UNIT 2 2D TRANSFORMATIONS
|
|
- Michael Reynard Phillips
- 5 years ago
- Views:
Transcription
1 UNIT 2 2D TRANSFORMATIONS Introduction With the procedures for displaying output primitives and their attributes, we can create variety of pictures and graphs. In many applications, there is also a need for altering or manipulating displays. Design applications and facility layouts are created by arranging the orientations and sizes of the component parts of the scene. And animations are produced by moving the "camera" or the objects in a scene along animation paths. Changes in orientation, size, and shape are accomplished with geometric transformations that alter the coordinate descriptions of objects. The basic geometric transformations are translation, rotation, and scaling. Other transformations that are often applied to objects include reflection and shear. 2D TRANSFORMATIONS AND MATRICES Representation of Points: 2 x 1 matrix: x y General Problem: B = T A T represents a generic operator to be applied to the points in A. T is the geometric transformation matrix. If A & T are known, the transformed points are obtained by calculating B. General Transformation of 2D points Solid body (size and shape does not change) transformations - above equation is valid for all set of points and lines of the object being transformed. 1
2 Special cases of 2D transforms T= identity matrix, a=d=1, b=c=0 => x'=x, y'=y Scaling A scaling transformation alters the size of an object b=0, c=0 => x' = a.x, y' = d.y; This is scaling by a in x, d in y. If, a=d >1,we have enlargement; If, 0< a=d < 1,we have compression; If a=d, we have uniform scaling, else non uniform scaling or differential scaling. Scale matrix: let Sx =a, Sy =d Sx 0 0 Sy Example of Scaling 2
3 If Sx and/ or Sy< 0 are negative reflections are obtained through an axis or plane. Only diagonal terms involved in scaling and reflections. Reflection A transformation that produces a mirror image of an object - Shearing - Shear transformation distorts the shape of the object. - This effect causes the object to be pushed to one side as it it was constructed of layers that slide on top of each other. - Off diagonal terms are involved in Shearing. 3
4 a = d = 1 let, c = 0, b = 2 x' = x y' = 2x+ y y' depends linearly on x; This effect is called shear. Similarly for b=0, c not equal to zero. The shear in this case is proportional to y- coordinate. Rotation A two-dimensional rotation is applied to an object by repositioning it along a circular path in the xy plane. x' = xcos(θ) - ysin(θ) y' = xsin(θ) + ycos(θ) In matrix form, this is : cos(θ) -sin(θ) sin(θ) cos(θ) 4
5 Positive Rotations can be achieved if rotation is in counter clockwise about the origin Negative rotations can be achieved if rotation is in clockwise about the origin For rotations, det T = 1 and T T = T -1, Rotation matrices are orthogonal. Transformation of a unit square Take a square with coordinates (0,0),(1,0),(1,1),(0,1) Then the matrix will be represented as The transformation on the matrix S with a Transformation matrix T = 5
6 S = S * T The unit square after applying transformation is shown in fig. as is changed to rhombus structure. Area of the unit square after transformation = ad bc = T This can be extended to any arbitrary area of any polygon. Translation 6
7 The translations are introduced with 1) Rotations -when object not centered at the origin. 2) Scaling -when objects / lines not centered at the origin. If line intersects the origin, no translation we cannot directly represent translations as matrix multiplication, as we can rotations and scalings. we can represent translations in our general transformation matrix by using homogeneous coordinates. 7
8 HOMOGENEOUS COORDINATES Each point is now represented by a triple: (x, y, W) ( x/w, y/w) are called the Cartesian coordinates of the homogeneous points. 8
9 Interpretation of homogeneous coordinates Triples of coordinates typically represent points in 3D space but here we are using them to represent points in 2D space. If we take all triples representing the same point that is all triples of the form (tx,ty,tw) with w 0, we get a line in 3D space. Thus each homogeneous point represents a line in 3D space. If we homogenize the point (divide by w), we get a point of the form(x,y,1). Thus the homogenized points form the plane defined by the equation w=1 in (x,y,w) space. The following fig. shows this relationship. Fig. The XYW homogeneous coordinate space with the w=1 plane and point P(X,Y,W) projected onto the W=1 plane. Two homogeneous coordinates (x1, y1, w1) & (x2,y2, w2) may represent the same point, iff they are multiples of one another: (1,2,3) & (3,6,9). There is no unique homogeneous representation of a point. All triples of the form (tx, ty, tw) form a line in x,y,w space. Cartesian coordinates are just the plane w=1 in this space. W=0, are the points at infinity 9
10 General purpose 2D transformations in homogeneous coordinate representation Parameters involved in scaling, rotation, reflection and shear are a, b, c, d If B = T.A then Translation parameters: ( p, q ) If B = A. T then Translation parameters: ( m, n ) COMPOSITION OF TRANSFORMATIONS If we want to apply a series of transformations T1, T2, T3 to a set of points, We can do it 2 ways: 1) We can calculate p'=t1*p, p'' = T2*p', p'''=t3*p'' 2) Calculate T= T1*T2*T3, then p'''= T*p. Method 2, saves large number of adds and multiplies. Approximately 1/3 as many operations. Therefore, we concatenate or compose the matrices into one final transformation matrix that we apply to the points. Translation Translate the points by tx1, ty1, then by tx2, ty2: 10
11 Scaling Scale the point by sx1,sy1 then by sx2,sy2. Rotation Rotate by θ1, then by θ2, 1) stick the (θ1+ θ2) in for θ, or 2) calculate T1 for θ1, then T2 for θ2 & multiply them. Gives same result Rotation about an arbitrary point P in space As we mentioned before, rotations are about the origin. So to rotate about a point P in space, translate so that P coincides with the origin, then rotate, then translate back. Steps are Translate by (-Px, -Py) Rotate Translate by (Px, Py) 11
12 T = T1(Px,Py) * T2(θ) * T3(-Px, -Py) Scaling about an arbitrary point in Space Again, Translate P to the origin Scale Translate P back T = T1(Px,Py) * T2(sx, sy)* T3(-Px, -Py) Reflection through an arbitrary line Reflections about any line y = mx + b in the xy plane can be accomplished with a combination of translate rotate-reflect transformations. In general, we first translate the Line so that it passes through the origin. Then we can rotate the line onto one of the coordinate axes and reflect about that axis. Finally, we restore the line to its original position with the inverse rotation and translation transformations. Steps: Translate line to the origin Rotation about the origin Reflection matrix Reverse the rotation Translate line back 12
13 Commutivity of Transformations Matrix multiplication is associative. For any three matrices, A, B, and C, the matrix product A. B. C can be performed by first multiplying A and B or by first multiplying B and C: A. B.C = (A.B).C = A. (B. C) Therefore, we can evaluate matrix products using either a left-to-right or a right to left associative grouping. On the other hand, transformation products may not be commutative: The matrix product A. B is not equal to B. A, in general. This means that if we want to translate and rotate an object, we must be careful about the order in which the composite matrix is evaluated. For some special cases, such as a sequence of transformations all of the same kind, the multiplication of transformation matrices is commutative. The following table shows the cases where T1 * T2 = T2 * T1 T1 translation scale rotation scale(uniform) T2 translation scale rotation rotation 13
14 Example: Order: R G B Fig.a fig.b In Figure a, the object is first scaled and then translated. In figure b, the object is first translated and then scaled. As seen in figs. both are not identical. Fig.c Fig.d In Figure c, the object is first rotated and then differential scaling is applied. In figure d, the object is first differentially scaled and then rotated. As seen in figs. both are not identical. In general, multiplication of transformation matrices is not commutative. 14
15 COORDINATE SYSTEMS Screen Coordinates: The coordinate system used to address the screen ( device coordinates) World Coordinates: A user-defined application specific coordinate system having its own units of measure, axis, origin, etc. Window: The rectangular region of the world that is visible. Viewport: The rectangular region of the screen space that is used to display the window. WINDOW TO VIEWPORT TRANSFORMATION The purpose is to find the transformation matrix that maps the window in world coordinates to the viewport in screen coordinates. Viewport: (u, v space) denoted by: u min, v min, u max, v max Window: (x, y space) denoted by: x min, y min, x max, y max 15
16 The overall transformation: Translate the window to the origin Scale it to the size of the viewport Translate it to the viewport location M WV -> Transformation matrix to convert Window to view port. Summary: The basic geometric transformations are translation, rotation, and scaling. Translation moves an object in a straight-line path from one position to another. Rotation moves an object from one position to another in a circular path around a specified pivot point (rotation point). Scaling changes the dimensions of an object relative to a specified fixed point. We can express two-dimensional geometric transformations as 3 by 3 matrix operators, so that sequences of transformations can be concatenated into a single composite matrix. This is an efficient formulation, since it allows us to reduce computations by applying the composite matrix to the initial coordinate positions of an object to obtain the final transformed positions. 16
17 To do this, we also need to express two-dimensional coordinate positions as threeelement column or row matrices. We choose a column-matrix representation for coordinate points because this is the standard mathematical convention and because many graphics packages also follow this convention. For twodimensional transformations, coordinate positions are: then represented with three-element homogeneous coordinates with the third (homogeneous) coordinate assigned the value. I. Composite transformations are formed as multiplications of any combination of translation, rotation, and scaling matrices. Other transformations include reflections and shears. Reflections are transformations that rotate an object 180" about a reflection axis. This produces a mirror image of the object with respect to that axis. When the reflection axis is perpendicular to the xy plane, the reflection is obtained as a rotation n the xy plane. When the reflection axis is in the xy plane, the reflection is obtained as a rotation in a plane that is perpendicular to the xy plane. Shear transformations distort the shape of an object by shifting x or y coordinate values by an amount to the coordinate distance from a shear reference line 2 Marks 1. What is Transformation? Transformation is the process of introducing changes in the shape size and orientation of the object using scaling rotation reflection shearing & translation etc. 2. Write short notes on active and passive transformations? In the active transformation the points x and x represent different coordinates of the same coordinate system. Here all the points are acted upon by the same transformation and hence the shape of the object is not distorted. In a passive transformation the points x and x represent same points in the space but in a different coordinate system. Here the change in the coordinates is merely due to the change in the type of the user coordinate system. 3. What is translation? Translation is the process of changing the position of an object in a straight-line path from one coordinate location to another. Every point (x, y) in the object must under go a displacement to (x,y ). the transformation is: x = x + tx ; y = y + ty 4. What is rotation? A 2-D rotation is done by repositioning the coordinates along a circular path, in the x-y plane by making an angle with the axes. The transformation is given by: x' = xcos(θ) - ysin(θ) y' = xsin(θ) + ycos(θ) 17
18 5. What is scaling? The scaling transformations changes the shape of an object and can be carried out by multiplying each vertex (x,y) by scaling factor Sx,Sy where Sx is the scaling factor of x and Sy is the scaling factor of y. 6. What is shearing? The shearing transformation actually slants the object along the X direction or the Y direction as required.ie; this transformation slants the shape of an object along a required plane. 7. What is reflection? The reflection is actually the transformation that produces a mirror image of an object. For this use some angles and lines of reflection. 8. Distinguish between window port & view port? A portion of a picture that is to be displayed by a window is known as window port. The display area of the part selected or the form in which the selected part is viewed is known as view port. 9. Define clipping? Clipping is the method of cutting a graphics display to neatly fit a predefined graphics region or the view port. 10. What is covering (exterior clipping)? This is just opposite to clipping. This removes the lines coming inside the windows and displays the remaining. Covering is mainly used to make labels on the complex pictures. 11. What is the need of homogeneous coordinates? To perform more than one transformation at a time, use homogeneous coordinates or matrixes. They reduce unwanted calculations intermediate steps saves time and memory and produce a sequence of transformations. 12. Distinguish between uniform scaling and differential scaling? When the scaling factors sx and sy are assigned to the same value, a uniform scaling is produced that maintains relative object proportions. Unequal values for sx and sy result in a differential scaling that is often used in design application. 13. What is fixed point scaling? The location of a scaled object can be controlled by a position called the fixed point that is to remain unchanged after the scaling transformation..14. Determine a sequence of basic transformations that are equivalent to the x-direction shearing matrix. 18
19 15. Why are matrices used for implementing transformations. Matrices allow arbitrary linear transformations to be represented in a consistent format, suitable for computation. This also allows transformations to be concatenated easily (by multiplying their matrices). Review Questions: 1. Explain reflection and shear? 2. Explain Basic Transformations with procedure 3. Explain LCD and LED. 4. Derive the rotation transformation for rotating a point w.r.t. an arbitrary point. 5. Transform a square with vertices A(2,0), B(6,0), C(6,6) and D(2,6) to half its size and placed at location such that the centre of square moves to (0,0). What will be the effect of shearing this transformed square with x-shearing factor as 2? 6. What is the difference between a window and viewport? Derive the window-toviewport transformation of a point P(4,4) inside a circular window of radius 6 and centre (2,2) transformed onto a circular viewport of radius 3 and centre (-1,-1). 7. The triangle position is A(1,1),B(3,5) and C(2,2), translate the given triangle with respect to its centroid with A in the centroid position and rotate it by an angle 30.after shrinking the object half the size. 8. Describe the matrix formulation of 2-D transformations; translation, rotation and scaling. 9. Device the window-to-view port transformation and Elaborate What is shearing transformation? Write down the X-direction shearing matrix for two dimension. Determine a sequence of basic transformations that are equivalent to the X-direction shearing. 19
2D TRANSFORMATIONS AND MATRICES
2D TRANSFORMATIONS AND MATRICES Representation of Points: 2 x 1 matrix: x y General Problem: B = T A T represents a generic operator to be applied to the points in A. T is the geometric transformation
More informationSection III: TRANSFORMATIONS
Section III: TRANSFORMATIONS in 2-D 2D TRANSFORMATIONS AND MATRICES Representation of Points: 2 x 1 matrix: X Y General Problem: [B] = [T] [A] [T] represents a generic operator to be applied to the points
More informationDHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI
DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI Department of Computer Science and Engineering CS6504-COMPUTER GRAPHICS Anna University 2 & 16 Mark Questions & Answers Year / Semester: III / V Regulation:
More informationComputer Graphics: Geometric Transformations
Computer Graphics: Geometric Transformations Geometric 2D transformations By: A. H. Abdul Hafez Abdul.hafez@hku.edu.tr, 1 Outlines 1. Basic 2D transformations 2. Matrix Representation of 2D transformations
More informationObject Representation Affine Transforms. Polygonal Representation. Polygonal Representation. Polygonal Representation of Objects
Object Representation Affine Transforms Polygonal Representation of Objects Although perceivable the simplest form of representation they can also be the most problematic. To represent an object polygonally,
More informationHello, welcome to the video lecture series on Digital Image Processing. So in today's lecture
Digital Image Processing Prof. P. K. Biswas Department of Electronics and Electrical Communications Engineering Indian Institute of Technology, Kharagpur Module 02 Lecture Number 10 Basic Transform (Refer
More informationUnit 3 Transformations and Clipping
Transformation Unit 3 Transformations and Clipping Changes in orientation, size and shape of an object by changing the coordinate description, is known as Geometric Transformation. Translation To reposition
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 informationModeling Transformations
Modeling Transformations Connelly Barnes CS 4810: Graphics Acknowledgment: slides by Connelly Barnes, Misha Kazhdan, Allison Klein, Tom Funkhouser, Adam Finkelstein and David Dobkin Modeling Transformations
More informationPart 3: 2D Transformation
Part 3: 2D Transformation 1. What do you understand by geometric transformation? Also define the following operation performed by ita. Translation. b. Rotation. c. Scaling. d. Reflection. 2. Explain two
More informationHomogeneous Coordinates and Transformations of the Plane
2 Homogeneous Coordinates and Transformations of the Plane 2. Introduction In Chapter planar objects were manipulated by applying one or more transformations. Section.7 identified the problem that the
More information2D and 3D Transformations AUI Course Denbigh Starkey
2D and 3D Transformations AUI Course Denbigh Starkey. Introduction 2 2. 2D transformations using Cartesian coordinates 3 2. Translation 3 2.2 Rotation 4 2.3 Scaling 6 3. Introduction to homogeneous coordinates
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 informationCoordinate transformations. 5554: Packet 8 1
Coordinate transformations 5554: Packet 8 1 Overview Rigid transformations are the simplest Translation, rotation Preserve sizes and angles Affine transformation is the most general linear case Homogeneous
More informationGraphics and Interaction Transformation geometry and homogeneous coordinates
433-324 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationCOMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates
COMP30019 Graphics and Interaction Transformation geometry and homogeneous coordinates Department of Computer Science and Software Engineering The Lecture outline Introduction Vectors and matrices Translation
More informationComputer Graphics with OpenGL ES (J. Han) Chapter IV Spaces and Transforms
Chapter IV Spaces and Transforms Scaling 2D scaling with the scaling factors, s x and s y, which are independent. Examples When a polygon is scaled, all of its vertices are processed by the same scaling
More informationTwo 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
More informationOverview. Affine Transformations (2D and 3D) Coordinate System Transformations Vectors Rays and Intersections
Overview Affine Transformations (2D and 3D) Coordinate System Transformations Vectors Rays and Intersections ITCS 4120/5120 1 Mathematical Fundamentals Geometric Transformations A set of tools that aid
More informationGame Engineering: 2D
Game Engineering: 2D CS420-2010F-07 Objects in 2D David Galles Department of Computer Science University of San Francisco 07-0: Representing Polygons We want to represent a simple polygon Triangle, rectangle,
More information2D transformations: An introduction to the maths behind computer graphics
2D transformations: An introduction to the maths behind computer graphics Lecturer: Dr Dan Cornford d.cornford@aston.ac.uk http://wiki.aston.ac.uk/dancornford CS2150, Computer Graphics, Aston University,
More informationFigure 1. Lecture 1: Three Dimensional graphics: Projections and Transformations
Lecture 1: Three Dimensional graphics: Projections and Transformations Device Independence We will start with a brief discussion of two dimensional drawing primitives. At the lowest level of an operating
More information3D Mathematics. Co-ordinate systems, 3D primitives and affine transformations
3D Mathematics Co-ordinate systems, 3D primitives and affine transformations Coordinate Systems 2 3 Primitive Types and Topologies Primitives Primitive Types and Topologies 4 A primitive is the most basic
More informationTransformations: 2D Transforms
1. Translation Transformations: 2D Transforms Relocation of point WRT frame Given P = (x, y), translation T (dx, dy) Then P (x, y ) = T (dx, dy) P, where x = x + dx, y = y + dy Using matrix representation
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 information2D and 3D Coordinate Systems and Transformations
Graphics & Visualization Chapter 3 2D and 3D Coordinate Systems and Transformations Graphics & Visualization: Principles & Algorithms Introduction In computer graphics is often necessary to change: the
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More informationMET71 COMPUTER AIDED DESIGN
UNIT - II BRESENHAM S ALGORITHM BRESENHAM S LINE ALGORITHM Bresenham s algorithm enables the selection of optimum raster locations to represent a straight line. In this algorithm either pixels along X
More information1 Affine and Projective Coordinate Notation
CS348a: Computer Graphics Handout #9 Geometric Modeling Original Handout #9 Stanford University Tuesday, 3 November 992 Original Lecture #2: 6 October 992 Topics: Coordinates and Transformations Scribe:
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 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 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 informationUNIT 2. Translation/ Scaling/ Rotation. Unit-02/Lecture-01
UNIT 2 Translation/ Scaling/ Rotation Unit-02/Lecture-01 Translation- (RGPV-2009,10,11,13,14) Translation is to move, or parallel shift, a figure. We use a simple point as a starting point. This is a simple
More informationTransformations. Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico
Transformations Ed Angel Professor of Computer Science, Electrical and Computer Engineering, and Media Arts University of New Mexico Angel: Interactive Computer Graphics 4E Addison-Wesley 25 1 Objectives
More informationMODULE - 7. Subject: Computer Science. Module: Other 2D Transformations. Module No: CS/CGV/7
MODULE - 7 e-pg Pathshala Subject: Computer Science Paper: Computer Graphics and Visualization Module: Other 2D Transformations Module No: CS/CGV/7 Quadrant e-text Objectives: To get introduced to the
More information(Refer Slide Time: 00:04:20)
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 8 Three Dimensional Graphics Welcome back all of you to the lectures in Computer
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 informationLast week. Machiraju/Zhang/Möller/Fuhrmann
Last week Machiraju/Zhang/Möller/Fuhrmann 1 Geometry basics Scalar, point, and vector Vector space and affine space Basic point and vector operations Sided-ness test Lines, planes, and triangles Linear
More information3D Computer Graphics. Jared Kirschner. November 8, 2010
3D Computer Graphics Jared Kirschner November 8, 2010 1 Abstract We are surrounded by graphical displays video games, cell phones, television sets, computer-aided design software, interactive touch screens,
More informationAH Matrices.notebook November 28, 2016
Matrices Numbers are put into arrays to help with multiplication, division etc. A Matrix (matrices pl.) is a rectangular array of numbers arranged in rows and columns. Matrices If there are m rows and
More information2D Object Definition (1/3)
2D Object Definition (1/3) Lines and Polylines Lines drawn between ordered points to create more complex forms called polylines Same first and last point make closed polyline or polygon Can intersect itself
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 informationComputer Graphics Hands-on
Computer Graphics Hands-on Two-Dimensional Transformations Objectives Visualize the fundamental 2D geometric operations translation, rotation about the origin, and scale about the origin Learn how to compose
More informationCSE528 Computer Graphics: Theory, Algorithms, and Applications
CSE528 Computer Graphics: Theory, Algorithms, and Applications Hong Qin Stony Brook University (SUNY at Stony Brook) Stony Brook, New York 11794-2424 Tel: (631)632-845; Fax: (631)632-8334 qin@cs.stonybrook.edu
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 informationChapter 5. Transforming Shapes
Chapter 5 Transforming Shapes It is difficult to walk through daily life without being able to see geometric transformations in your surroundings. Notice how the leaves of plants, for example, are almost
More informationComputer Graphics: Two Dimensional Viewing
Computer Graphics: Two Dimensional Viewing Clipping and Normalized Window By: A. H. Abdul Hafez Abdul.hafez@hku.edu.tr, 1 Outlines 1. End 2 Transformation between 2 coordinate systems To transform positioned
More informationInteractive Computer Graphics. Hearn & Baker, chapter D transforms Hearn & Baker, chapter 5. Aliasing and Anti-Aliasing
Interactive Computer Graphics Aliasing and Anti-Aliasing Hearn & Baker, chapter 4-7 D transforms Hearn & Baker, chapter 5 Aliasing and Anti-Aliasing Problem: jaggies Also known as aliasing. It results
More informationGeometric transformations assign a point to a point, so it is a point valued function of points. Geometric transformation may destroy the equation
Geometric transformations assign a point to a point, so it is a point valued function of points. Geometric transformation may destroy the equation and the type of an object. Even simple scaling turns a
More informationAffine Transformation. Edith Law & Mike Terry
Affine Transformation Edith Law & Mike Terry Graphic Models vs. Images Computer Graphics: the creation, storage and manipulation of images and their models Model: a mathematical representation of an image
More informationTransformations Week 9, Lecture 18
CS 536 Computer Graphics Transformations Week 9, Lecture 18 2D Transformations David Breen, William Regli and Maxim Peysakhov Department of Computer Science Drexel University 1 3 2D Affine Transformations
More informationChapter 5. Projections and Rendering
Chapter 5 Projections and Rendering Topics: Perspective Projections The rendering pipeline In order to view manipulate and view a graphics object we must find ways of storing it a computer-compatible way.
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics 3D Viewing and Projection Yong Cao Virginia Tech Objective We will develop methods to camera through scenes. We will develop mathematical tools to handle perspective projection.
More information2D/3D Geometric Transformations and Scene Graphs
2D/3D Geometric Transformations and Scene Graphs Week 4 Acknowledgement: The course slides are adapted from the slides prepared by Steve Marschner of Cornell University 1 A little quick math background
More informationTHE VIEWING TRANSFORMATION
ECS 178 Course Notes THE VIEWING TRANSFORMATION Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview One of the most important
More informationCSE328 Fundamentals of Computer Graphics
CSE328 Fundamentals of Computer Graphics Hong Qin State University of New York at Stony Brook (Stony Brook University) Stony Brook, New York 794--44 Tel: (63)632-845; Fax: (63)632-8334 qin@cs.sunysb.edu
More informationHomework 5: Transformations in geometry
Math 21b: Linear Algebra Spring 2018 Homework 5: Transformations in geometry This homework is due on Wednesday, February 7, respectively on Thursday February 8, 2018. 1 a) Find the reflection matrix at
More informationSection 12.1 Translations and Rotations
Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry. We look at two types of isometries in this section: translations and rotations. Translations A
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 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 informationTherefore, after becoming familiar with the Matrix Method, you will be able to solve a system of two linear equations in four different ways.
Grade 9 IGCSE A1: Chapter 9 Matrices and Transformations Materials Needed: Straightedge, Graph Paper Exercise 1: Matrix Operations Matrices are used in Linear Algebra to solve systems of linear equations.
More informationRectangular Coordinates in Space
Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11 Introduction We quickly review one and two-dimensional spaces and then
More informationGEOMETRIC OBJECTS AND TRANSFORMATIONS I
Computer UNIT Graphics - 4 and Visualization 6 Hrs GEOMETRIC OBJECTS AND TRANSFORMATIONS I Scalars Points, and vectors Three-dimensional primitives Coordinate systems and frames Modelling a colored cube
More informationLinear algebra deals with matrixes: two-dimensional arrays of values. Here s a matrix: [ x + 5y + 7z 9x + 3y + 11z
Basic Linear Algebra Linear algebra deals with matrixes: two-dimensional arrays of values. Here s a matrix: [ 1 5 ] 7 9 3 11 Often matrices are used to describe in a simpler way a series of linear equations.
More informationCALCULATING TRANSFORMATIONS OF KINEMATIC CHAINS USING HOMOGENEOUS COORDINATES
CALCULATING TRANSFORMATIONS OF KINEMATIC CHAINS USING HOMOGENEOUS COORDINATES YINGYING REN Abstract. In this paper, the applications of homogeneous coordinates are discussed to obtain an efficient model
More informationSpecifying Complex Scenes
Transformations Specifying Complex Scenes (x,y,z) (r x,r y,r z ) 2 (,,) Specifying Complex Scenes Absolute position is not very natural Need a way to describe relative relationship: The lego is on top
More informationMatrices. Chapter Matrix A Mathematical Definition Matrix Dimensions and Notation
Chapter 7 Introduction to Matrices This chapter introduces the theory and application of matrices. It is divided into two main sections. Section 7.1 discusses some of the basic properties and operations
More informationCOMP30019 Graphics and Interaction Three-dimensional transformation geometry and perspective
COMP30019 Graphics and Interaction Three-dimensional transformation geometry and perspective Department of Computing and Information Systems The Lecture outline Introduction Rotation about artibrary axis
More informationModels and The Viewing Pipeline. Jian Huang CS456
Models and The Viewing Pipeline Jian Huang CS456 Vertex coordinates list, polygon table and (maybe) edge table Auxiliary: Per vertex normal Neighborhood information, arranged with regard to vertices and
More informationINTRODUCTION TO COMPUTER GRAPHICS. It looks like a matrix Sort of. Viewing III. Projection in Practice. Bin Sheng 10/11/ / 52
cs337 It looks like a matrix Sort of Viewing III Projection in Practice / 52 cs337 Arbitrary 3D views Now that we have familiarity with terms we can say that these view volumes/frusta can be specified
More informationCs602-computer graphics MCQS MIDTERM EXAMINATION SOLVED BY ~ LIBRIANSMINE ~
Cs602-computer graphics MCQS MIDTERM EXAMINATION SOLVED BY ~ LIBRIANSMINE ~ Question # 1 of 10 ( Start time: 08:04:29 PM ) Total Marks: 1 Sutherland-Hodgeman clipping algorithm clips any polygon against
More informationS56 (5.3) Higher Straight Line.notebook June 22, 2015
Daily Practice 5.6.2015 Q1. Simplify Q2. Evaluate L.I: Today we will be revising over our knowledge of the straight line. Q3. Write in completed square form x 2 + 4x + 7 Q4. State the equation of the line
More information3D Polygon Rendering. Many applications use rendering of 3D polygons with direct illumination
Rendering Pipeline 3D Polygon Rendering Many applications use rendering of 3D polygons with direct illumination 3D Polygon Rendering What steps are necessary to utilize spatial coherence while drawing
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 informationGRAFIKA KOMPUTER. ~ M. Ali Fauzi
GRAFIKA KOMPUTER ~ M. Ali Fauzi Drawing 2D Graphics VIEWPORT TRANSFORMATION Recall :Coordinate System glutreshapefunc(reshape); void reshape(int w, int h) { glviewport(0,0,(glsizei) w, (GLsizei) h); glmatrixmode(gl_projection);
More informationCSC 305 The Graphics Pipeline-1
C. O. P. d y! "#"" (-1, -1) (1, 1) x z CSC 305 The Graphics Pipeline-1 by Brian Wyvill The University of Victoria Graphics Group Perspective Viewing Transformation l l l Tools for creating and manipulating
More informationCOMP3421. Vector geometry, Clipping
COMP3421 Vector geometry, Clipping Transformations Object in model co-ordinates Transform into world co-ordinates Represent points in object as 1D Matrices Multiply by matrices to transform them Coordinate
More informationCS602 MCQ,s for midterm paper with reference solved by Shahid
#1 Rotating a point requires The coordinates for the point The rotation angles Both of above Page No 175 None of above #2 In Trimetric the direction of projection makes unequal angle with the three principal
More informationComputer Vision cmput 428/615
Computer Vision cmput 428/615 Basic 2D and 3D geometry and Camera models Martin Jagersand The equation of projection Intuitively: How do we develop a consistent mathematical framework for projection calculations?
More information2009 GCSE Maths Tutor All Rights Reserved
2 This book is under copyright to GCSE Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents angles 3 bearings 8 triangle similarity 9 triangle congruency 11 Pythagoras
More informationRepresenting 2D Transformations as Matrices
Representing 2D Transformations as Matrices John E. Howland Department of Computer Science Trinity University One Trinity Place San Antonio, Texas 78212-7200 Voice: (210) 999-7364 Fax: (210) 999-7477 E-mail:
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 informationLecture 9: Transformations. CITS3003 Graphics & Animation
Lecture 9: Transformations CITS33 Graphics & Animation E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 212 Objectives Introduce standard transformations Rotation Translation Scaling
More informationViewing with Computers (OpenGL)
We can now return to three-dimension?', graphics from a computer perspective. Because viewing in computer graphics is based on the synthetic-camera model, we should be able to construct any of the classical
More informationToday. Parity. General Polygons? Non-Zero Winding Rule. Winding Numbers. CS559 Lecture 11 Polygons, Transformations
CS559 Lecture Polygons, Transformations These are course notes (not used as slides) Written by Mike Gleicher, Oct. 005 With some slides adapted from the notes of Stephen Chenney Final version (after class)
More informationChapter 5. Transformations of Objects
Chapter 5. Transformations of Objects Minus times minus is plus, the reason for this we need not discuss W.H.Auden If I eat one of these cakes, she thought, it's sure to make some change in my size. So
More informationN-Views (1) Homographies and Projection
CS 4495 Computer Vision N-Views (1) Homographies and Projection Aaron Bobick School of Interactive Computing Administrivia PS 2: Get SDD and Normalized Correlation working for a given windows size say
More informationComputer Aided Design (CAD)
CAD/CAM Dr. Ibrahim Al-Naimi Chapter two Computer Aided Design (CAD) The information-processing cycle in a typical manufacturing firm. PRODUCT DESIGN AND CAD Product design is a critical function in the
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 informationCHAPTER - 10 STRAIGHT LINES Slope or gradient of a line is defined as m = tan, ( 90 ), where is angle which the line makes with positive direction of x-axis measured in anticlockwise direction, 0 < 180
More informationMotion Control (wheeled robots)
Motion Control (wheeled robots) Requirements for Motion Control Kinematic / dynamic model of the robot Model of the interaction between the wheel and the ground Definition of required motion -> speed control,
More informationLecture 5 2D Transformation
Lecture 5 2D Transformation What is a transformation? In computer graphics an object can be transformed according to position, orientation and size. Exactly what it says - an operation that transforms
More informationLinear and Affine Transformations Coordinate Systems
Linear and Affine Transformations Coordinate Systems Recall A transformation T is linear if Recall A transformation T is linear if Every linear transformation can be represented as matrix Linear Transformation
More informationChapter 2 - Basic Mathematics for 3D Computer Graphics
Chapter 2 - Basic Mathematics for 3D Computer Graphics Three-Dimensional Geometric Transformations Affine Transformations and Homogeneous Coordinates Combining Transformations Translation t + t Add a vector
More informationGeometry Vocabulary. acute angle-an angle measuring less than 90 degrees
Geometry Vocabulary acute angle-an angle measuring less than 90 degrees angle-the turn or bend between two intersecting lines, line segments, rays, or planes angle bisector-an angle bisector is a ray that
More informationLecture 3 Sections 2.2, 4.4. Mon, Aug 31, 2009
Model s Lecture 3 Sections 2.2, 4.4 World s Eye s Clip s s s Window s Hampden-Sydney College Mon, Aug 31, 2009 Outline Model s World s Eye s Clip s s s Window s 1 2 3 Model s World s Eye s Clip s s s Window
More informationOverview of Transformations (18 marks) In many applications, changes in orientations, size, and shape are accomplished with
Two Dimensional Transformations In many applications, changes in orientations, size, and shape are accomplished with geometric transformations that alter the coordinate descriptions of objects. Basic geometric
More informationLet a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines, a line which passes through the point P
Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines, a line which passes through the point P and parallel to l, can be drawn. A triangle can be
More informationComputer Graphics Hands-on
Computer Graphics Hands-on Two-Dimensional Transformations Objectives Visualize the fundamental 2D geometric operations translation, rotation about the origin, and scale about the origin Experimentally
More informationPRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES
UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,
More information