Ppt on application of linear algebra in computer graphics

CSE 690 General-Purpose Computation on Graphics Hardware (GPGPU) Courtesy David Luebke, University of Virginia.

GPUs –Level sets on GPUs Intended Audience Anyone interested in accelerated computing –from all academic disciplines No graphics background required –will cover all necessary and relevant detail in the course What is required… –working knowledge of C/C++ –linear algebra –enthusiasm and an open mind Course Schedule Understanding the fabric: computer graphics basics Overview of GPUs –architecture –features –programming model –some simple applications System issues –cache and data management, –languages and/


Voices of the Partner Disciplines: CRAFTY’s Curriculum Foundations Project.

simply teach mathematics. Move heavily to the application of the concepts within the mathematics classroom. Mechanical and Manufacturing Technology Some Major Themes in Technology 6. Technology in the Mathematics Classroom Students should develop proficiency with at least one [computation software] program, as well as a working knowledge of spreadsheets. Electronics and Telecommunications Students should use a variety of software packages in mathematics classes. … students should be able to/


1 A Look at Library Software for Linear Algebra: Past, Present, and Future Jack Dongarra University of Tennessee and Oak Ridge National Laboratory.

Computation, Vol II »Landmark in the development of numerical algorithms and software »Basis for a number of software projects EISPACK, a number of linear algebra routines in IMSL and the F chapters of NAG –IBM 370/195 »Pipelined architecture; Out of/64 »More computing power than a Cray 1 and much much better graphics  1997 –MPI-2 Finished –Fortran 95  1998 –Issues of parallel and/HPF model  A group of 30-40 “experts” in message- passing: –MPP vendors, –CS researchers, –Application developers.  Met 3 days/


High Performance Computing – CISC 811 Dr Rob Thacker Dept of Physics (308A)

in this topic because of its relation to parallel computing Many researchers are interested in this topic because of its relation to parallel computing Graphics processors provide the most number of flops per dollar (by an order of magnitude) Graphics processors provide the most number of flops per dollar (by an order of/! Case study: Linear Algebra on GPUs Linear algebra is a problem that does not immediately map well to modern cache based CPUs (need to optimize for cache) Linear algebra is a problem that/


Communication-Avoiding Algorithms for Linear Algebra and Beyond Jim Demmel Math & EECS Departments UC Berkeley.

featured/brave-new-hair/ – graphics.pixar.com/library/CurlyHairA/paper.pdf graphics.pixar.com/library/CurlyHairA/paper./applications: – 3D LMC (a low-mach number combustion code) 2.5x in bottom solve, 1.5x overall GMG solve – 3D Nyx (an N-body and gas dynamics code) 2x in bottom solve, 1.15x overall GMG solve Summary of Iterative Linear Algebra New lower bounds, optimal algorithms, big speedups in theory and practice Lots of/. For each input vector: Dot products are computed using 1, 2, 3 or 4 threads /


Introduction to Computer Vision Dr. Chang Shu COMP 4900C Winter 2008.

SRI Applications: Robotics Applications: Surveillance Mathematical tools Linear algebra Vector calculus Euclidean geometry Projective geometry Differential geometry Differential equations Numerical analysis Probability and statistics Programming tools OpenCV – an open source library for computer vision. Ch – a C interpretation environment. Course Organization Textbook: Introductory Techniques for 3-D Computer Vision, by Trucco and Verri Two parts: Part I (Chang Shu) – Introduction, Review of linear algebra/


Chapter 6 More on Matrices

elimination. Many linear algebra textbooks incorrectly assert that such techniques are better suited for implementation on a computer because they require fewer arithmetic operations. This is true for large matrices, or for matrices with a structure that can be exploited. However, for arbitrary matrices of smaller order like the 2 x 2, 3 x 3, and 4 x 4 used most often in geometric applications, the/


OPS 301 Module B and Additional Topics in Linear Programming

B and Additional Topics in Linear Programming Dr. Steven Harrod Topics Definition of Linear Programming Steps to Formulate a Problem Applications Mathematical Requirements Linear functions Convex, non-negative feasible region Single objective Steps to Formulate a Problem Solution Techniques Graphical Simplex Linear Programming A mathematical tool to make multiple decisions affecting a single outcome or objective Guaranteed to find the optimal solution (best possible set of decisions) relative to/


Introduction to 2D Graphics

. When we say “transformation,” we usually mean a composition of scale, rotate and translate transforms Object Coordinates Display Application Coordinates 2D Graphics using OpenGL – 9/9/2014 Transformations (1/3) We will use GLM to do linear algebra for us and build the mapping matrix In addition to the projection matrix mentioned earlier, also keep track of a model and a view matrix. More about the significance/


Chapter 8 Rotation in Three Dimensions

in place, we are now free to perform rotations using the matrix at a higher level of abstraction. Our interface functions match exactly the high-level intentions of the programmer. Chapter 8 Notes 3D Math Primer for Graphics & Game Dev 3D Math Primer for Graphics & Game Dev Some Justification Furthermore, we have removed the confusing linear algebra/ operations can make very quick work of matrix multiplication. Quaternion conjugate provides a way to compute the opposite angular displacement very efficiently./


1 Precise Voronoi Cell Extraction of Free-form Rational Planar Closed Curves Iddo Hanniel, Ramanathan Muthuganapathy, Gershon Elber Department of Computer.

and Patrikalakis 1992, Held 1998, Ju-Hsein Kao 1999]. Center for Graphics and Geometric Computing, Technion 7 Our approach  Using the original representation of the input curves (with no linear / circular approximation).  Generate an accurate implicit representation of the Voronoi cell. ExactApproximated Center for Graphics and Geometric Computing, Technion 8 Outline of the algorithm Implicit bisector function Application of constraints Lower envelope algorithm Splitting into monotone pieces tr-space/


Progressive Teaching with Tablet Technology Birgit Loch Mathematics & Computing

ONC, 110 EXT, 1 st  Engineering and Science Details of the study – course 1  S2, 2004. UQ  Calculus and Linear Algebra I (all)  320 students  Engineering and Science  One lecturer  Initially OHP and computerGraphics tablet (A6)  PDF  No technical problems  Notes made available on website afterwards Birgit Details of the study – course 2  Two lecturers  Linear Algebra taught with graphics tablet (A3)  Calculus taught writing on OHP  PDF/


Introduction to Computer Graphics CS 445 / 645 David Brogan

around a Harrier Jet (NASA Ames) Graphics Applications Computer Aided Design (CAD) Graphics Applications TrainingTraining Designing Effective Step-By-Step Assembly Instructions (Maneesh Agrawala et. al) Graphics Applications Entertainment: Games GT Racer 3 Polyphony Digital: Gran Turismo 3, A Spec Games Circus Atari (Atari) Education Outside In (Geometry Center, University of Minnesota) The Basics Computer graphics: generating 2D images of a 3D world represented in a computer. Main tasks: modeling: (shape/


Communication-Avoiding Algorithms for Linear Algebra and Beyond Jim Demmel EECS & Math Departments UC Berkeley bebop.cs.berkeley.edu.

, cloud, heterogeneous, low-energy, … Integration into applications – ASPIRE Lab – computer vision video background subtraction, optical flow, … – AMPLab – building library using Spark for data analysis – CTF (with ANL): symmetric tensor contractions, on BG/Q Outline Survey state of the art of CA (Comm-Avoiding) algorithms – TSQR: Tall-Skinny QR – CA O(n 3 ) 2.5D Matmul – Sparse Matrices Beyond linear algebra – Extending lower bounds to any algorithm/


2009 Mathematics Standards of Learning Training Institutes Algebra II Virginia Department of Education.

Categories Expressions and Operations Equations and Inequalities Functions Statistics Major Changes to Algebra II Content moved to Mathematical Analysis  matrices  conic sections Content eliminated from Algebra II  linear programming (it resides still in AFDA) Content moved to Algebra I  Inverse variation  Systems of Linear Equations and Inequalities Content added  permutations and combinations  the normal distribution Algebra II Expressions and Operations STANDARD AII.1 The student, given rational/


Art and design Key Stage 3 Range In art and design, pupils at Key Stage 3 should develop their understanding and investigating skills in order to enrich.

in a variety of ways – verbally, graphically, using informal written methods. Skills Mathematics Key Stage 3 3. Reason mathematically Pupils should be given opportunities to: extend mental methods of computation to consolidate a range of non-calculator methods justify how they arrived at a conclusion to a problem; give solutions in the context of the problem; confirm that results are of the right order of magnitude interpret and use simple algebraic/


Computing & Information Sciences Kansas State University CIS 536/636 Introduction to Computer Graphics William H. Hsu Department of Computing and Information.

CIS 536/636 Introduction to Computer Graphics Overview: First Month (Weeks 2-5 of Course)  Review of mathematical foundations of CG: analytic geometry, linear algebra  Line and polygon rendering  Matrix transformations  Graphical interfaces Line and Polygon Rendering (Week 3)  Basic line drawing and 2-D clipping  Bresenham’s algorithm  Follow-up: 3-D clipping, z-buffering (painter’s algorithm) Matrix Transformations (Week 4)  Application of linear transformations to rendering  Basic operations/


CS 480/680 Computer Graphics Course Overview Dr. Frederick C Harris, Jr. Fall 2012.

Harris, Jr. Fall 2012 Objectives Broad introduction to Computer Graphics –Software –Hardware –Applications Top-down approach Shader-Based OpenGL –OpenGL 3.1 –Open GL ES 2.0 –webGL Prerequisites Good programming skills in C (or C++) Basic Data Structures –Linked lists –Arrays Geometry Simple Linear Algebra Resources Can run OpenGL on any system –Windows: check graphics card properties for level of OpenGL supported –Linux –Mac: need extensions for/


CS123 | INTRODUCTION TO COMPUTER GRAPHICS Andries van Dam © 1/32 Introduction to 2D Graphics Using OpenGL 2D Graphics using OpenGL – 9/10/2015.

use of computer graphics, e.g. in UI/UX, documents, browsers Why Learn 2D first? 2D Graphics using OpenGL – 9/10/2015 CS123 | INTRODUCTION TO COMPUTER GRAPHICS Andries van Dam © 4/32 Graphics Platforms (1/4)  Applications that only write pixels are rare  Application /(OpenGL Mathematics) to do our linear algebra instead of using the Fixed-function API OpenGL (3/3) 2D Graphics using OpenGL – 9/10/2015 CS123 | INTRODUCTION TO COMPUTER GRAPHICS Andries van Dam © 18/32  In future labs and your final project/


1 Introduction to Computer Graphics SEN 991 - Introduction to OpenGL Graphics Applications.

) 3 Objectives Broad introduction to Computer Graphics ­Software ­Hardware ­Applications Top-down approach OpenGL 4 Prerequisites Good programming skills in C (or C++) Basic Data Structures ­Linked lists ­Arrays Geometry Simple Linear Algebra 5 Requirements 4 Assigned Projects ­/of examples and information at: www.opengl.orgwww.opengl.org PowerPoint lecture notes at www.bhecker.com www.bhecker.com 9 Outline: Part 1 Part 1: Introduction Text: Chapter 1 Lectures 1-3 ­What is Computer Graphics? ­Applications /


1 Thoughts on Rationalizing Algebra 1 and 2 in Ways that Serve Kids, Not Universities CAMT – July 16, 2010 Steve Leinwand American Institutes for Research.

variables The student will recognize and apply real-world functions in a variety of representations and translate among verbal, tabular, graphic, and algebraic representations of functions. The student will recognize an example of a function; identify the role of independent and dependent variables in a function; determine the domain and range of a linear function; find the slope and intercepts of a linear function; and use function notation to evaluate a function/


1 Disentangling Age-Period-Cohort Effects: New Models, Methods, and Empirical Applications Kenneth C. Land, Duke University PRI Summer Methodology Workshop.

or “ net install apc” on the Stata 9.2 command line on any computer connected to the Internet  Download from the Statistical Software Components archive at http:///of individuals in the samples. Solution: Use of different temporal groupings for the A, P, and C dimensions breaks the linear dependency:  Single year of age  Time periods correspond to years in which the surveys are conducted  Cohorts can be defined either by five- or ten-year intervals that are conventional in demography or by application of/


Mathematics & Biology The New Synergy College Algebra, Precalculus, and Up Sheldon P. Gordon

algebra courses. It can be the unifying theme that links functions, the real world, and the other disciplines. Integrating Statistics in Precalculus 3.The normal distribution function is It makes for an excellent example involving both stretching and shifting functions and a function of a function. Integrating Statistics in Precalculus 4. The z-value associated with a measurement x is a nice application of a linear function of/


Results Goals and Objectives ALADDIN (The UAB Alice, Linear Algebra, Dynamic Dimensional Information Network) Drawing the GENIEous out of the kid Empowering.

state objectives for linear algebra and counts as a math credit towards graduation. 15 teachers and 5 librarians have attended linear algrebra workshops to facilate future classes. 3. Computer Graphics and Visualization I & II These two classes are taught during the summer of the student’s rising junior and senior years at the UAB ETLab. In Workshop-I, students learn about the fundamental principles and applications in computer graphics and scientific/


Communication-Avoiding Algorithms for Linear Algebra and Beyond Jim Demmel EECS & Math Departments UC Berkeley.

CHOMBO AMR frameworks Speedups for two BoxLib applications: – 3D LMC (a low-mach number combustion code) 2.5x in bottom solve, 1.5x overall GMG solve – 3D Nyx (an N-body and gas dynamics code) 2x in bottom solve, 1.15x overall GMG solve Summary of Iterative Linear Algebra New lower bounds, optimal algorithms, big speedups in theory and practice Lots of other progress, open problems – Many different/


Yann LeCun Energy-Based Learning. Structured Output Models Energy-Based Learning. Structured Output Models Yann LeCun, The Courant Institute of Mathematical.

Graphical Models (non-probabilistic factor graphs) Latent variable models Linear factors: Conditional Random Fields and Maximum Margin Markov Nets Gradient-based learning with non-linear factors Applications: supervised and unsupervised learning Integrated segmentation/recognition in /LeCun Graph Composition, Transducers. The composition of two graphs can be computed, the same way the dot product between two vectors can be computed. General theory: semi-ring algebra on weighted finite- state transducers and/


Yann LeCun Energy-Based Models: Structured Learning Beyond Likelihoods Yann LeCun, The Courant Institute of Mathematical Sciences New York University Yann.

Graphical Models (non-probabilistic factor graphs) Latent variable models Linear factors: Conditional Random Fields and Maximum Margin Markov Nets Gradient-based learning with non-linear factors Applications: supervised and unsupervised learning Integrated segmentation/recognition in /LeCun Graph Composition, Transducers. The composition of two graphs can be computed, the same way the dot product between two vectors can be computed. General theory: semi-ring algebra on weighted finite- state transducers and/


The first-generation Cell Broadband Engine (BE) processor is a multi-core chip comprised of a 64-bit Power Architecture processor core and eight synergistic.

account the attributes of both the application and target system. A successful implementation lies on deep understanding of data access patterns, computation properties, available hardware resources where it can take advantage of generalized performance planning techniques to produce successful implementation across a wide variety of multicore architectures. Combinatorial algorithms play important role in scientific computing for efficient parallelization of linear algebra, computational physics, numerical/


Integrated Design of Mechatronic Systems using Bond Graphs.

project management Which kind of tool I is needed ? Structured, unified, generic, Prof. Belkacem Ould BOUAMAMA, Polytech’Lille What is Mechatronic Systems Mecatronics (« Meca »+ « Tronics » Engineering systems putting in evidence multiple skills Mechanics : Hydraulics, Thermal engineering, Mechanism, pneumatic Electronics : power electronics, Networks, converters AN/NA, Micro controllers, Automatic control : Linear and nonlinear control, Advanced control, Stability, … Computer Engineering : Real time/


Refocusing the Courses Below Calculus A Joint Initiative of MAA, AMATYC & NCTM.

at http://www.mathsci.appstate.edu/~wmcb/ICTCM CRAFTY & College Algebra The Guidelines: Course Objectives College algebra through applications/modeling Meaningful & appropriate use of technology Course Goals Challenge, develop, and strengthen students’ understanding and skills mastery CRAFTY & College Algebra The Guidelines: Student Competencies - Problem solving - Functions and Equations - Data Analysis Pedagogy - Algebra in context - Technology for exploration and analysis Assessment - Extended set/


MATLAB for Scientist and Engineers

Introduction with Applications, Amos Gilat, John Wiley & Sons, Inc., 2004 Graphics and GUIs with MATLAB, 3rd Ed, Patrick Marchand and O. Thomas Holland, Chapman & Hall/CRC, 2003 MATLAB for Scientist and Engineers Using Symbolic Toolbox Course Introductions MATLAB for Scientist and Engineers Using Symbolic Toolbox Old History of MATLAB 1967: "Computer solution of linear algebraic equations", Forsythe and Moler 1976: "Matrix Eigensystem Routines, EISPACK Guide" in FORTRAN 1976/


Practical Application of Coordinate and Dot Transformations Topics: linear algebra, programming with Delphi, computer graphics, OO conception Sándor Kaczur.

Application of Coordinate and Dot Transformations Topics: linear algebra, programming with Delphi, computer graphics, OO conception Sándor Kaczur kaczur@gdf.hu 2 Circumstances 1. semester –Mathematics 1. Linear algebra, matrices System of equations 3. semester –Computer graphics, Digital image processing Homogeneous coordinates Coordinate and dot transformations Projections 3 Problems They can solve exercises from linear algebra/Let be a dot P in 3D with 3 coodinates: If let coordinates of dot P: If than let/


1 Lecture 12 Resource Allocation Part II (involving Continuous Variable (Linear Programming, continued) Samuel Labi and Fred Moavenzadeh Massachusetts.

are few decision variables and even fewer constraints. Linear Programming II 12 Method 4: Using Linear Algebra (Matrices) In this method, the vertices of the feasible region are determined as follows: - Set up the objective function and constraints as a set of linear algebra equations - Develop the corresponding matrices. - Solve the set of linear equations using vector algebra. This yields the optimal value of the objective function. - Simplex method can be employed/


1 Structure of Computer Systems Course 2 Computer performance and optimality.

numerical linear algebra performs numerical linear algebra 22 Examples of benchmark programs  SPEC - Standard Performance Evaluation Corporation a non-profit international organization focused on developing standard tools for measuring the performance of computer systems a non-profit international organization focused on developing standard tools for measuring the performance of computer systems www.spec.org www.spec.org www.spec.org develops standard sets of benchmarks based on real applications develops/


Quantitative Education for Life Sciences: BIO2010 and Beyond Louis J. Gross Departments of Ecology and Evolutionary Biology and Mathematics, The Institute.

, and graphically display data in a variety of representations. Students should also become skilled at using the Internet to carry out literature searches, locate published articles, and access major databases. Concepts of Mathematics and Computer Science Calculus Complex numbers Functions Limits Continuity The integral The derivative and linearization Elementary functions Fourier series Multi-dimensional calculus: linear approximations, integration over multiple variables Linear Algebra Scalars, vectors/


Big Data Open Source Software and Projects ABDS in Summary XXIII: Layer 16 Part 1 Data Science Curriculum March 1 2015 Geoffrey Fox

Hourglass was created to make these computations more efficient, yielding sometimes 50-95% reductions in computational resources R R is GPL Open Source http://en.wikipedia.org/wiki/R_(programming_language) and many books and online resources and is widely used by statistics communityhttp://en.wikipedia.org/wiki/R_(programming_language) R provides a wide variety of statistical and graphical techniques, including linear and nonlinear modeling, classical statistical/


Introduction to Computer Graphics: ITCS 4120/5120 Dr. Zachary Wartell Revision 1.2 2/16/07 Copyright 2006, Dr. Zachary Wartell, UNCC, All Rights Reserved.

graphics (ITCS 4120) – linear/vector algebra, geometry & trig. -advanced graphics advanced calculus, computational geometry, differential geometry, topology, …..  optics (very approximate in ITCS 4120)  software engineering and programming  hardware engineering  psychophysics (branch of/Fetter, 1960, Boeing Aircraft Co. “Boeing Man”, human figure simulation, credited with “computer graphics” ©Zachary Wartell In what way do CG applications differ?  2D versus 3D  Speed – Frames Per Second (FPS)  Realism/


Transformation Wen-Chieh (Steve) Lin Department of Computer Science & Institute of Multimedia Engineering Rich Riesenfeld’s CG Slides, Shirley, Fundamentals.

RV 2 ILE5014 Computer Graphics 10F 24 Composition of Transformation Any sequence of linear transformations can be /of function application.) Premultiply: right-to-left (same as function application.) Postmultiply: left-to-right (reverse of function application.) Premultiply: right-to-left (same as function application.) M  D M  CM M  BM M  AM M  A M  MB M  MC M  MD or both give the same result. Premultiply Postmultiply ILE5014 Computer Graphics 10F 28 Column Vector Convention The convention in/


GEOMETRIC OPTICS.

In computations , it is often easier to work with the refractive index of a material than directly with the speed of light . n = speed of light in vacuum speed of light in medium refractive index is quite sensitive to a material’s chemical composition . a small amount of salt or sugar dissolved in/. The clinical application of wavefront aberrometry makes it possible to measure higher order aberrations , which were previously lumped into a catchall term – irregular astigmatism . Examples of higher -order /


Computational Algebraic Problems in Variational PDE Image Processing Tony F. Chan Department of Mathematics, UCLA International Summer School in Numerical.

Computational Algebraic Problems in Variational PDE Image Processing Tony F. Chan Department of Mathematics, UCLA International Summer School in Numerical Linear Algebra Chinese University of Hong Kong Reports: www./Vector-Valued * Compression * Still vs Video * Segmentation * Registration Related fields: Computer Graphics, Computer Vision. IP And Applied Math Important applications: –Medical, astronomy, Comp. Vision/Comp. Graphics, Math Models: –standard or create your own Math Tools: –harmonic analysis,/


EECS Electrical Engineering and Computer Sciences B ERKELEY P AR L AB P A R A L L E L C O M P U T I N G L A B O R A T O R Y EECS Electrical Engineering.

Widely applicable: all linear algebra, Health app… 29 EECS Electrical Engineering and Computer Sciences B ERKELEY P AR L AB Communication-Avoiding QR Decomposition for GPUs 30 The QR decomposition of tall-skinny matrices is a key computation in many applications Linear least/ resources to execute components in parallel 32 EECS Electrical Engineering and Computer Sciences B ERKELEY P AR L AB 33 OS-multiplexed Efficient Parallel Composition of Libraries is Hard Physics AI Graphics Audio OpenMP Scheduler TBB /


The Berkeley View: A New Framework & a New Platform for Parallel Research David Patterson and a cast of thousands Pardee Professor of Computer Science,

other fields 1. Embedded Computing (EEMBC benchmark) 2. Desktop/Server Computing (SPEC2006) 3. Machine Learning  Advice from colleagues Mike Jordan and Dan Klein 4. Games/Graphics/Vision 5. Data Base Software  Advice from Jim Gray of Microsoft and colleague Joe Hellerstein Result: Added 7 more dwarfs, revised 2 original dwarfs, renumbered list 15 Final 14 Dwarfs 1. Dense Linear Algebra 2. Sparse Linear Algebra 3. Spectral Methods 4/


Evolution of the Graphics Process Units Dr. Zhijie Xu

have enabled the use of GPU processors for general purpose computation.  Applications in: Linear algebra Geometric Computing Database and Stream Mining GPU Ray Tracing Advanced Image Processing Computational Fluid dynamics (CFD) and Finite Element Analysis Problems Need to be Solved  Significant barriers exist for the developer who wishes to use the inexpensive power of commodity graphics hardware, whether for in-game simulation of physics or for conventional computational science.  These chips are/


Mathematics for the Laboratory Sciences: College Algebra, Precalculus, and Up Sheldon P. Gordon

algebra courses. It can be the unifying theme that links functions, the real world, and the other disciplines. Integrating Statistics in Precalculus 3.The normal distribution function is It makes for an excellent example involving both stretching and shifting functions and a function of a function. Integrating Statistics in Precalculus 4. The z-value associated with a measurement x is a nice application of a linear function of/


Zhang & Liang, Computer Graphics Using Java 2D and 3D (c) 2007 Pearson Education, Inc. All rights reserved. 1 Chapter 1 Overview of Computer Graphics.

F To understand the basic objectives and scope of computer graphics F To identify computer graphics applications F To understand the basic structures of 2D and 3D graphics systems F To understand evolution of graphics programming environments F To identify common graphics APIs F To understand the roles of Java language, Java 2D and Java 3D packages F To identify computer graphics related fields Zhang & Liang, Computer Graphics Using Java 2D and 3D (c) 2007 Pearson/


2IV60 Computer Graphics Basic Math for CG Jack van Wijk TU/e.

of linear equations: Such systems occur in many, many applications. They are studied in Linear Algebra. Very short intro to Linear Algebra H&B A-5 System of linear equations: Typical questions: -Given u, v, w, what are x, y, z? -Can we find a unique solution? H&B A-5 Very short intro to Linear Algebra System of linear equations: Crucial in computer graphics: -Transforming geometric objects -Change of coordinates H&B A-5 Very short intro to Linear Algebra/


Comprehensive Curriculum Algebra 1 Presented by Kim Melancon Geometry Presented by Mandy Boudwin.

graphically? Unit 3: Linear Functions and Their Graphs, Rates of Change, and Applications Time Frame: Approximately five weeks Unit Description This unit leads to the investigation of the role of functions in the development of algebraic thinking and modeling. Heavy emphasis is given in this unit to understanding rates of/ and the computations that can be performed using them. Student Understandings Students should be able to find the precision of an instrument and determine the accuracy of a given /


Unit 1: Representing relationships mathematically 450 mins~ 5.63 days In this unit, students solidify their previous work with functional relationships.

. Students will investigate key features, domains, and ranges of linear functions as described in F-IF.B.4 and F-IF.B.5; write linear functions to model relationships between two quantities as in F-BF.A1a; and compare properties of linear functions as in F-IF.C.9. Interpreting Functions-F-IF B. Interpret functions that arise in applications in terms of the context 6. Calculate and interpret the average rate/


Computer Graphics Sense from Symbol. Applications of Graphics Entertainment consumer enjoyment Design of physical objects to be built Visualization of.

scaling Shearing Geometry change in perpendicularity introduces slant Algebra in X —x’ = x + y * Sh x —y’ = x * Sh y + x Display of Motion Create visual perception of motion movies, TV, interactive graphics sequence of snap shots : frames played back rapidly at a constant frame rate Variety of frame rates Film : 24 fps (frames per second) S-TV: 30 fps HD-TV: 60 fps Computers : 60-120 fps Interactive/


Computing & Information Sciences Kansas State University Advanced CG 1 of 8: TexturingCIS 636/736: (Introduction to) Computer Graphics CIS 736 Computer.

25 Feb 2008  Preparing term project presentations and demos for graphics – April Computing & Information Sciences Kansas State University Advanced CG 1 of 8: TexturingCIS 636/736: (Introduction to) Computer Graphics Background Expected Both Courses  Proficiency in C/C++ or strong proficiency in Java and ability to learn  Strongly recommended: matrix theory or linear algebra (e.g., Math 551)  At least 120 hours for semester (up to 150 depending on/


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