COS702; Assignment 6. Point Cloud Data Surface Interpolation University of Southern Missisippi Tyler Reese December 3, 2012

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "COS702; Assignment 6. Point Cloud Data Surface Interpolation University of Southern Missisippi Tyler Reese December 3, 2012"

Transcription

1 COS702; Assignment 6 Point Cloud Data Surface Interpolation University of Southern Missisippi Tyler Reese December 3, 2012

2 The Problem COS 702, Assignment 6: Given appropriate sets of Point Cloud data, use Radial Basis Functions to recreate the surface through interpolation. The first set of data to be used includes coordinates of points on the original surface and the normal vector at each of those points for 453 points. Using Inverse MQ, adjust the shape parameter and resolution of the 3D mesh used to recreate the surface, and find a combination that yields optimal results. The second set of data is of the same format but includes 1889 points. The larger data set is expected to require the use of Compactly Supported Radial Basis Functions. Using a simple CS-RBF (φ = (1 r) 2 ), adjust the support parameter and resolution of the 3D mesh, and find a combination that yields optimal results for this data set as well. If this generates well behaved results, attempt to use Point Cloud data for 8171 points. Results are to be judged visually. Provided below in Figure 1 is the original object whose surface is to be recreated. Figure 1: Original bunny statue from which the Point Cloud data was generated. [1] 1

3 Methods Overview Surface Interpolation with Radial Basis Functions (RBFs) uses data points for which function values are known to construct an interpolation matrix based on the distance from each point to the other points. A more thorough discussion of this concept can be found in references [2] and [3]. Cases where a large number of data points are being used can result in a high density of data points and lead to ill-conditioned Distance Matrices. In these cases, Compactly Supported RBFs (CS-RBFs) can be utilized which are restricted to only being concerned with points that fall within a designated support radius. More information on this concept can be found in references [2] and [4]. CS-RBFs allow for the matrix calculations to be done with sparse matrices increasing computational efficiency and helping to reduce the time required. When given a Point Cloud data set with coordinates and normals, one can effectively use RBFs to reconstruct the original surface. One distinction between recreating the surface given by a function (i.e. f = f(x, y)) and recreating a surface given by Point Cloud data, is that for the Point Cloud data there is not a functional dependence of the z-coordinate on the x- and y-coordinates. There is no longer any required functional dependence among any of the coordinate components. This means, for example, that for a given x,y pair there could be more than one appropriate z-coordinate, and one can no longer reliably predict the what the z-coordinate should be. One method to resolve this issue is to create a functional dependence based on all three coordinate values, i.e. f = f(x, y, z). [5] To achieve this, one needs to generate two additional sets of point coordinates using the provided Point Cloud data points and the normal vectors associated with each point. The normal vectors provide a direction pointing out from the surface. Using this information, one can create one point inside the surface and another point outside the surface for every data point in the Point Cloud set. The distance from the surface, δ, needs to be specified such that a line connecting these three points only crosses the surface once at the original point and does not encounter any other portions of the surface. In this work δ was specified to be 1/100th the overall length scale, meaning the longest distance spanned along the x, y, or z direction in the original coordinate set divided by 100. Once the original N point data set is turned into a 3N point data set, the functional values are arbitrarily set equal to the following with x i representing the coordinate (x i, y i, z i ) and n i representing 2

4 the normal vector with i going from 1 to N: f(x i δn i ) = 1 f(x i ) = 0 f(x i + δn i ) = 1. (1) With this relationship, one can use RBFs to interpolate the values of f over a 3D mesh. Understanding that the original surface points coorresponded to f = 0, one can locate an approximation of the surface between the original point by identifying all the interpolated points that also result in a value of f = 0. Matlab has a built in functionality to accomplish this called isosurface. When provided with the x, y, and z grid spacing, the matrix containing the values of f at each grid point, and the value of f to locate the surface (in this case f = 0), the isosurface function plots a surface that intersects with all the (x, y, z) coordinates that result in f = 0. Unlike previous work, the quality of the results from this method can not be easily judged by numerical means such as RMSE or Max Error. Instead, a visual comparison to the original bunny statue pictured in Figure 1 will be the basis of the relative success of these methods. In the case of global RBFs, adjustment of the shape parameter affects the ability to recreate the original surface while adjusting the resolution of the 3D mesh, as expected, can affect the detail possible in the resulting surface. In the case of CS-RBFs, adjusting 1 the support parameter (i.e. ) has a similar affect to the shape support radius parameter affecting the ability to recreate the surface, and the affect of the resolution of the 3D mesh not only affects the level of detail possible directly but also as a consequence of the support parameter as will be demonstrated in the results section below. To accomplish these tasks, modifications were made to a functioning Matlab script that worked for 2D Point Cloud data that was provided. These modifications included adding a third dimension, inputting the Point Cloud data from files, removing data points associated with zero normals, and saving a *.jpg & *.fig file of the recreated surface with a filename that included the parameters used as well as the time required to run, e.g. ep150 neval25 time10p422.jpg. This coorrespondes to a shape parameter of 150, a 25x25x25 grid, and a run time of seconds. This script is named PointCloud3D Bunny.m and has function calls to rbf.m and DistanceMatrix.m within the script. 3

5 An augmentation of this script was then saved under the name Point- Cloud3D BunnyCS.m that was used to handle the data set including 1889 points. Using that many original data points results in total points and requires the use of CS-RBFs. This Compact Support (CS) version of the script calls the function DistanceMatrixCSRBF.m which incorporates kdtree.dll and kdrangequeary.dll to allow for a more efficient implementation of the kd-tree algorithm to determine which points are within the support radius than is possible when executing the kd-tree methods using matlab scripts. Similar to the global case, *.jpg and *.fig files are saved with information about the run contained within the filename as is shown in this example: ep100 neval40 time15p406 shift0.jpg. This denotes a support parameter of 100, a 40x40x40 grid, a run time of seconds, and a shift value of zero. The presence of the shift value in the file is a result of a brief investigation into any effects of shift the right hand side (f = 1, 0, 1) by some nonzero integer value (e.g. f = 1, 2, 3) and shifting the isosurface value by the same amount (in this example putting the surface at all values equal to 2). As expected, this generated the same surface and was deemed to be a relatively trivial exercise, and while the designation is still in the filename, it can be ignored. PointCloud3D BunnyCSLong.m is a duplicate copy of the first CS version except for the fact that it uses the largest data set of 8171 points. 4

6 Results As was previously discussed, the task of optimizing the shape parameter and reconstruction resolution for the globally supported RBF and optimizing the support parameter and reconstruction resolution for the CS-RBF was achieved by the relatively crude approach of running the scripts and comparing each surface generated, attempting to identify which most closely resembled the original bunny. Globally Supported IMQ: Finding the Shape Parameter Figure 2 shows the provided point cloud of 453 points to be interpolated into a surface. Figure 2: Point Cloud of 453 points on the surface of the bunny. 5

7 Figures 3 and 4 demonstrate the effect of changing the shape parameter. (a) IMQ Shape Parameter equal to 50. (b) IMQ Shape Parameter equal to 75. (c) IMQ Shape Parameter equal to 100. (d) IMQ Shape Parameter equal to 125. Figure 3: Adjusting the Shape Parameter with IMQ RBF (50, 75, 100, 125) interpolating of a 40x40x40 mesh. 6

8 (a) IMQ Shape Parameter equal to 150. (b) IMQ Shape Parameter equal to 175. (c) IMQ Shape Parameter equal to 200. (d) IMQ Shape Parameter equal to 225. Figure 4: Adjusting the Shape Parameter with IMQ RBF (150, 175, 200, 225) interpolating of a 40x40x40 mesh. Globally Supported IMQ: Finding an Appropriate Mesh Resolution Figures 5 and 6 demonstrate the effect of changing the resolution of the 3D mesh over which the surface is interpolated. 7

9 (a) IMQ interpolation over a 15x15x15 mesh. (b) IMQ interpolation over a 20x20x20 mesh. (c) IMQ interpolation over a 25x25x25 mesh. (d) IMQ interpolation over a 30x30x30 mesh. Figure 5: Adjusting the mesh resolution (15, 20, 25, 30) while interpolating with IMQ and a shape parameter value of

10 (a) IMQ interpolation over a 35x35x35 mesh. (b) IMQ interpolation over a 40x40x40 mesh. (c) IMQ interpolation over a 45x45x45 mesh. (d) IMQ interpolation over a 50x50x50 mesh. Figure 6: Adjusting the mesh resolution (35, 40, 45, 50) while interpolating with IMQ and a shape parameter value of

11 Figure 7: IMQ interpolation over a 40x40x40 mesh with a shape parameter of 150. Run time = seconds. 10

12 Globally Supported IMQ: Runtimes for a variety of shape parameters and mesh resolutions Table 1: Shape parameter, Mesh Resolution, and Run time for the interpolations shown in Figures 3, 4, 5, and 6 Shape Parameter Mesh Resolution Runtime (seconds)

13 Compactly Supported RBF: Finding an Appropriate Mesh Resolution Figure 8 shows the provided point cloud of 1889 points to be interpolated into a surface using CS-RBF. Figure 8: Point Cloud of 1889 points on the surface of the bunny. Figures 9 and 10 demonstrate the effect of changing the resolution of the 3D mesh over which the surface is interpolated in the case of the 1889 point set using a simple CS-RBF (i.e. (1 r) 2 ). 12

14 (a) CS-RBF over a 20x20x20 mesh. (b) CS-RBF over a 25x25x25 mesh. (c) CS-RBF over a 30x30x30 mesh. (d) CS-RBF over a 35x35x35 mesh. Figure 9: Adjusting the mesh resolution (20, 25, 30, 35) while interpolating with CS-RBF and a support parameter value of

15 (a) CS-RBF over a 40x40x40 mesh. (b) CS-RBF over a45x45x45 mesh. (c) CS-RBF over a 50x50x50 mesh. (d) CS-RBF over a 100x100x100 mesh. Figure 10: Adjusting the mesh resolution (40, 45, 50, 100) while interpolating withcs-rbf and a support parameter value of 50. Compactly Supported RBF: Finding an Appropriate Support Parameter Figures 11 and 12 demonstrate one effect of changing support parameter in the case of the 1889 point set using a simple CS-RBF (i.e. (1 r) 2 ). Increases in the support parameter result in decreases in the support radius, radius = 1 parameter. 14

16 (a) CS-RBF with support parameter 30. (b) CS-RBF with support parameter 40. (c) CS-RBF with support parameter 50. (d) CS-RBF with support parameter 60. (e) CS-RBF with support parameter 70. (f) CS-RBF with support parameter 80. Figure 11: Adjusting the support parameter for the CS-RBF (30, 40, 50, 60, 70, 80) while interpolating over a 40x40x40 3D mesh. 15

17 (a) CS-RBF with support parameter 100. (b) CS-RBF with support parameter 120. (c) CS-RBF with support parameter 140. (d) CS-RBF with support parameter 160. (e) CS-RBF with support parameter 180. (f) CS-RBF with support parameter 200. Figure 12: Adjusting the support parameter for the CS-RBF (100, 120, 140, 160, ) while interpolating over a 40x40x40 3D mesh. 16

18 Figure 13: CSRBF Interpolation of Point Cloud of 1889 points with Support Parameter of 60 and a mesh resolution of Run time = seconds. 17

19 CS-RBF Run Times for various Support Parameters and 3D Mesh resolutions: Table 2: Support parameter, Mesh Resolution, and Run time for the interpolations shown in Figures 9, 10, 11, and 12 Support Parameter Mesh Resolution Runtime (seconds) CS-RBF Continued manipulation of support parameter and 3D mesh resolution In the course of investigating which support parameter and mesh resolution yielded the optimal results, some potentially useful behaviors were observed. Shown below in Figures 14 and 15 are some examples of this information. 18

20 (a) CS-RBF param.=500, on 10 3 mesh. (b) CS-RBF param.=500, on 20 3 mesh. (c) CS-RBF param.=500, on mesh. (d) CS-RBF param.=1000, on mesh. Figure 14: The effects of having a very large support parameter resulting in a very small support radius on a variety of 3D mesh resolutions. Table 3: Support parameter, Mesh Resolutions, and Run time for the above interpolations in Figure 14 Support Parameter Mesh Resolution Runtime (seconds)

21 (a) CS-RBF param.=500, on mesh. (b) CS-RBF param.=250, on mesh. (c) CS-RBF param.=200, on mesh. (d) CS-RBF param.=100, on mesh. Figure 15: The process of starting from a large support parameter on a fine 3D mesh and incrementally reducing the parameter. Table 4: Support parameter, Mesh Resolutions, and Run time for the above interpolations in Figure 15 Support Parameter Mesh Resolution Runtime (seconds)

22 Using a Compactly Supported RBF to interpolate the surface from a Point Cloud of 8171 points Figure 16 shows the provided point cloud of 8171 points to be interpolated into a surface using CS-RBF. Figure 16: Point Cloud of 8171 points on the surface of the bunny. Figures 17 and 18 demonstrate one effect of changing support parameter in the case of the 1889 point set using a simple CS-RBF (i.e. (1 r) 2 ). Increases in the support parameter result in decreases in the support radius, radius = 1 parameter. 21

23 (a) CS-RBF param.=500, on 50 3 mesh. (b) CS-RBF param.=500, on 75 3 mesh. (c) CS-RBF param.=500, on mesh. (d) CS-RBF param.=500, on mesh. (e) CS-RBF param.=500, on mesh. (f) CS-RBF param.=500, on mesh. Figure 17: Adjusting the mesh resolution for the CS-RBF (50 3, 75 3, 100 3, 125 3, 150 3, ) while interpolating with a support parameter of

24 (a) CS-RBF param.=500, on mesh. (b) CS-RBF param.=400, on mesh. (c) CS-RBF param.=300, on mesh. (d) CS-RBF param.=200, on mesh. (e) CS-RBF param.=175, on mesh. (f) CS-RBF param.=150, on mesh. Figure 18: Adjusting the support parameter for the CS-RBF (500, 400, 300, 200, 175, 150) while interpolating over a 150x150x150 3D mesh. 23

25 Table 5: Support parameter, Mesh Resolutions, and Run time for the above interpolations in Figures 17 and 18 Support Parameter Mesh Resolution Runtime (seconds) Figure 19: CSRBF Interpolation of Point Cloud of 8171 points with Support Parameter of 250 and a mesh resolution of Run time = seconds. 24

26 Conclusions It can be seen that from each Point Cloud data set one can recreate a surface that is readily recognizable as a bunny. As expected, when more points are provided, the reconstructed surface has more detail and more closely resembles the original bunny statue. While the 453 point data set does yield a bunny with some noticeable issues (i.e. the ears), this is also expected behavior because these issues occur in places where the surface changes abruptly and has sharper corners/edges and with fewer points comes less information to interpolate from. Other than these issues, adjusting to find suitable parametrization was relatively straight forward, and run time was not a significant issue. For the second Point Cloud data set with 1889 points, run time became a bigger concern. Though the process of investigating which combination of support parameter and mesh resolution generated a satisfactory reconstruction of the bunny, it was discovered that starting with an excessively large support parameter would drastically reduce the run time (as expected) and allow for a more systematic approach to a support parameter small enough to provide a smooth interpolation over a suitable mesh. This approach was used for the third data set of 8171 points. This allowed for an initial assessment of the feasibility of various mesh resolutions and led to a more efficient approach to reconstructing a high level of detail. As expected, even the process of using CS-RBFs to interpolate over this many points (3x8171=24513) over a fine enough mesh to reflect the detail of information provided can take a significant amount of time to run. The matlab script developed for this task used a for loop to iteratively change the grid resolution and save the *.jpg and *.fig files. While this did not reduce the run time for the individual runs, this did remove the need for the user to be monitoring progress the whole time. As was discussed, the evaluation of success was based solely on visual inspection. Several other figures were generated than the ones that were included in this report. Also tested for the largest data set was one of Wendland s shifted CS-RBFs (φ(r) = r+( r + 35r 2 )). Submitted with this report separate from the folder containing the files for the Latex version of the report in a folder labeled figures is a complete collection of the figures generated. This may provided useful to allow one the opportunity to see which other combinations of parameters were tested and the results generated from these tests. The subdirectory IMQ contains results from the 25

27 453 point data set, CSRBF contains results from the 1889 point data set, and CSRBF2 contains results from the 8171 point data set. In this last set, the figure filenames starting with W coorresponde to those using the Wendland s CS-RBF. 26

28 References [1] turk/bunny/bunny.html, Greg Turk, Computer Graphics Group, Georgia Institute of Technology, USA. [2] C.S. Chen, Y.C.Hon, R.A. Schaback, Scientific Computing with Radial Basis Functions, Department of Mathematics, University of Southern Mississippi, USA.S [3] Tyler Reese, COS702 Assignment 1; Radial Basis Functions, University of Southern Mississippi, Septmeber 22nd, [4] Tyler Reese, COS702 Assignment 3; Compactly Supported Radial Basis Functions, University of Southern Mississippi, October 16th, [5] C.S. Chen, Lecture Notes (presentation slides); COS 702; Point Cloud Data, Department of Mathematics, University of Southern Mississippi, USA.S 27

29 28

WEEK #3, Lecture 1: Sparse Systems, MATLAB Graphics

WEEK #3, Lecture 1: Sparse Systems, MATLAB Graphics WEEK #3, Lecture 1: Sparse Systems, MATLAB Graphics Visualization of Matrices Good visuals anchor any presentation. MATLAB has a wide variety of ways to display data and calculation results that can be

More information

Beginner s Matlab Tutorial

Beginner s Matlab Tutorial Christopher Lum lum@u.washington.edu Introduction Beginner s Matlab Tutorial This document is designed to act as a tutorial for an individual who has had no prior experience with Matlab. For any questions

More information

Computer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI)

Computer Graphics CS 543 Lecture 12 (Part 1) Curves. Prof Emmanuel Agu. Computer Science Dept. Worcester Polytechnic Institute (WPI) Computer Graphics CS 54 Lecture 1 (Part 1) Curves Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI) So Far Dealt with straight lines and flat surfaces Real world objects include

More information

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc.

Linear Programming for Optimization. Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1. Introduction Linear Programming for Optimization Mark A. Schulze, Ph.D. Perceptive Scientific Instruments, Inc. 1.1 Definition Linear programming is the name of a branch of applied mathematics that

More information

We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model

We can display an object on a monitor screen in three different computer-model forms: Wireframe model Surface Model Solid model CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant

More information

We want to define smooth curves: - for defining paths of cameras or objects. - for defining 1D shapes of objects

We want to define smooth curves: - for defining paths of cameras or objects. - for defining 1D shapes of objects lecture 10 - cubic curves - cubic splines - bicubic surfaces We want to define smooth curves: - for defining paths of cameras or objects - for defining 1D shapes of objects We want to define smooth surfaces

More information

AMATH 352 Lecture 3 MATLAB Tutorial Starting MATLAB Entering Variables

AMATH 352 Lecture 3 MATLAB Tutorial Starting MATLAB Entering Variables AMATH 352 Lecture 3 MATLAB Tutorial MATLAB (short for MATrix LABoratory) is a very useful piece of software for numerical analysis. It provides an environment for computation and the visualization. Learning

More information

Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

More information

Interpolating and Approximating Implicit Surfaces from Polygon Soup

Interpolating and Approximating Implicit Surfaces from Polygon Soup Interpolating and Approximating Implicit Surfaces from Polygon Soup 1. Briefly summarize the paper s contributions. Does it address a new problem? Does it present a new approach? Does it show new types

More information

Volume visualization I Elvins

Volume visualization I Elvins Volume visualization I Elvins 1 surface fitting algorithms marching cubes dividing cubes direct volume rendering algorithms ray casting, integration methods voxel projection, projected tetrahedra, splatting

More information

2.1: MATRIX OPERATIONS

2.1: MATRIX OPERATIONS .: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

More information

Factoring Trinomials: The ac Method

Factoring Trinomials: The ac Method 6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

More information

the points are called control points approximating curve

the points are called control points approximating curve Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

More information

AFM Image Deconvolution Software User s Manual

AFM Image Deconvolution Software User s Manual M. C. Goh Research Group Version 2.0 AFM Image Deconvolution Software User s Manual Table of Content Introduction Contact M. C. Goh Research Group... 3 Zod Information... 3 Installing Zod Downloading files...

More information

Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

More information

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are

More information

OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

More information

Least-Squares Intersection of Lines

Least-Squares Intersection of Lines Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

DELFT UNIVERSITY OF TECHNOLOGY

DELFT UNIVERSITY OF TECHNOLOGY DELFT UNIVERSITY OF TECHNOLOGY REPORT 12-06 ADJOINT SENSITIVITY IN ELECTRICAL IMPEDANCE TOMOGRAPHY USING COMSOL MULTIPHYSICS W. MULCKHUYSE, D. LAHAYE, A. BELITSKAYA ISSN 1389-6520 Reports of the Department

More information

MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem

MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and

More information

MATH 590: Meshfree Methods

MATH 590: Meshfree Methods MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter

More information

Generate Cryptographic key using generated 3D- Digital Image

Generate Cryptographic key using generated 3D- Digital Image Dr.Hala B. Abdul Wahab*, Dr.Suhad M. Kadhum* & Ekhlas K. Gbashi* Received on: 20/ 1 /2008 Accepted on: 4/9 /2008 Abstract Every few years, computer security has to re-invent itself. New technologies and

More information

Solutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the

Solutions to Linear Algebra Practice Problems 1. form (because the leading 1 in the third row is not to the right of the Solutions to Linear Algebra Practice Problems. Determine which of the following augmented matrices are in row echelon from, row reduced echelon form or neither. Also determine which variables are free

More information

Anatomic Modeling from Unstructured Samples Using Variational Implicit Surfaces

Anatomic Modeling from Unstructured Samples Using Variational Implicit Surfaces Studies in Health Technology and Informatics, vol. 81 (Proceedings of Medicine Meets Virtual Reality 2001. J. D. Westwood, et al., eds.), Amsterdam: IOS Press, pp. 594-600. Anatomic Modeling from Unstructured

More information

1 Cubic Hermite Spline Interpolation

1 Cubic Hermite Spline Interpolation cs412: introduction to numerical analysis 10/26/10 Lecture 13: Cubic Hermite Spline Interpolation II Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mark Cowlishaw, Nathanael Fillmore 1 Cubic Hermite

More information

LINEAR SYSTEMS. Consider the following example of a linear system:

LINEAR SYSTEMS. Consider the following example of a linear system: LINEAR SYSTEMS Consider the following example of a linear system: Its unique solution is x +2x 2 +3x 3 = 5 x + x 3 = 3 3x + x 2 +3x 3 = 3 x =, x 2 =0, x 3 = 2 In general we want to solve n equations in

More information

Scattered Node Compact Finite Difference-Type Formulas Generated from Radial Basis Functions

Scattered Node Compact Finite Difference-Type Formulas Generated from Radial Basis Functions Scattered Node Compact Finite Difference-Type Formulas Generated from Radial Basis Functions Grady B. Wright a,,, Bengt Fornberg b,2 a Department of Mathematics, University of Utah, Salt Lake City, UT

More information

CS3220 Lecture Notes: QR factorization and orthogonal transformations

CS3220 Lecture Notes: QR factorization and orthogonal transformations CS3220 Lecture Notes: QR factorization and orthogonal transformations Steve Marschner Cornell University 11 March 2009 In this lecture I ll talk about orthogonal matrices and their properties, discuss

More information

Constrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume *

Constrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume * Constrained Tetrahedral Mesh Generation of Human Organs on Segmented Volume * Xiaosong Yang 1, Pheng Ann Heng 2, Zesheng Tang 3 1 Department of Computer Science and Technology, Tsinghua University, Beijing

More information

A Direct Numerical Method for Observability Analysis

A Direct Numerical Method for Observability Analysis IEEE TRANSACTIONS ON POWER SYSTEMS, VOL 15, NO 2, MAY 2000 625 A Direct Numerical Method for Observability Analysis Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper presents an algebraic method

More information

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

A Novel Method to Improve Resolution of Satellite Images Using DWT and Interpolation

A Novel Method to Improve Resolution of Satellite Images Using DWT and Interpolation A Novel Method to Improve Resolution of Satellite Images Using DWT and Interpolation S.VENKATA RAMANA ¹, S. NARAYANA REDDY ² M.Tech student, Department of ECE, SVU college of Engineering, Tirupati, 517502,

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

More information

POISSON AND LAPLACE EQUATIONS. Charles R. O Neill. School of Mechanical and Aerospace Engineering. Oklahoma State University. Stillwater, OK 74078

POISSON AND LAPLACE EQUATIONS. Charles R. O Neill. School of Mechanical and Aerospace Engineering. Oklahoma State University. Stillwater, OK 74078 21 ELLIPTICAL PARTIAL DIFFERENTIAL EQUATIONS: POISSON AND LAPLACE EQUATIONS Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078 2nd Computer

More information

P164 Tomographic Velocity Model Building Using Iterative Eigendecomposition

P164 Tomographic Velocity Model Building Using Iterative Eigendecomposition P164 Tomographic Velocity Model Building Using Iterative Eigendecomposition K. Osypov* (WesternGeco), D. Nichols (WesternGeco), M. Woodward (WesternGeco) & C.E. Yarman (WesternGeco) SUMMARY Tomographic

More information

1.2 Solving a System of Linear Equations

1.2 Solving a System of Linear Equations 1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

More information

Numerical Analysis. Professor Donna Calhoun. Fall 2013 Math 465/565. Office : MG241A Office Hours : Wednesday 10:00-12:00 and 1:00-3:00

Numerical Analysis. Professor Donna Calhoun. Fall 2013 Math 465/565. Office : MG241A Office Hours : Wednesday 10:00-12:00 and 1:00-3:00 Numerical Analysis Professor Donna Calhoun Office : MG241A Office Hours : Wednesday 10:00-12:00 and 1:00-3:00 Fall 2013 Math 465/565 http://math.boisestate.edu/~calhoun/teaching/math565_fall2013 What is

More information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

A Short Introduction to Computer Graphics

A Short Introduction to Computer Graphics A Short Introduction to Computer Graphics Frédo Durand MIT Laboratory for Computer Science 1 Introduction Chapter I: Basics Although computer graphics is a vast field that encompasses almost any graphical

More information

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year. This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

More information

Interior-Point Algorithms for Quadratic Programming

Interior-Point Algorithms for Quadratic Programming Interior-Point Algorithms for Quadratic Programming Thomas Reslow Krüth Kongens Lyngby 2008 IMM-M.Sc-2008-19 Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800

More information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

07-Nodal Analysis Text: ECEGR 210 Electric Circuits I

07-Nodal Analysis Text: ECEGR 210 Electric Circuits I 07Nodal Analysis Text: 3.1 3.4 ECEGR 210 Electric Circuits I Overview Introduction Nodal Analysis Nodal Analysis with Voltage Sources Dr. Louie 2 Basic Circuit Laws Ohm s Law Introduction Kirchhoff s Voltage

More information

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli

Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Situation 23: Simultaneous Equations Prepared at the University of Georgia EMAT 6500 class Date last revised: July 22 nd, 2013 Nicolina Scarpelli Prompt: A mentor teacher and student teacher are discussing

More information

5.7 Maximum and Minimum Values

5.7 Maximum and Minimum Values 5.7 Maximum and Minimum Values Objectives Icandefinecriticalpoints. I know the di erence between local and absolute minimums/maximums. I can find local maximum(s), minimum(s), and saddle points for a given

More information

Scheduling Algorithm with Optimization of Employee Satisfaction

Scheduling Algorithm with Optimization of Employee Satisfaction Washington University in St. Louis Scheduling Algorithm with Optimization of Employee Satisfaction by Philip I. Thomas Senior Design Project http : //students.cec.wustl.edu/ pit1/ Advised By Associate

More information

A Prototype System for Character Animation Based on Real-time Deformations

A Prototype System for Character Animation Based on Real-time Deformations A Prototype System for Character Animation Based on Real-time Deformations Nikita Kojekine, Vladimir Savchenko*, Mikhail Senin**, Ichiro Hagiwara Faculty of Engineering, Tokyo Institute of Technology,

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

Lecture 1: Systems of Linear Equations

Lecture 1: Systems of Linear Equations MTH Elementary Matrix Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Lecture 1 Systems of Linear Equations ² Systems of two linear equations with two variables

More information

( ) which must be a vector

( ) which must be a vector MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

More information

3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES

3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES 3D POINT CLOUD CONSTRUCTION FROM STEREO IMAGES Brian Peasley * I propose an algorithm to construct a 3D point cloud from a sequence of stereo image pairs that show a full 360 degree view of an object.

More information

Level Set Framework, Signed Distance Function, and Various Tools

Level Set Framework, Signed Distance Function, and Various Tools Level Set Framework Geometry and Calculus Tools Level Set Framework,, and Various Tools Spencer Department of Mathematics Brigham Young University Image Processing Seminar (Week 3), 2010 Level Set Framework

More information

Natural Neighbour Interpolation

Natural Neighbour Interpolation Natural Neighbour Interpolation DThe Natural Neighbour method is a geometric estimation technique that uses natural neighbourhood regions generated around each point in the data set. The method is particularly

More information

Robert Collins CSE598G. More on Mean-shift. R.Collins, CSE, PSU CSE598G Spring 2006

Robert Collins CSE598G. More on Mean-shift. R.Collins, CSE, PSU CSE598G Spring 2006 More on Mean-shift R.Collins, CSE, PSU Spring 2006 Recall: Kernel Density Estimation Given a set of data samples x i ; i=1...n Convolve with a kernel function H to generate a smooth function f(x) Equivalent

More information

A Learning Based Method for Super-Resolution of Low Resolution Images

A Learning Based Method for Super-Resolution of Low Resolution Images A Learning Based Method for Super-Resolution of Low Resolution Images Emre Ugur June 1, 2004 emre.ugur@ceng.metu.edu.tr Abstract The main objective of this project is the study of a learning based method

More information

Solving Systems of Linear Equations

Solving Systems of Linear Equations LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

More information

Module1. x 1000. y 800.

Module1. x 1000. y 800. Module1 1 Welcome to the first module of the course. It is indeed an exciting event to share with you the subject that has lot to offer both from theoretical side and practical aspects. To begin with,

More information

Vector and Matrix Norms

Vector and Matrix Norms Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

More information

Using MuPAD and JavaView to Visualize Mathematics on the Internet

Using MuPAD and JavaView to Visualize Mathematics on the Internet Using MuPAD and JavaView to Visualize Mathematics on the Internet M. Majewski K. Polthier Zayed University, UAE Zuse Institute Berlin (ZIB) majewski@mupad.com polthier@zib.de Abstract Mathematics education

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

What s the difference between Bitmap and Vector?

What s the difference between Bitmap and Vector? Colourgen Limited Tel: +44 (0)1628 588700 The AmBer Centre Sales: +44 (0)1628 588733 Oldfield Road, Maidenhead Support: +44 (0)1628 588755 Berkshire, SL6 1TH Accounts: +44 (0)1628 588766 United Kingdom

More information

Surface Modeling. Polygon Surfaces. Types: Generating models: Polygon Tables. Set of surface polygons that enclose an object interior

Surface Modeling. Polygon Surfaces. Types: Generating models: Polygon Tables. Set of surface polygons that enclose an object interior Surface Modeling Types: Polygon surfaces Curved surfaces Volumes Generating models: Interactive Procedural Polygon Surfaces Set of surface polygons that enclose an object interior Slide 1 Slide 2 Polygon

More information

3. INNER PRODUCT SPACES

3. INNER PRODUCT SPACES . INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

More information

α = u v. In other words, Orthogonal Projection

α = u v. In other words, Orthogonal Projection Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

More information

DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

More information

MATLAB and Big Data: Illustrative Example

MATLAB and Big Data: Illustrative Example MATLAB and Big Data: Illustrative Example Rick Mansfield Cornell University August 19, 2014 Goals Use a concrete example from my research to: Demonstrate the value of vectorization Introduce key commands/functions

More information

Visualization Plugin for ParaView

Visualization Plugin for ParaView Alexey I. Baranov Visualization Plugin for ParaView version 1.3 Springer Contents 1 Visualization with ParaView..................................... 1 1.1 ParaView plugin installation.................................

More information

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are

x y The matrix form, the vector form, and the augmented matrix form, respectively, for the system of equations are Solving Sstems of Linear Equations in Matri Form with rref Learning Goals Determine the solution of a sstem of equations from the augmented matri Determine the reduced row echelon form of the augmented

More information

CME 192: Introduction to Matlab

CME 192: Introduction to Matlab CME 192: Introduction to Matlab Matlab Basics Brett Naul January 7, 2015 What is Matlab? Computing & programming environment Visualization and mathematical prototyping tool Why Matlab? Comparison with

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

Method of Data Center Classifications

Method of Data Center Classifications Method of Data Center Classifications Krisztián Kósi Óbuda University, Bécsi út 96/B, H-1034 Budapest, Hungary kosi.krisztian@phd.uni-obuda.hu Abstract: This paper is about the Classification of big data

More information

Multi scale random field simulation program

Multi scale random field simulation program Multi scale random field simulation program 1.15. 2010 (Updated 12.22.2010) Andrew Seifried, Stanford University Introduction This is a supporting document for the series of Matlab scripts used to perform

More information

b) lower case always use lower case for all matlab commands. This is what matlab recognizes.

b) lower case always use lower case for all matlab commands. This is what matlab recognizes. 1 Matlab 1) Fundamentals a) Getting Help for more detailed help on any topic, typing help, then a space, and then the matlab command brings up a detailed page on the command or topic. For really difficult

More information

2+2 Just type and press enter and the answer comes up ans = 4

2+2 Just type and press enter and the answer comes up ans = 4 Demonstration Red text = commands entered in the command window Black text = Matlab responses Blue text = comments 2+2 Just type and press enter and the answer comes up 4 sin(4)^2.5728 The elementary functions

More information

Edmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H).

Edmund Li. Where is defined as the mutual inductance between and and has the SI units of Henries (H). INDUCTANCE MUTUAL INDUCTANCE If we consider two neighbouring closed loops and with bounding surfaces respectively then a current through will create a magnetic field which will link with as the flux passes

More information

Blind Deconvolution of Corrupted Barcode Signals

Blind Deconvolution of Corrupted Barcode Signals Blind Deconvolution of Corrupted Barcode Signals Everardo Uribe and Yifan Zhang Advisors: Ernie Esser and Yifei Lou Interdisciplinary Computational and Applied Mathematics Program University of California,

More information

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

More information

Leapfrog : new software for faster and better 3D geological modelling

Leapfrog : new software for faster and better 3D geological modelling Leapfrog : new software for faster and better 3D geological modelling Paul Hodkiewicz, Principal Consultant (Geology), SRK Consulting, 10 Richardson Street, West Perth WA 6005, Australia, phodkiewicz@srk.com.au

More information

Groups 1. Definition 1 A Group G is a set with an operation which satisfies the following: e a = a e = e. a a 1 = a 1 a = e.

Groups 1. Definition 1 A Group G is a set with an operation which satisfies the following: e a = a e = e. a a 1 = a 1 a = e. Groups 1 1 Introduction to Groups Definition 1 A Group G is a set with an operation which satisfies the following: 1. there is an identity element e G, such that for every a G e a = a e = e 2. every element

More information

CS 4620 Practicum Programming Assignment 6 Animation

CS 4620 Practicum Programming Assignment 6 Animation CS 4620 Practicum Programming Assignment 6 Animation out: Friday 14th November 2014 due: : Monday 24th November 2014 1 Introduction In this assignment, we will explore a common topic in animation: key

More information

Name: Section Registered In:

Name: Section Registered In: Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

More information

Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li. Advised by: Dave Mount. May 22, 2014

Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li. Advised by: Dave Mount. May 22, 2014 Euclidean Minimum Spanning Trees Based on Well Separated Pair Decompositions Chaojun Li Advised by: Dave Mount May 22, 2014 1 INTRODUCTION In this report we consider the implementation of an efficient

More information

MATHEMATICS CLASS - XII BLUE PRINT - II. (1 Mark) (4 Marks) (6 Marks)

MATHEMATICS CLASS - XII BLUE PRINT - II. (1 Mark) (4 Marks) (6 Marks) BLUE PRINT - II MATHEMATICS CLASS - XII S.No. Topic VSA SA LA TOTAL ( Mark) (4 Marks) (6 Marks). (a) Relations and Functions 4 () 6 () 0 () (b) Inverse trigonometric Functions. (a) Matrices Determinants

More information

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical

More information

An Overview of the Finite Element Analysis

An Overview of the Finite Element Analysis CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry

More information

CS Master Level Courses and Areas COURSE DESCRIPTIONS. CSCI 521 Real-Time Systems. CSCI 522 High Performance Computing

CS Master Level Courses and Areas COURSE DESCRIPTIONS. CSCI 521 Real-Time Systems. CSCI 522 High Performance Computing CS Master Level Courses and Areas The graduate courses offered may change over time, in response to new developments in computer science and the interests of faculty and students; the list of graduate

More information

C A R I B B E A N E X A M I N A T I O N S C O U N C I L REPORT ON CANDIDATES WORK IN THE SECONDARY EDUCATION CERTIFICATE EXAMINATION MAY/JUNE 2011

C A R I B B E A N E X A M I N A T I O N S C O U N C I L REPORT ON CANDIDATES WORK IN THE SECONDARY EDUCATION CERTIFICATE EXAMINATION MAY/JUNE 2011 C A R I B B E A N E X A M I N A T I O N S C O U N C I L REPORT ON CANDIDATES WORK IN THE SECONDARY EDUCATION CERTIFICATE EXAMINATION MAY/JUNE 2011 MATHEMATICS GENERAL PROFICIENCY EXAMINATION Copyright

More information

Avizo AvizoFire - The 3D visualization Software for NDT & Materials Science

Avizo AvizoFire - The 3D visualization Software for NDT & Materials Science Avizo AvizoFire - The 3D visualization Software for NDT & Materials Science Peter Westenberger Application Enginieer May 7, 2010 Avizo Visualize to Understand Avizo software is a powerful, multifaceted

More information

Scan Conversion of Filled Primitives Rectangles Polygons. Many concepts are easy in continuous space - Difficult in discrete space

Scan Conversion of Filled Primitives Rectangles Polygons. Many concepts are easy in continuous space - Difficult in discrete space Walters@buffalo.edu CSE 480/580 Lecture 7 Slide 1 2D Primitives I Point-plotting (Scan Conversion) Lines Circles Ellipses Scan Conversion of Filled Primitives Rectangles Polygons Clipping In graphics must

More information

Lecture 6. Inverse of Matrix

Lecture 6. Inverse of Matrix Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

More information

Orthogonal Projections and Orthonormal Bases

Orthogonal Projections and Orthonormal Bases CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).

More information

Colour Image Segmentation Technique for Screen Printing

Colour Image Segmentation Technique for Screen Printing 60 R.U. Hewage and D.U.J. Sonnadara Department of Physics, University of Colombo, Sri Lanka ABSTRACT Screen-printing is an industry with a large number of applications ranging from printing mobile phone

More information

http://school-maths.com Gerrit Stols

http://school-maths.com Gerrit Stols For more info and downloads go to: http://school-maths.com Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It

More information

Visualization with OpenDX

Visualization with OpenDX Alexey I. Baranov Visualization with OpenDX User s Guide Springer Contents 1 Visualization with OpenDX..................................... 1 1.1 OpenDX module installation.................................

More information

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3

MATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3 MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................

More information

G.H. Raisoni College of Engineering, Nagpur. Department of Information Technology

G.H. Raisoni College of Engineering, Nagpur. Department of Information Technology Practical List 1) WAP to implement line generation using DDA algorithm 2) WAP to implement line using Bresenham s line generation algorithm. 3) WAP to generate circle using circle generation algorithm

More information

The development of an information modelling system for regional water resource assessments

The development of an information modelling system for regional water resource assessments FRIEND 2002 Regional Hydrology 1 : Bridging the Gap between Research and Practice (Proceedings ofthe fourth International FRII-ND Conference held al Cape Town. South Africa. March 2002). IAI IS Publ. no.

More information

MATLAB Tutorial. Chapter 6. Writing and calling functions

MATLAB Tutorial. Chapter 6. Writing and calling functions MATLAB Tutorial Chapter 6. Writing and calling functions In this chapter we discuss how to structure a program with multiple source code files. First, an explanation of how code files work in MATLAB is

More information

INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

More information