Prof. Feng Liu. Fall /25/2018
|
|
- Calvin Beasley
- 5 years ago
- Views:
Transcription
1 Prof. Feng Liu Fall /5/08
2 Last time Clipping
3 Toda Rasterization In-class Mid-term November Close-book eam Notes on page of A or Letter size paper
4 Where We Stand At this point we know how to: Convert points from local to screen coordinates Clip polgons and lines to the view volume Net thing: Determine which piels to fill for an given point, line or polgon
5 Drawing Points When points are mapped into window coordinates, the could land anwhere not just at a piel center Solution is the simple, obvious one Map to window space Fill the closest piel Can also specif a radius fill a square of that size, or fill a circle Square is faster 5
6 Drawing Points ()
7 Drawing Lines Task: Decide which piels to fill (samples to use) to represent a line We know that all of the line lies inside the visible region (clipping gave us this!) 7
8 Line Drawing Algorithms Consider lines of the form =m + c, where m=/, 0<m<, integer coordinates All others follow b smmetr, modif for real numbers Variet of slow algorithms (Wh slow?): step, compute new at each step b equation, rounding: step, compute new at each step b adding m to old, rounding: i, i round ( mi b), round ( ) i i i i i m 8
9 Bresenham s Algorithm Overview Aim: For each, plot the piel whose -value is closest to the line Given ( i, i ), must choose from either ( i +, i +) or ( i +, i ) Idea: compute a decision variable Value that will determine which piel to draw Eas to update from one piel to the net Bresenham s algorithm is the midpoint algorithm for lines Other midpoint algorithms for conic sections (circles, ellipses) 9
10 Midpoint Methods Consider the midpoint between ( i +, i +) and ( i +, i ) If it s above the line, we choose ( i +, i ), otherwise we choose ( i +, i +) i + i + i i i i + i i + Choose ( i +, i ) Choose ( i +, i +) 0
11 Midpoint Decision Variable Write the line from (, ) to (, ) in implicit form: F, a b c Assume <= = -, = - The value of F(,) tells us where points are with respect to the line F(,)=0: the point is on the line F(,)>0: The point is above the line F(,)<0: The point is below the line The decision variable is the value of d i = F( i +, i +0.5) The factor of two makes the math easier
12 What Can We Decide? d i i ( i ) d i positive=> net point at ( i +, i ) d i negative => net point at ( i +, i +) At each point, we compute d i and decide which piel to draw How do we update it? What is d i+?
13 Updating The Decision Variable d k+ is the old value, d k, plus an increment: If we chose i+ = i +: d d k dk ( dk dk k dk ) If we chose i+ = i : d k dk What is d (assuming integer endpoints)? Notice that we don t need c an more d
14 Bresenham s Algorithm For integers, slope between 0 and : =, =, d= d -d, draw (, ) until = =+ If d<0 then { =+, draw (, ), d=d- + } If d>0 then { =, draw (, ), d=d- } Compute the constants (- and ) once at the start Inner loop does onl adds and comparisons For floating point, initialization is harder, and will be floating point, but still no rounding required
15 Eample: (,) to (7,6) =5, = d
16 Eample: (,) to (7,6) =5, = d
17 Eample: (,) to (7,6) =5, = d
18 Eample: (,) to (7,6) =5, = d
19 Eample: (,) to (7,6) =5, = d
20 Eample: (,) to (7,6) =5, = d
21 Filling Triangles
22 Filling Triangles
23 Algorithm Decide which piels to fill (samples to use) to represent a triangle? Calculate the color for each piel?
24 Barcentric coordinates P0=(0, 0) P P P P 0 0,, P=(, ) P=(, )
25 Barcentric coordinates ) ( ) (, ) ( ) (, ) ( ) (,, /,, /,, /, f f f f f f f f f
26 Rasterizing Triangle min =min( 0,, ), ma =ma( 0,, ) min =min( 0,, ), ma =ma( 0,, ) for = min to ma for = min to ma calculate,, and if 0,,and c c 0 c c draw (, ) with color c 6
27 Anti-Aliasing Recall: We can t sample and then accuratel reconstruct an image that is not band-limited Infinite Nquist frequenc Attempting to sample sharp edges gives jaggies, or stairstep lines Solution: Band-limit b filtering (pre-filtering) What sort of filter will give a band-limited result? In practice, difficult to do for graphics rendering 7
28 Alpha-based Anti-Aliasing Set the of a piel to simulate a thick line The piel gets the line color, but with <= This supports the correct drawing of primitives one on top of the other Draw back to front, and composite each primitive over the eisting image Onl some hidden surface removal algorithms support it /8 0 / /8 / 0.9/ 0 /8.9 / /
29 Calculating Consider a line as having thickness (all good drawing programs do this) Consider piels as little squares Set according to the proportion of the square covered b the line The sub-piel coverage interpretation of / /.9 /8 0 /.9 / 0 /8.9 / /
30 Weighted Sampling Instead of using the proportion of the area covered b the line, use convolution to do the sampling Equivalent to filtering the line then point sampling the result Place the filter at each piel, and integrate product of piel and line Common filters are cones (like Bartlett) or Gaussians 0
31 Post-Filtering (Supersampling) Sample at a higher resolution than required for displa, and filter image down Eas to implement in hardware Tpical is sampling per piel, with simple averaging to get final What kind of filter? More advanced methods generate different samples (eg. not on regular grid) and filter properl Issues of which samples to take, and how to filter them
32 Net Time Hidden Surface Removal
Raster Displays and Scan Conversion. Computer Graphics, CSCD18 Fall 2008 Instructor: Leonid Sigal
Raster Displays and Scan Conversion Computer Graphics, CSCD18 Fall 28 Instructor: Leonid Sigal Rater Displays Screen is represented by 2D array of locations called piels y Rater Displays Screen is represented
More informationOUTPUT PRIMITIVES. CEng 477 Introduction to Computer Graphics METU, 2007
OUTPUT PRIMITIVES CEng 477 Introduction to Computer Graphics METU, 007 Recap: The basic forward projection pipeline: MCS Model Model Modeling Transformations M M 3D World Scene Viewing Transformations
More informationGraphics Output Primitives
Important Graphics Output Primitives Graphics Output Primitives in 2D polgons, circles, ellipses & other curves piel arra operations in 3D triangles & other polgons Werner Purgathofer / Computergraphik
More informationRasterization: Geometric Primitives
Rasterization: Geometric Primitives Outline Rasterizing lines Rasterizing polygons 1 Rasterization: What is it? How to go from real numbers of geometric primitives vertices to integer coordinates of pixels
More informationCSC Computer Graphics
7//7 CSC. Computer Graphics Lecture Kasun@dscs.sjp.ac.l Department of Computer Science Universit of Sri Jaewardanepura Line drawing algorithms DDA Midpoint (Bresenham s) Algorithm Circle drawing algorithms
More informationGRAPHICS OUTPUT PRIMITIVES
CHAPTER 3 GRAPHICS OUTPUT PRIMITIVES LINE DRAWING ALGORITHMS DDA Line Algorithm Bresenham Line Algorithm Midpoint Circle Algorithm Midpoint Ellipse Algorithm CG - Chapter-3 LINE DRAWING Line drawing is
More information0. Introduction: What is Computer Graphics? 1. Basics of scan conversion (line drawing) 2. Representing 2D curves
CSC 418/2504: Computer Graphics Course web site (includes course information sheet): http://www.dgp.toronto.edu/~elf Instructor: Eugene Fiume Office: BA 5266 Phone: 416 978 5472 (not a reliable way) Email:
More informationLine Drawing. Introduction to Computer Graphics Torsten Möller / Mike Phillips. Machiraju/Zhang/Möller
Line Drawing Introduction to Computer Graphics Torsten Möller / Mike Phillips Rendering Pipeline Hardware Modelling Transform Visibility Illumination + Shading Perception, Color Interaction Texture/ Realism
More informationScan Conversion. Drawing Lines Drawing Circles
Scan Conversion Drawing Lines Drawing Circles 1 How to Draw This? 2 Start From Simple How to draw a line: y(x) = mx + b? 3 Scan Conversion, a.k.a. Rasterization Ideal Picture Raster Representation Scan
More informationRasterization, or What is glbegin(gl_lines) really doing?
Rasterization, or What is glbegin(gl_lines) really doing? Course web page: http://goo.gl/eb3aa February 23, 2012 Lecture 4 Outline Rasterizing lines DDA/parametric algorithm Midpoint/Bresenham s algorithm
More informationComputer Graphics. Modelling in 2D. 2D primitives. Lines and Polylines. OpenGL polygon primitives. Special polygons
Computer Graphics Modelling in D Lecture School of EECS Queen Mar, Universit of London D primitives Digital line algorithms Digital circle algorithms Polgon filling CG - p.hao@qmul.ac.uk D primitives Line
More informationLine Drawing. Foundations of Computer Graphics Torsten Möller
Line Drawing Foundations of Computer Graphics Torsten Möller Rendering Pipeline Hardware Modelling Transform Visibility Illumination + Shading Perception, Interaction Color Texture/ Realism Reading Angel
More informationComputer Graphics. Lecture 3 Graphics Output Primitives. Somsak Walairacht, Computer Engineering, KMITL
Computer Graphics Lecture 3 Graphics Output Primitives Somsa Walairacht, Computer Engineering, KMITL Outline Line Drawing Algorithms Circle-, Ellipse-Generating Algorithms Fill-Area Primitives Polgon Fill
More informationRasterization and Graphics Hardware. Not just about fancy 3D! Rendering/Rasterization. The simplest case: Points. When do we care?
Where does a picture come from? Rasterization and Graphics Hardware CS559 Course Notes Not for Projection November 2007, Mike Gleicher Result: image (raster) Input 2D/3D model of the world Rendering term
More informationCS 450: COMPUTER GRAPHICS RASTERIZING LINES SPRING 2016 DR. MICHAEL J. REALE
CS 45: COMPUTER GRAPHICS RASTERIZING LINES SPRING 6 DR. MICHAEL J. REALE OBJECT-ORDER RENDERING We going to start on how we will perform object-order rendering Object-order rendering Go through each OBJECT
More informationOpenGL Graphics System. 2D Graphics Primitives. Drawing 2D Graphics Primitives. 2D Graphics Primitives. Mathematical 2D Primitives.
D Graphics Primitives Eye sees Displays - CRT/LCD Frame buffer - Addressable pixel array (D) Graphics processor s main function is to map application model (D) by projection on to D primitives: points,
More informationCS559: Computer Graphics. Lecture 12: Antialiasing & Visibility Li Zhang Spring 2008
CS559: Computer Graphics Lecture 12: Antialiasing & Visibility Li Zhang Spring 2008 Antialising Today Hidden Surface Removal Reading: Shirley ch 3.7 8 OpenGL ch 1 Last time A 2 (x 0 y 0 ) (x 1 y 1 ) P
More informationUNIT -8 IMPLEMENTATION
UNIT -8 IMPLEMENTATION 1. Discuss the Bresenham s rasterization algorithm. How is it advantageous when compared to other existing methods? Describe. (Jun2012) 10M Ans: Consider drawing a line on a raster
More informationScan Conversion. CMP 477 Computer Graphics S. A. Arekete
Scan Conversion CMP 477 Computer Graphics S. A. Areete What is Scan-Conversion? 2D or 3D objects in real world space are made up of graphic primitives such as points, lines, circles and filled polygons.
More informationDerivatives 3: The Derivative as a Function
Derivatives : The Derivative as a Function 77 Derivatives : The Derivative as a Function Model : Graph of a Function 9 8 7 6 5 g() - - - 5 6 7 8 9 0 5 6 7 8 9 0 5 - - -5-6 -7 Construct Your Understanding
More informationComputer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 14
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture - 14 Scan Converting Lines, Circles and Ellipses Hello everybody, welcome again
More informationCPSC / Scan Conversion
CPSC 599.64 / 601.64 Computer Screens: Raster Displays pixel rasters (usually) square pixels in rectangular raster evenly cover the image problem no such things such as lines, circles, etc. scan conversion
More informationCS 4731: Computer Graphics Lecture 21: Raster Graphics: Drawing Lines. Emmanuel Agu
CS 4731: Computer Graphics Lecture 21: Raster Graphics: Drawing Lines Emmanuel Agu 2D Graphics Pipeline Clipping Object World Coordinates Applying world window Object subset window to viewport mapping
More informationFrom Vertices to Fragments: Rasterization. Reading Assignment: Chapter 7. Special memory where pixel colors are stored.
From Vertices to Fragments: Rasterization Reading Assignment: Chapter 7 Frame Buffer Special memory where pixel colors are stored. System Bus CPU Main Memory Graphics Card -- Graphics Processing Unit (GPU)
More informationCS 543: Computer Graphics. Rasterization
CS 543: Computer Graphics Rasterization Robert W. Lindeman Associate Professor Interactive Media & Game Development Department of Computer Science Worcester Polytechnic Institute gogo@wpi.edu (with lots
More informationRenderer Implementation: Basics and Clipping. Overview. Preliminaries. David Carr Virtual Environments, Fundamentals Spring 2005
INSTITUTIONEN FÖR SYSTEMTEKNIK LULEÅ TEKNISKA UNIVERSITET Renderer Implementation: Basics and Clipping David Carr Virtual Environments, Fundamentals Spring 2005 Feb-28-05 SMM009, Basics and Clipping 1
More informationIn today s lecture we ll have a look at: A simple technique The mid-point circle algorithm
Drawing Circles In today s lecture we ll have a look at: Circle drawing algorithms A simple technique The mid-point circle algorithm Polygon fill algorithms Summary raster drawing algorithms A Simple Circle
More informationMATH 115: Review for Chapter 1
MATH 115: Review for Chapter 1 Can you use the Distance Formula to find the distance between two points? (1) Find the distance d P, P between the points P and 1 1, 6 P 10,9. () Find the length of the line
More informationAlternate Angles. Clip 67. Mathswatch
Clip 67 Alternate Angles ) Line PQ is parallel to line RS If angle PQR is equal to 6 a) What is the size of angle QRS? b) Give a reason for ou answer. P 6 Q R S ) Line DCE is parallel to line AB a) Find
More informationScan Converting Lines
Scan Conversion 1 Scan Converting Lines Line Drawing Draw a line on a raster screen between two points What s wrong with the statement of the problem? it doesn t say anything about which points are allowed
More informationAnnouncements. Midterms graded back at the end of class Help session on Assignment 3 for last ~20 minutes of class. Computer Graphics
Announcements Midterms graded back at the end of class Help session on Assignment 3 for last ~20 minutes of class 1 Scan Conversion Overview of Rendering Scan Conversion Drawing Lines Drawing Polygons
More informationRasterization. CS4620/5620: Lecture 12. Announcements. Turn in HW 1. PPA 1 out. Friday lecture. History of graphics PPA 1 in 4621.
CS4620/5620: Lecture 12 Rasterization 1 Announcements Turn in HW 1 PPA 1 out Friday lecture History of graphics PPA 1 in 4621 2 The graphics pipeline The standard approach to object-order graphics Many
More informationMATH STUDENT BOOK. 10th Grade Unit 9
MATH STUDENT BOOK 10th Grade Unit 9 Unit 9 Coordinate Geometr MATH 1009 Coordinate Geometr INTRODUCTION 3 1. ORDERED PAIRS 5 POINTS IN A PLANE 5 SYMMETRY 11 GRAPHS OF ALGEBRAIC CONDITIONS 19 SELF TEST
More informationComputer Graphics: Graphics Output Primitives Line Drawing Algorithms
Computer Graphics: Graphics Output Primitives Line Drawing Algorithms By: A. H. Abdul Hafez Abdul.hafez@hku.edu.tr, 1 Outlines 1. Basic concept of lines in OpenGL 2. Line Equation 3. DDA Algorithm 4. DDA
More informationComputer Graphics : Bresenham Line Drawing Algorithm, Circle Drawing & Polygon Filling
Computer Graphics : Bresenham Line Drawing Algorithm, Circle Drawing & Polygon Filling Downloaded from :www.comp.dit.ie/bmacnamee/materials/graphics/006- Contents In today s lecture we ll have a loo at:
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More information12.4 The Ellipse. Standard Form of an Ellipse Centered at (0, 0) (0, b) (0, -b) center
. The Ellipse The net one of our conic sections we would like to discuss is the ellipse. We will start b looking at the ellipse centered at the origin and then move it awa from the origin. Standard Form
More informationSection 4.2 Graphing Lines
Section. Graphing Lines Objectives In this section, ou will learn to: To successfull complete this section, ou need to understand: Identif collinear points. The order of operations (1.) Graph the line
More informationLine Drawing Week 6, Lecture 9
CS 536 Computer Graphics Line Drawing Week 6, Lecture 9 David Breen, William Regli and axim Peysakhov Department of Computer Science Drexel University Outline Line drawing Digital differential analyzer
More informationRendering approaches. 1.image-oriented. 2.object-oriented. foreach pixel... 3D rendering pipeline. foreach object...
Rendering approaches 1.image-oriented foreach pixel... 2.object-oriented foreach object... geometry 3D rendering pipeline image 3D graphics pipeline Vertices Vertex processor Clipper and primitive assembler
More informationComputer Graphics. Attributes of Graphics Primitives. Somsak Walairacht, Computer Engineering, KMITL 1
Computer Graphics Chapter 4 Attributes of Graphics Primitives Somsak Walairacht, Computer Engineering, KMITL 1 Outline OpenGL State Variables Point Attributes t Line Attributes Fill-Area Attributes Scan-Line
More informationChapter - 2: Geometry and Line Generations
Chapter - 2: Geometry and Line Generations In Computer graphics, various application ranges in different areas like entertainment to scientific image processing. In defining this all application mathematics
More informationLines and Their Slopes
8.2 Lines and Their Slopes Linear Equations in Two Variables In the previous chapter we studied linear equations in a single variable. The solution of such an equation is a real number. A linear equation
More informationFrom Ver(ces to Fragments: Rasteriza(on
From Ver(ces to Fragments: Rasteriza(on From Ver(ces to Fragments 3D vertices vertex shader rasterizer fragment shader final pixels 2D screen fragments l determine fragments to be covered l interpolate
More informationRasterization. CS 4620 Lecture Kavita Bala w/ prior instructor Steve Marschner. Cornell CS4620 Fall 2015 Lecture 16
Rasterization CS 4620 Lecture 16 1 Announcements A3 due on Thu Will send mail about grading once finalized 2 Pipeline overview you are here APPLICATION COMMAND STREAM 3D transformations; shading VERTEX
More informationGraphics (Output) Primitives. Chapters 3 & 4
Graphics (Output) Primitives Chapters 3 & 4 Graphic Output and Input Pipeline Scan conversion converts primitives such as lines, circles, etc. into pixel values geometric description a finite scene area
More informationCS 450: COMPUTER GRAPHICS REVIEW: DRAWING LINES AND CIRCLES SPRING 2015 DR. MICHAEL J. REALE
CS 450: COMPUTER GRAPHICS REVIEW: DRAWING LINES AND CIRCLES SPRING 2015 DR. MICHAEL J. REALE DRAWING PRIMITIVES: LEGACY VS. NEW Legacy: specify primitive in glbegin() glbegin(gl_points); glvertex3f(1,5,0);
More informationUnit 2: Function Transformation Chapter 1
Basic Transformations Reflections Inverses Unit 2: Function Transformation Chapter 1 Section 1.1: Horizontal and Vertical Transformations A of a function alters the and an combination of the of the graph.
More informationCS 325 Computer Graphics
CS 325 Computer Graphics 02 / 06 / 2012 Instructor: Michael Eckmann Today s Topics Questions? Comments? Antialiasing Polygons Interior points Fill areas tiling halftoning dithering Antialiasing Aliasing
More informationOutput Primitives. Dr. S.M. Malaek. Assistant: M. Younesi
Output Primitives Dr. S.M. Malaek Assistant: M. Younesi Output Primitives Output Primitives: Basic geometric structures (points, straight line segment, circles and other conic sections, quadric surfaces,
More informationTransformation of curve. a. reflect the portion of the curve that is below the x-axis about the x-axis
Given graph of y f = and sketch:. Linear Transformation cf ( b + a) + d a. translate a along the -ais. f b. scale b along the -ais c. scale c along the y-ais d. translate d along the y-ais Transformation
More informationSLOPE A MEASURE OF STEEPNESS through 7.1.5
SLOPE A MEASURE OF STEEPNESS 7.1. through 7.1.5 Students have used the equation = m + b throughout this course to graph lines and describe patterns. When the equation is written in -form, the m is the
More informationCS 130 Final. Fall 2015
CS 130 Final Fall 2015 Name Student ID Signature You may not ask any questions during the test. If you believe that there is something wrong with a question, write down what you think the question is trying
More informationGRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM
FOM 11 T7 GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM 1 1 GRAPHING QUADRATIC FUNCTIONS IN STANDARD FORM I) THE STANDARD FORM OF A QUADRATIC FUNCTION (PARABOLA) IS = a +b +c. To graph a quadratic function
More information(Refer Slide Time: 00:03:51)
Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 17 Scan Converting Lines, Circles and Ellipses Hello and welcome everybody
More informationRasterization. CS4620 Lecture 13
Rasterization CS4620 Lecture 13 2014 Steve Marschner 1 The graphics pipeline The standard approach to object-order graphics Many versions exist software, e.g. Pixar s REYES architecture many options for
More informationLESSON 3.1 INTRODUCTION TO GRAPHING
LESSON 3.1 INTRODUCTION TO GRAPHING LESSON 3.1 INTRODUCTION TO GRAPHING 137 OVERVIEW Here s what ou ll learn in this lesson: Plotting Points a. The -plane b. The -ais and -ais c. The origin d. Ordered
More informationSurface shading: lights and rasterization. Computer Graphics CSE 167 Lecture 6
Surface shading: lights and rasterization Computer Graphics CSE 167 Lecture 6 CSE 167: Computer Graphics Surface shading Materials Lights Rasterization 2 Scene data Rendering pipeline Modeling and viewing
More informationUNIT 2 GRAPHIC PRIMITIVES
UNIT 2 GRAPHIC PRIMITIVES Structure Page Nos. 2.1 Introduction 46 2.2 Objectives 46 2.3 Points and Lines 46 2.4 Line Generation Algorithms 48 2.4.1 DDA Algorithm 49 2.4.2 Bresenhams Line Generation Algorithm
More informationR asterisation. Part I: Simple Lines. Affine transformation. Transform Render. Rasterisation Line Rasterisation 2/16
ECM2410:GraphicsandAnimation R asterisation Part I: Simple Lines Rasterisation 1/16 Rendering a scene User space Device space Affine transformation Compose Transform Render Com pose from primitives (lines,
More informationRasterization. COMP 575/770 Spring 2013
Rasterization COMP 575/770 Spring 2013 The Rasterization Pipeline you are here APPLICATION COMMAND STREAM 3D transformations; shading VERTEX PROCESSING TRANSFORMED GEOMETRY conversion of primitives to
More informationMidpoint and Distance Formulas
Midpoint and Distance Formulas Find the midpoint of a segment on the coordinate plane. Find the distance between two points on the coordinate plane. Fremont are the Midpoint and Distance Formulas used
More information= secant lines of the n+1 unit circle divisions (d)
The logarithm of is a complex number such that. In my theory, I represent as the complex number, which is the same as Euler s, where Taking it a step further with, you get a complex number and its conjugate
More informationCS 548: COMPUTER GRAPHICS DRAWING LINES AND CIRCLES SPRING 2015 DR. MICHAEL J. REALE
CS 548: COMPUTER GRAPHICS DRAWING LINES AND CIRCLES SPRING 05 DR. MICHAEL J. REALE OPENGL POINTS AND LINES OPENGL POINTS AND LINES In OenGL, there are different constants used to indicate what ind of rimitive
More informationThe graphics pipeline. Pipeline and Rasterization. Primitives. Pipeline
The graphics pipeline Pipeline and Rasterization CS4620 Lecture 9 The standard approach to object-order graphics Many versions exist software, e.g. Pixar s REYES architecture many options for quality and
More informationShape 3 Assessment Calculator allowed for all questions
Shape Assessment Calculator allowed for all questions Foundation Higher All questions Time for the test: 50 minutes Name: MATHSWATCH ANSWERS Grade Title of clip Marks Score Percentage Clip 7 D Area of
More informationMidterm Review. Wen-Chieh (Steve) Lin Department of Computer Science
Midterm Review Wen-Chieh (Steve) Lin Department of Computer Science Administration Assignment due on /3 :59 PM Midterm eam on /6 (Monda) Lecture slides Chapter 3 ecluding 3.6 & 3.8 Chapter 6, 7, 8 Chapter
More informationTopic #1: Rasterization (Scan Conversion)
Topic #1: Rasterization (Scan Conversion) We will generally model objects with geometric primitives points, lines, and polygons For display, we need to convert them to pixels for points it s obvious but
More informationDisplay Technologies: CRTs Raster Displays
Rasterization Display Technologies: CRTs Raster Displays Raster: A rectangular array of points or dots Pixel: One dot or picture element of the raster Scanline: A row of pixels Rasterize: find the set
More informationCOMP30019 Graphics and Interaction Scan Converting Polygons and Lines
COMP30019 Graphics and Interaction Scan Converting Polygons and Lines Department of Computer Science and Software Engineering The Lecture outline Introduction Scan conversion Scan-line algorithm Edge coherence
More informationCS Rasterization. Junqiao Zhao 赵君峤
CS10101001 Rasterization Junqiao Zhao 赵君峤 Department of Computer Science and Technology College of Electronics and Information Engineering Tongji University Vector Graphics Algebraic equations describe
More informationComputer Graphics. Chapter 4 Attributes of Graphics Primitives. Somsak Walairacht, Computer Engineering, KMITL 1
Computer Graphics Chapter 4 Attributes of Graphics Primitives Somsak Walairacht, Computer Engineering, KMITL 1 Outline OpenGL State Variables Point Attributes Line Attributes Fill-Area Attributes Scan-Line
More informationReteaching Golden Ratio
Name Date Class Golden Ratio INV 11 You have investigated fractals. Now ou will investigate the golden ratio. The Golden Ratio in Line Segments The golden ratio is the irrational number 1 5. c On the line
More informationMODULE - 4. e-pg Pathshala
e-pg Pathshala MODULE - 4 Subject : Computer Science Paper: Computer Graphics and Visualization Module: Midpoint Circle Drawing Procedure Module No: CS/CGV/4 Quadrant 1 e-text Before going into the Midpoint
More information1 Introduction to Graphics
1 1.1 Raster Displays The screen is represented by a 2D array of locations called pixels. Zooming in on an image made up of pixels The convention in these notes will follow that of OpenGL, placing the
More informationAlgebra I. Linear Equations. Slide 1 / 267 Slide 2 / 267. Slide 3 / 267. Slide 3 (Answer) / 267. Slide 4 / 267. Slide 5 / 267
Slide / 67 Slide / 67 lgebra I Graphing Linear Equations -- www.njctl.org Slide / 67 Table of ontents Slide () / 67 Table of ontents Linear Equations lick on the topic to go to that section Linear Equations
More informationPainter s HSR Algorithm
Painter s HSR Algorithm Render polygons farthest to nearest Similar to painter layers oil paint Viewer sees B behind A Render B then A Depth Sort Requires sorting polygons (based on depth) O(n log n) complexity
More informationScan Conversion- Polygons
Scan Conversion- olgons Flood Fill Algorithm Chapter 9 Scan Conversion (part ) Drawing olgons on Raster Displa Input polgon with rasterized edges = (x,) point inside Goal: Fill interior with specified
More informationPipeline and Rasterization. COMP770 Fall 2011
Pipeline and Rasterization COMP770 Fall 2011 1 The graphics pipeline The standard approach to object-order graphics Many versions exist software, e.g. Pixar s REYES architecture many options for quality
More informationGraphing Calculator Graphing with the TI-86
Graphing Calculator Graphing with the TI-86 I. Introduction The TI-86 has fift kes, man of which perform multiple functions when used in combination. Each ke has a smbol printed on its face. When a ke
More information1) y = 2x 7 2) (-2, 3) ( 3, -1) 3) table. 4) y 5 = ½ ( x 4) 5) 2x + 4y = 7 6) y = 5 7) 8) 9) (-1, 5) (0, 4) 10) y = -3x 7. 11) 2y = -3x 5 12) x = 5
I SPY Slope! Geometr tetbook 3-6, pg 165 (), pg 172 (calculator) Name: Date: _ Period: Strategies: On a graph or a table rise ( Δ) Slope = run Δ ( ) Given 2 points Slope = 2 2 In an equation 1 1 1) = 2
More informationComputer Graphics. - Rasterization - Philipp Slusallek
Computer Graphics - Rasterization - Philipp Slusallek Rasterization Definition Given some geometry (point, 2D line, circle, triangle, polygon, ), specify which pixels of a raster display each primitive
More informationAliasing. Can t draw smooth lines on discrete raster device get staircased lines ( jaggies ):
(Anti)Aliasing and Image Manipulation for (y = 0; y < Size; y++) { for (x = 0; x < Size; x++) { Image[x][y] = 7 + 8 * sin((sqr(x Size) + SQR(y Size)) / 3.0); } } // Size = Size / ; Aliasing Can t draw
More informationCS 335 Graphics and Multimedia. Geometric Warping
CS 335 Graphics and Multimedia Geometric Warping Geometric Image Operations Eample transformations Straightforward methods and their problems The affine transformation Transformation algorithms: Forward
More informationComputer Graphics. Rasterization. Teacher: A.prof. Chengying Gao( 高成英 ) School of Data and Computer Science
Rasterization Teacher: A.prof. Chengying Gao( 高成英 ) E-mail: mcsgcy@mail.sysu.edu.cn School of Data and Computer Science To make an image, we can... Drawing Photography 2 Two Ways to Render an Image In
More information2D rendering takes a photo of the 2D scene with a virtual camera that selects an axis aligned rectangle from the scene. The photograph is placed into
2D rendering takes a photo of the 2D scene with a virtual camera that selects an axis aligned rectangle from the scene. The photograph is placed into the viewport of the current application window. A pixel
More informationTest Name: Chapter 3 Review
Test Name: Chapter 3 Review 1. For the following equation, determine the values of the missing entries. If needed, write your answer as a fraction reduced to lowest terms. 10x - 8y = 18 Note: Each column
More informationIntermediate Algebra. Gregg Waterman Oregon Institute of Technology
Intermediate Algebra Gregg Waterman Oregon Institute of Technolog c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license
More informationMath 125 Little Book Homework Chapters 7, 10, 11, and 12
Math 125 Little Book Homework Chapters 7, 10, 11, and 12 Do NOT copy the book follow the guidelines given for each section. NO CREDIT will be given if you copy the book! You earn 2 points if you turn in
More informationLesson 11 Skills Maintenance. Activity 1. Model. The addition problem is = 4. The subtraction problem is 5 9 = 4.
Lesson Skills Maintenance Lesson Planner Vocabular Development -coordinate -coordinate point of origin Skills Maintenance ddition and Subtraction of Positive and Negative Integers Problem Solving: We look
More informationEinführung in Visual Computing
Einführung in Visual Computing 186.822 Rasterization Werner Purgathofer Rasterization in the Rendering Pipeline scene objects in object space transformed vertices in clip space scene in normalized device
More informationChapter 2: Introduction to Functions
Chapter 2: Introduction to Functions Lesson 1: Introduction to Functions Lesson 2: Function Notation Lesson 3: Composition of Functions Lesson 4: Domain and Range Lesson 5: Restricted Domain Lesson 6:
More informationLecture 6 of 41. Scan Conversion 1 of 2: Midpoint Algorithm for Lines and Ellipses
Scan Conversion 1 of 2: Midpoint Algorithm for Lines and Ellipses William H. Hsu Department of Computing and Information Sciences, KSU KSOL course pages: http://bit.ly/hgvxlh / http://bit.ly/evizre Public
More informationLecture 6 of 41. Scan Conversion 1 of 2: Midpoint Algorithm for Lines and Ellipses
Scan Conversion 1 of 2: Midpoint Algorithm for Lines and Ellipses William H. Hsu Department of Computing and Information Sciences, KSU KSOL course pages: http://bit.ly/hgvxlh / http://bit.ly/evizre Public
More informationx=2 26. y 3x Use calculus to find the area of the triangle with the given vertices. y sin x cos 2x dx 31. y sx 2 x dx
4 CHAPTER 6 APPLICATIONS OF INTEGRATION 6. EXERCISES 4 Find the area of the shaded region.. =5-. (4, 4) =. 4. = - = (_, ) = -4 =œ + = + =.,. sin,. cos, sin,, 4. cos, cos, 5., 6., 7.,, 4, 8., 8, 4 4, =_
More informationComputer Graphics (CS 543) Lecture 10: Rasterization and Antialiasing
Computer Graphics (CS 543) Lecture 10: Rasterization and Antialiasing Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) Recall: Rasterization Rasterization (scan conversion)
More informationCS-321 Thursday 12 September 2002 Quiz (3 pts.) What is the purpose of a control grid in a cathode ray tube (CRT)?
Name CS-321 Thursday 12 September 2002 Quiz 1 1. (3 pts.) What is the purpose of a control grid in a cathode ray tube (CRT)? 2. (7 pts.) For the same resolution in pixels (for example, 640x480), why does
More informationTranslations, Reflections, and Rotations
Translations, Reflections, and Rotations The Marching Cougars Lesson 9-1 Transformations Learning Targets: Perform transformations on and off the coordinate plane. Identif characteristics of transformations
More information1.2 Visualizing and Graphing Data
6360_ch01pp001-075.qd 10/16/08 4:8 PM Page 1 1 CHAPTER 1 Introduction to Functions and Graphs 9. Volume of a Cone The volume V of a cone is given b V = 1 3 pr h, where r is its radius and h is its height.
More informationMidpoint of a Line Segment. INVESTIGATE the Math
.1 Midpoint of a Line Segment YOU WILL NEED grid paper, ruler, and compass, or dnamic geometr software GOAL Develop and use the formula for the midpoint of a line segment. INVESTIGATE the Math Ken s circular
More information