model Based Pose Estimation For Omnidiretional Stereovision Guillaume Caron, Eri Marhand and El Mustapha Mouaddib Abstrat Robot vision has a lot to win as well with wide field of view indued by atadioptri
More informationMotion Correction Structured Light using Pattern Interleaving Technique
Universit of Kentuk UKnoledge Universit of Kentuk Master's Theses Graduate Shool 28 Motion Corretion Strutured Light using Pattern Interleaving Tehnique Raja Kalan Ra Cavaturu Universit of Kentuk, rajaavaturu@gail.o
More informationA Unified Subdivision Scheme for Polygonal Modeling
EUROGRAPHICS 2 / A. Chalmers and T.-M. Rhyne (Guest Editors) Volume 2 (2), Number 3 A Unified Subdivision Sheme for Polygonal Modeling Jérôme Maillot Jos Stam Alias Wavefront Alias Wavefront 2 King St.
More informationModeling 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
More information3D 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
More informationCleanUp: Improving Quadrilateral Finite Element Meshes
CleanUp: Improving Quadrilateral Finite Element Meshes Paul Kinney MD-10 ECC P.O. Box 203 Ford Motor Company Dearborn, MI. 8121 (313) 28-1228 pkinney@ford.om Abstrat: Unless an all quadrilateral (quad)
More informationCS 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
More informationGraph-Based vs Depth-Based Data Representation for Multiview Images
Graph-Based vs Depth-Based Data Representation for Multiview Images Thomas Maugey, Antonio Ortega, Pasal Frossard Signal Proessing Laboratory (LTS), Eole Polytehnique Fédérale de Lausanne (EPFL) Email:
More informationHidden-Surface Removal.
Hidden-Surface emoval. Here we need to discover whether an object is visible or another one obscures it. here are two fundamental approaches to remove the hidden surfaces: ) he object-space approach )
More informationSTRAND I: Geometry and Trigonometry. UNIT 37 Further Transformations: Student Text Contents. Section Reflections. 37.
MEP Jamaica: STRN I UNIT 7 Further Transformations: Student Tet ontents STRN I: Geometr and Trigonometr Unit 7 Further Transformations Student Tet ontents Section 7. Reflections 7. Rotations 7. Translations
More information3D 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
More informationC URVES AND V ECTORS
96-ch- SB5-Ostebee June 6, 9:57 C H A P T E R O U T L I N E. Three-Dimensional Space. Curves and Parametric Equations. Polar Coordinates and Polar Curves.4 Vectors.5 Vector-Valued Functions, Derivatives,
More informationIntroduction to Computer Graphics 4. Viewing in 3D
Introduction to Computer Graphics 4. Viewing in 3D National Chiao Tung Univ, Taiwan By: I-Chen Lin, Assistant Professor Textbook: E.Angel, Interactive Computer Graphics, 5 th Ed., Addison Wesley Ref: Hearn
More informationTriangle LMN and triangle OPN are similar triangles. Find the angle measurements for x, y, and z.
1 Use measurements of the two triangles elow to find x and y. Are the triangles similar or ongruent? Explain. 1a Triangle LMN and triangle OPN are similar triangles. Find the angle measurements for x,
More informationImage 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
More informationProperties Transformations
9 Properties of Transformations 9. Translate Figures and Use Vectors 9.2 Use Properties of Matrices 9.3 Perform Reflections 9.4 Perform Rotations 9.5 ppl ompositions of Transformations 9.6 Identif Smmetr
More informationUniform spherical grids via equal area projection from the cube to the sphere
Uniform spherical grids via equal area projection from the cube to the sphere Daniela Roşca Gerlind Plonka April 4, 0 Abstract We construct an area preserving map from the cube to the unit sphere S, both
More informationWhat 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
More informationViewing/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
More informationUniversal Turing Machine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice s Theorem Decidability Continued
CD5080 AUBER odels of Computation, anguages and Automata ecture 14 älardalen University Content Universal Turing achine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice s Decidability
More information9.4. Perform Rotations. Draw a rotation. STEP 1 Draw a segment from A to P. STEP 2 Draw a ray to form a 1208 angle with } PA.
40 40 50 30 9.4 erform otations efore You rotated figures about the origin. Now You will rotate figures about a point. Wh? So ou can classif transformations, as in Es. 3 5. Ke Vocabular center of rotation
More informationAbstract. We describe a parametric hybrid Bezier patch that, in addition. schemes are local in that changes to part of the data only aect portions of
A Parametri Hyrid Triangular Bezier Path Stephen Mann and Matthew Davidhuk Astrat. We desrie a parametri hyrid Bezier path that, in addition to lending interior ontrol points, lends oundary ontrol points.
More information1-2 Geometric vectors
1-2 Geometric ectors We are going to start simple, by defining 2-dimensional ectors, the simplest ectors there are. Are these the ectors that can be defined by to numbers only? Yes, and here is a formal
More informationTransformations. Examples of transformations: shear. scaling
Transformations Eamples of transformations: translation rotation scaling shear Transformations More eamples: reflection with respect to the y-ais reflection with respect to the origin Transformations Linear
More informationSTAT, GRAPH, TA- BLE, RECUR
Chapter Sketch Function The sketch function lets you dra lines and graphs on an existing graph. Note that Sketch function operation in the STAT, GRAPH, TA- BLE, RECUR and CONICS Modes is different from
More informationWeek 3. Topic 5 Asymptotes
Week 3 Topic 5 Asmptotes Week 3 Topic 5 Asmptotes Introduction One of the strangest features of a graph is an asmptote. The come in three flavors: vertical, horizontal, and slant (also called oblique).
More information