AFFINE COMBINATIONS, BARYCENTRIC COORDINATES, AND CONVEX COMBINATIONS
|
|
- Emil Stevenson
- 6 years ago
- Views:
Transcription
1 On-Line Geometric Modeling Notes AFFINE COMBINATIONS, BARYCENTRIC COORDINATES, AND CONVEX COMBINATIONS Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis 1 Affine Combinations of Points Points in an affine space are utilized to position ourselves within the space. The operations on the vectors of an affine space are numerous addition, scalar multiplication, dot products, cross products but the operations on the points are limited. In this section we discuss the basic operations on points affine combinations. Let P 1 and P 2 be points in an affine space. Consider the expression P = P 1 + t(p 2 P 1 ) This equation is meaningful, as P 2 P 1 is a vector, and thus so is t(p 2 P 1 ). Therefore P is the sum of a point and a vector which is again a point (see the notes on Points and Vectors. This point P represents, in the affine space of two-dimensional points and vectors, a point on the line that passes through P 1 and P 2. We note that if 0 t 1 then P is somewhere on the line segment joining P 1 and P 2. This expression allows us to define a basic operation on points. We utilize the following notation P = (1 t)p 1 + tp 2
2 to mean that P is the point defined by P = P 1 + t(p 2 P 1 ) We can then define an affine combination of two points P 1 and P 2 to be P = α 1 P 1 + α 2 P 2 where α 1 + α 2 = 1. The form P = (1 t)p 1 + tp 2 is shown to be an affine transformation by setting α 2 = t. We can generalize this to define an affine combination of an arbitrary number of points. If P 1, P 2,..., P n are points and α 1, α 2,..., α n are scalars such that α 1 + α α n = 1, then α 1 P 1 + α 2 P α n P n is defined to be the point P 1 + α 2 (P 2 P 1 ) + + α n (P n P 1 ) To construct an excellent example of an affine combination consider three points P 1, P 2 and P 3. A point P defined by P = α 1 P 1 + α 2 P 2 + α 3 P 3 where α 1 + α 2 + α 3 = 1, gives a point in the triangle P 1 P 2 P 3. We note that the definition of affine combination defines this point to be P = P 1 + α 2 (P 2 P 1 ) + α 3 (P 3 P 1 ) The following illustration shows the point P generated when α 1 = α 2 = 1 4 and α 3 =
3 In fact, it can be easily shown that if 0 α1, α2, α3 1 then the point P will be within (or on the boundary) of the triangle. If any α i is less than zero or greater than one, then the point will lie outside the triangle. If any α i is zero, then the point will lie on the boundary of the triangle. 2 Barycentric Coordinates Given a frame ( v 1, v 2,..., v n, O) for an affine space A, we can write any point P uniquely as P = p 1 v 1 + p 2 v p n v n + O If we define points P i by P 0 = O P 1 = O + v 1 P 2 = O + v 2. P n = O + v n and define p 0 to be p 0 = 1 (p 1 + p p n ) 3
4 then we can see that P can be equivalently written as P = p 0 P 0 + p 1 P 1 + p 2 P p n P n where p 0 + p 1 + p p n = 1 In this form, the values (p 0, p 1, p 2,..., p n ) are called the barycentric coordinates of P relative to the points (P 0, P 1, P 2,..., P n ) Vectors can also be expressed in barycentric form by letting u 0 = (u 1 + u u n ) Then we have u = u 0 P 0 + u 1 P 1 + u 2 P u n P n where now we have that u 0 + u 1 + u u n = 0. To give a simple example of barycentric coordinates, consider two points P 1 and P 2 in the plane. If α 1 and α 2 are scalars such that α 1 + α 2 = 1, then the point P defined by P = α 1 P 1 + α 2 P 2 is a point on the line that passes through P 1 and P 2. If 0 α 1, α 2 1 then the point P is on the line segment joining P 1 and P 2. The following figure shows an example of a line and three points P, Q and R. These points were generated using the following αs: P : α 1 = 1 3, α 2 = 2 3 Q : α 1 = 3 4, α 2 = 1 4 R : α 1 = 4 3, α 2 = 1 3 4
5 To give a slightly more complex example of barycentric coordinates, consider three points P 1, P 2, P 3 in the plane. If α 1, α 2, α 3 are scalars such that α 1 + α 2 + α 3 = 1, then the point P defined by P = α 1 P 1 + α 2 P 2 + α 3 P 3 is a point on the plane of the triangle formed by P 1, P 2, P 3. The point is within the triangle P 1 P 2 P 3 if 0 α 1, α 2, α 3 1. If any of the α s is less than zero or greater than one, the point P is outside the triangle. If any of the α s is zero, we reduce to the example above and note that P is on one of the lines joining the vertices of the triangle. The following figure shows an example of such a triangle and three points P, Q and R, these points were calculated using the following α s: P : α 1 = α 2 = 1 4, α 3 = 1 2. Q : α 1 = 1 2, α 2 = 3 4, α 3 = 1 4. R : α 1 = 0, α 2 = 3 4, α 3 =
6 Thus barycentric coordinates are another method of introducing coordinates into an affine space. If the coordinates sum to one, they represent a point ; if the coordinates sum to zero, they represent a vector. 3 Convex Combinations Given a set of points P 0, P 1,..., P n, we can form affine combinations of these points by selecting α 0, α 1,..., α n, with α 0 + α α n = 1 and form the point P = α 0 P 0 + α 1 P α n P n If each α i is such that 0 α i 1, then the points P is called a convex combination of the points P 0, P 1,..., P n. To give a simple example of this, consider two points P 0 and P 1. Any point P on the line passing through these two points can be written as P = α 0 P 0 + α 1 P 1 which is an affine combination of the two points. The points Q and R in the following figure are affine combinations of P 0 and P 1. However, the point Q is a convex combination, as 0 α 0, α 1 1, and any point on the line segment joining P 0 and P 1 can be written in this way. Given any set of points, we say that the set is a convex set, if given any two points of the set, any convex combination of these two points is also in the set. The following figure illustrates both a convex set (on the left) and a non-convex set (on the right). 6
7 This concept is actually quite intuitive, in that if one can draw a straight line between two points of the set that is not completely contained within the set, the the set is non-convex. The set of all points P that can be written as convex combinations of P 0, P 1,..., P n is called the convex hull of the points P 0, P 1,..., P n. This convex hull is the smallest convex set that contains the set of points P 0, P 1,..., P n. The following figure illustrates the convex hull of a set of six points: One of the six points does not contribute to the boundary of the convex hull. If one looked closely at the coordinates of the point, one would find that this point could be written as a convex combination of the other five. 4 Acknowledgement Most of this material was adapted from Tony DeRose s wonderful treatment of affine spaces given in [1]. 7
8 References [1] Tony DeRose. Coordinate-free geometric programming. Technical Report , Department of Computer Science and Engineering, University of Washington, Seattle, Washington, All contents copyright (c) 1996, 1997, 1998, 1999 Computer Science Department, University of California, Davis All rights reserved. 8
BARYCENTRIC COORDINATES
Computer Graphics Notes BARYCENTRIC COORDINATES Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview If we are given a frame
More informationCONVERSION OF COORDINATES BETWEEN FRAMES
ECS 178 Course Notes CONVERSION OF COORDINATES BETWEEN FRAMES Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview Frames
More informationOn-Line Geometric Modeling Notes FRAMES
On-Line Geometric Modeling Notes FRAMES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis In computer graphics we manipulate objects.
More informationOn-Line Geometric Modeling Notes
On-Line Geometric Modeling Notes CUBIC BÉZIER CURVES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview The Bézier curve representation
More informationSlope Fields and Differential Equations. Copyright Cengage Learning. All rights reserved.
Slope Fields and Differential Equations Copyright Cengage Learning. All rights reserved. Objectives Review verifying solutions to differential equations. Review solving differential equations. Review using
More informationOn-Line Geometric Modeling Notes VECTOR SPACES
On-Line Geometric Modeling Notes VECTOR SPACES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis These notes give the definition of
More informationECS 178 Course Notes QUATERNIONS
ECS 178 Course Notes QUATERNIONS Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview The quaternion number system was discovered
More informationCONTROL POLYGONS FOR CUBIC CURVES
On-Line Geometric Modeling Notes CONTROL POLYGONS FOR CUBIC CURVES Kenneth I. Joy Visualization and Graphics Research Group Department of Computer Science University of California, Davis Overview B-Spline
More informationComputer Graphics MTAT Raimond Tunnel
Computer Graphics MTAT.03.015 Raimond Tunnel Points and Vectors In computer graphics we distinguish: Point a location in space (location vector, kohavektor) Vector a direction in space (direction vector,
More informationCSE 167: Lecture 11: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2012
CSE 167: Introduction to Computer Graphics Lecture 11: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2012 Announcements Homework project #5 due Nov. 9 th at 1:30pm
More informationQuantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors.
Vectors summary Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors. AB is the position vector of B relative to A and is the vector
More informationEuclidean Spaces. Euclidean Spaces. Chapter 10 -S&B
Chapter 10 -S&B The Real Line: every real number is represented by exactly one point on the line. The plane (i.e., consumption bundles): Pairs of numbers have a geometric representation Cartesian plane
More informationChapter. Triangles. Copyright Cengage Learning. All rights reserved.
Chapter 3 Triangles Copyright Cengage Learning. All rights reserved. 3.5 Inequalities in a Triangle Copyright Cengage Learning. All rights reserved. Inequalities in a Triangle Important inequality relationships
More informationPart I, Number Systems CS131 Mathematics for Computer Scientists II Note 3 VECTORS
CS131 Part I, Number Systems CS131 Mathematics for Computer Scientists II Note 3 VECTRS Vectors in two and three dimensional space are defined to be members of the sets R 2 and R 3 defined by: R 2 = {(x,
More informationPractical Linear Algebra: A Geometry Toolbox
Practical Linear Algebra: A Geometry Toolbox Third edition Chapter 2: Here and There: Points and Vectors in 2D Gerald Farin & Dianne Hansford CRC Press, Taylor & Francis Group, An A K Peters Book www.farinhansford.com/books/pla
More informationCONVERGENCE OF A SPECIAL SET OF TRIANGLES
Journal of Mathematical Sciences: Advances and Applications Volume 50 08 Pages 59-70 Available at http://scientificadvances.co.in DOI: http://dx.doi.org/0.864/jmsaa_700938 COVERGECE OF A SPECIAL SET OF
More informationCS 4204 Computer Graphics
CS 4204 Computer Graphics Vector and Matrix Yong Cao Virginia Tech Vectors N-tuple: Vectors N-tuple tuple: Magnitude: Unit vectors Normalizing a vector Operations with vectors Addition Multiplication with
More informationNotes on Cellwise Data Interpolation for Visualization Xavier Tricoche
Notes on Cellwise Data Interpolation for Visualization Xavier Tricoche urdue University While the data (computed or measured) used in visualization is only available in discrete form, it typically corresponds
More information1 Convexity explains SVMs
1 Convexity explains SVMs The convex hull of a set is the collection of linear combinations of points in the set where the coefficients are nonnegative and sum to one. Two sets are linearly separable if
More informationThe Distance Formula & The Midpoint Formula
The & The Professor Tim Busken Mathematics Department Januar 14, 2015 Theorem ( : 1 dimension) If a and b are real numbers, then the distance between them on a number line is a b. a b : 2 dimensions Consider
More informationCMSC427 Parametric curves: Hermite, Catmull-Rom, Bezier
CMSC427 Parametric curves: Hermite, Catmull-Rom, Bezier Modeling Creating 3D objects How to construct complicated surfaces? Goal Specify objects with few control points Resulting object should be visually
More informationChapter 1E - Complex Numbers
Fry Texas A&M University Math 150 Spring 2015 Unit 4 20 Chapter 1E - Complex Numbers 16 exists So far the largest (most inclusive) number set we have discussed and the one we have the most experience with
More informationChapter 6. Additional Topics in Trigonometry. 6.6 Vectors. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 6 Additional Topics in Trigonometry 6.6 Vectors Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Obectives: Use magnitude and direction to show vectors are equal. Visualize scalar multiplication,
More informationInterpolation and polynomial approximation Interpolation
Outline Interpolation and polynomial approximation Interpolation Lagrange Cubic Splines Approximation B-Splines 1 Outline Approximation B-Splines We still focus on curves for the moment. 2 3 Pierre Bézier
More informationTHINKING IN PYRAMIDS
ECS 178 Course Notes THINKING IN PYRAMIDS Kenneth I. Joy Institute for Data Anaysis and Visuaization Department of Computer Science University of Caifornia, Davis Overview It is frequenty usefu to think
More informationChapter 9. Conic Sections and Analytic Geometry. 9.2 The Hyperbola. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 9 Conic Sections and Analytic Geometry 9. The Hyperbola Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Locate a hyperbola s vertices and foci. Write equations of hyperbolas in standard
More informationVectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its.
Fry Texas A&M University Fall 2016 Math 150 Notes Chapter 9 Page 248 Chapter 9 -- Vectors Remember that is the set of real numbers, often represented by the number line, 2 is the notation for the 2-dimensional
More informationCMSC427 Geometry and Vectors
CMSC427 Geometry and Vectors Review: where are we? Parametric curves and Hw1? Going beyond the course: generative art https://www.openprocessing.org Brandon Morse, Art Dept, ART370 Polylines, Processing
More informationwhich are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar.
It follows that S is linearly dependent since the equation is satisfied by which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar. is
More informationwhich has a check digit of 9. This is consistent with the first nine digits of the ISBN, since
vector Then the check digit c is computed using the following procedure: 1. Form the dot product. 2. Divide by 11, thereby producing a remainder c that is an integer between 0 and 10, inclusive. The check
More informationArsène Pérard-Gayot (Slides by Piotr Danilewski)
Computer Graphics - Splines - Arsène Pérard-Gayot (Slides by Piotr Danilewski) CURVES Curves Explicit y = f x f: R R γ = x, f x y = 1 x 2 Implicit F x, y = 0 F: R 2 R γ = x, y : F x, y = 0 x 2 + y 2 =
More informationMath (P)Review Part II:
Math (P)Review Part II: Vector Calculus Computer Graphics Assignment 0.5 (Out today!) Same story as last homework; second part on vector calculus. Slightly fewer questions Last Time: Linear Algebra Touched
More informationIntroduction. Chapter Points, Vectors and Coordinate Systems
Chapter 1 Introduction Computer aided geometric design (CAGD) concerns itself with the mathematical description of shape for use in computer graphics, manufacturing, or analysis. It draws upon the fields
More informationBernstein polynomials of degree N are defined by
SEC. 5.5 BÉZIER CURVES 309 5.5 Bézier Curves Pierre Bézier at Renault and Paul de Casteljau at Citroën independently developed the Bézier curve for CAD/CAM operations, in the 1970s. These parametrically
More informationLecture 8: Coordinate Frames. CITS3003 Graphics & Animation
Lecture 8: Coordinate Frames CITS3003 Graphics & Animation E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012 Objectives Learn how to define and change coordinate frames Introduce
More informationCSC 470 Introduction to Computer Graphics. Mathematical Foundations Part 2
CSC 47 Introduction to Computer Graphics Mathematical Foundations Part 2 Vector Magnitude and Unit Vectors The magnitude (length, size) of n-vector w is written w 2 2 2 w = w + w2 + + w n Example: the
More informationSimulation in Computer Graphics Elastic Solids. Matthias Teschner
Simulation in Computer Graphics Elastic Solids Matthias Teschner Outline Introduction Elastic forces Miscellaneous Collision handling Visualization University of Freiburg Computer Science Department 2
More informationConvex Sets with Applications to Economics
Convex Sets with Applications to Economics Debasis Mishra March 10, 2010 1 Convex Sets A set C R n is called convex if for all x, y C, we have λx+(1 λ)y C for all λ [0, 1]. The definition says that for
More informationR.2 Number Line and Interval Notation
8 R.2 Number Line and Interval Notation As mentioned in the previous section, it is convenient to visualise the set of real numbers by identifying each number with a unique point on a number line. Order
More informationThe Vector Space. [3] The Vector Space
The Vector Space [3] The Vector Space Linear Combinations An expression α 1 v 1 + + α n v n is a linear combination of the vectors v 1,..., v n. The scalars α 1,..., α n are the coefficients of the linear
More information22m:033 Notes: 6.1 Inner Product, Length and Orthogonality
m:033 Notes: 6. Inner Product, Length and Orthogonality Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman April, 00 The inner product Arithmetic is based on addition and
More informationDistance in the Plane
Distance in the Plane The absolute value function is defined as { x if x 0; and x = x if x < 0. If the number a is positive or zero, then a = a. If a is negative, then a is the number you d get by erasing
More informationIntroduction to Curves. Modelling. 3D Models. Points. Lines. Polygons Defined by a sequence of lines Defined by a list of ordered points
Introduction to Curves Modelling Points Defined by 2D or 3D coordinates Lines Defined by a set of 2 points Polygons Defined by a sequence of lines Defined by a list of ordered points 3D Models Triangular
More informationRIGID BODY MOTION (Section 16.1)
RIGID BODY MOTION (Section 16.1) There are cases where an object cannot be treated as a particle. In these cases the size or shape of the body must be considered. Rotation of the body about its center
More informationMA 323 Geometric Modelling Course Notes: Day 11 Barycentric Coordinates and de Casteljau s algorithm
MA 323 Geometric Modelling Course Notes: Day 11 Barycentric Coordinates and de Casteljau s algorithm David L. Finn December 16th, 2004 Today, we introduce barycentric coordinates as an alternate to using
More information3 Vectors. 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan
Chapter 3 Vectors 3 Vectors 18 October 2018 PHY101 Physics I Dr.Cem Özdoğan 2 3 3-2 Vectors and Scalars Physics deals with many quantities that have both size and direction. It needs a special mathematical
More informationChapter 5: Equilibrium of a Rigid Body
Chapter 5: Equilibrium of a Rigid Body Develop the equations of equilibrium for a rigid body Concept of the free-body diagram for a rigid body Solve rigid-body equilibrium problems using the equations
More informationChapter 8. Rigid transformations
Chapter 8. Rigid transformations We are about to start drawing figures in 3D. There are no built-in routines for this purpose in PostScript, and we shall have to start more or less from scratch in extending
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationChapter 1. Preliminaries
Chapter 1 Preliminaries 1.1 The Vector Concept Revisited The concept of a vector has been one of the most fruitful ideas in all of mathematics, and it is not surprising that we receive repeated exposure
More informationMatrix Basic Concepts
Matrix Basic Concepts Topics: What is a matrix? Matrix terminology Elements or entries Diagonal entries Address/location of entries Rows and columns Size of a matrix A column matrix; vectors Special types
More informationVectors and the Geometry of Space
Vectors and the Geometry of Space Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized by a single real number scaled to appropriate units of
More informationCreated by T. Madas VECTOR PRACTICE Part B Created by T. Madas
VECTOR PRACTICE Part B THE CROSS PRODUCT Question 1 Find in each of the following cases a) a = 2i + 5j + k and b = 3i j b) a = i + 2j + k and b = 3i j k c) a = 3i j 2k and b = i + 3j + k d) a = 7i + j
More informationComputergrafik. Matthias Zwicker Universität Bern Herbst 2016
Computergrafik Matthias Zwicker Universität Bern Herbst 2016 2 Today Curves Introduction Polynomial curves Bézier curves Drawing Bézier curves Piecewise curves Modeling Creating 3D objects How to construct
More informationThis pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication.
This pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication. Copyright Pearson Canada Inc. All rights reserved. Copyright Pearson
More informationSpring, 2006 CIS 610. Advanced Geometric Methods in Computer Science Jean Gallier Homework 1
Spring, 2006 CIS 610 Advanced Geometric Methods in Computer Science Jean Gallier Homework 1 January 23, 2006; Due February 8, 2006 A problems are for practice only, and should not be turned in. Problem
More informationGeometric approximation of curves and singularities of secant maps Ghosh, Sunayana
University of Groningen Geometric approximation of curves and singularities of secant maps Ghosh, Sunayana IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish
More informationMathematical review trigonometry vectors Motion in one dimension
Mathematical review trigonometry vectors Motion in one dimension Used to describe the position of a point in space Coordinate system (frame) consists of a fixed reference point called the origin specific
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationReadings for Unit I from Ryan (Topics from linear algebra)
Readings for Unit I from Ryan (Topics from linear algebra) I.0 : Background Suggested readings. Ryan : pp. 193 202 General convention. In most cases, the background readings for a section of the course
More informationPLC Papers Created For:
PLC Papers Created For: Daniel Inequalities Inequalities on number lines 1 Grade 4 Objective: Represent the solution of a linear inequality on a number line. Question 1 Draw diagrams to represent these
More information1 Overview. CS348a: Computer Graphics Handout #8 Geometric Modeling Original Handout #8 Stanford University Thursday, 15 October 1992
CS348a: Computer Graphics Handout #8 Geometric Modeling Original Handout #8 Stanford University Thursday, 15 October 1992 Original Lecture #1: 1 October 1992 Topics: Affine vs. Projective Geometries Scribe:
More informationExercises for Unit I (Topics from linear algebra)
Exercises for Unit I (Topics from linear algebra) I.0 : Background Note. There is no corresponding section in the course notes, but as noted at the beginning of Unit I these are a few exercises which involve
More informationThe Cross Product. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) The Cross Product Spring /
The Cross Product Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) The Cross Product Spring 2012 1 / 15 Introduction The cross product is the second multiplication operation between vectors we will
More informationA geometric interpretation of the homogeneous coordinates is given in the following Figure.
Introduction Homogeneous coordinates are an augmented representation of points and lines in R n spaces, embedding them in R n+1, hence using n + 1 parameters. This representation is useful in dealing with
More informationExercises for Unit I (Topics from linear algebra)
Exercises for Unit I (Topics from linear algebra) I.0 : Background This does not correspond to a section in the course notes, but as noted at the beginning of Unit I it contains some exercises which involve
More informationCharacterization of Planar Cubic Alternative curve. Perak, M sia. M sia. Penang, M sia.
Characterization of Planar Cubic Alternative curve. Azhar Ahmad α, R.Gobithasan γ, Jamaluddin Md.Ali β, α Dept. of Mathematics, Sultan Idris University of Education, 59 Tanjung Malim, Perak, M sia. γ Dept
More informationMath 341: Convex Geometry. Xi Chen
Math 341: Convex Geometry Xi Chen 479 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca CHAPTER 1 Basics 1. Euclidean Geometry
More informationMEI STRUCTURED MATHEMATICS CONCEPTS FOR ADVANCED MATHEMATICS, C2. Practice Paper C2-C
MEI Mathematics in Education and Industry MEI STRUCTURED MATHEMATICS CONCEPTS FOR ADVANCED MATHEMATICS, C Practice Paper C-C Additional materials: Answer booklet/paper Graph paper MEI Examination formulae
More informationEstimating Gaussian Mixture Densities with EM A Tutorial
Estimating Gaussian Mixture Densities with EM A Tutorial Carlo Tomasi Due University Expectation Maximization (EM) [4, 3, 6] is a numerical algorithm for the maximization of functions of several variables
More informationSmooth Path Generation Based on Bézier Curves for Autonomous Vehicles
Smooth Path Generation Based on Bézier Curves for Autonomous Vehicles Ji-wung Choi, Renwick E. Curry, Gabriel Hugh Elkaim Abstract In this paper we present two path planning algorithms based on Bézier
More information6.837 LECTURE 8. Lecture 8 Outline Fall '01. Lecture Fall '01
6.837 LECTURE 8 1. 3D Transforms; Part I - Priciples 2. Geometric Data Types 3. Vector Spaces 4. Basis Vectors 5. Linear Transformations 6. Use of Matrix Operators 7. How to Read a Matrix Expression 8.
More informationEnergy Efficiency, Acoustics & Daylighting in building Prof. B. Bhattacharjee Department of Civil Engineering Indian Institute of Technology, Delhi
Energy Efficiency, Acoustics & Daylighting in building Prof. B. Bhattacharjee Department of Civil Engineering Indian Institute of Technology, Delhi Lecture - 05 Introduction & Environmental Factors (contd.)
More informationThree-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems
To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate.
More informationHomogeneous Coordinates
Homogeneous Coordinates Basilio Bona DAUIN-Politecnico di Torino October 2013 Basilio Bona (DAUIN-Politecnico di Torino) Homogeneous Coordinates October 2013 1 / 32 Introduction Homogeneous coordinates
More informationChapter 2 One-Dimensional Kinematics
Review: Chapter 2 One-Dimensional Kinematics Description of motion in one dimension Copyright 2010 Pearson Education, Inc. Review: Motion with Constant Acceleration Free fall: constant acceleration g =
More informationVectors a vector is a quantity that has both a magnitude (size) and a direction
Vectors In physics, a vector is a quantity that has both a magnitude (size) and a direction. Familiar examples of vectors include velocity, force, and electric field. For any applications beyond one dimension,
More informationInsights into the Geometry of the Gaussian Kernel and an Application in Geometric Modeling
Insights into the Geometry of the Gaussian Kernel and an Application in Geometric Modeling Master Thesis Michael Eigensatz Advisor: Joachim Giesen Professor: Mark Pauly Swiss Federal Institute of Technology
More information1 Notes for lectures during the week of the strike Part 2 (10/25)
1 Notes for lectures during the week of the strike Part 2 (10/25) Last time, we stated the beautiful: Theorem 1.1 ( Three reflections theorem ). Any isometry (of R 2 ) is the composition of at most three
More information22m:033 Notes: 1.3 Vector Equations
m:0 Notes: Vector Equations Dennis Roseman University of Iowa Iowa City, IA http://wwwmathuiowaedu/ roseman January 7, 00 Algebra and Geometry We think of geometric things as subsets of the plane or of
More informationPhysics Lab 202P-3. Electric Fields and Superposition: A Virtual Lab NAME: LAB PARTNERS:
Physics Lab 202P-3 Electric Fields and Superposition: A Virtual Lab NAME: LAB PARTNERS: LAB SECTION: LAB INSTRUCTOR: DATE: EMAIL ADDRESS: Penn State University Created by nitin samarth Physics Lab 202P-3
More informationVectors and Matrices
Vectors and Matrices Scalars We often employ a single number to represent quantities that we use in our daily lives such as weight, height etc. The magnitude of this number depends on our age and whether
More informationMath 144 Activity #9 Introduction to Vectors
144 p 1 Math 144 ctiity #9 Introduction to Vectors Often times you hear people use the words speed and elocity. Is there a difference between the two? If so, what is the difference? Discuss this with your
More informationThe geometry of least squares
The geometry of least squares We can think of a vector as a point in space, where the elements of the vector are the coordinates of the point. Consider for example, the following vector s: t = ( 4, 0),
More informationMechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras
Mechanics, Heat, Oscillations and Waves Prof. V. Balakrishnan Department of Physics Indian Institute of Technology, Madras Lecture 08 Vectors in a Plane, Scalars & Pseudoscalers Let us continue today with
More informationVectors Coordinate frames 2D implicit curves 2D parametric curves. Graphics 2008/2009, period 1. Lecture 2: vectors, curves, and surfaces
Graphics 2008/2009, period 1 Lecture 2 Vectors, curves, and surfaces Computer graphics example: Pixar (source: http://www.pixar.com) Computer graphics example: Pixar (source: http://www.pixar.com) Computer
More informationBrief Review of Vector Algebra
APPENDIX Brief Review of Vector Algebra A.0 Introduction Vector algebra is used extensively in computational mechanics. The student must thus understand the concepts associated with this subject. The current
More informationLecture 2: Vector-Vector Operations
Lecture 2: Vector-Vector Operations Vector-Vector Operations Addition of two vectors Geometric representation of addition and subtraction of vectors Vectors and points Dot product of two vectors Geometric
More informationRules of Differentiation
Rules of Differentiation The process of finding the derivative of a function is called Differentiation. 1 In the previous chapter, the required derivative of a function is worked out by taking the limit
More informationSection 3.1. Best Affine Approximations. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section 3.1 Best Affine Approximations We are now in a position to discuss the two central problems of calculus as mentioned in Section 1.1. In this chapter
More informationCHAPTER II AFFINE GEOMETRY
CHAPTER II AFFINE GEOMETRY In the previous chapter we indicated how several basic ideas from geometry have natural interpretations in terms of vector spaces and linear algebra. This chapter continues the
More informationVector Geometry. Chapter 5
Chapter 5 Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at
More informationLab: Vectors. You are required to finish this section before coming to the lab. It will be checked by one of the lab instructors when the lab begins.
Lab: Vectors Lab Section (circle): Day: Monday Tuesday Time: 8:00 9:30 1:10 2:40 Name Partners Pre-Lab You are required to finish this section before coming to the lab. It will be checked by one of the
More information11.1 Vectors in the plane
11.1 Vectors in the plane What is a vector? It is an object having direction and length. Geometric way to represent vectors It is represented by an arrow. The direction of the arrow is the direction of
More informationLIMITS AND DERIVATIVES
2 LIMITS AND DERIVATIVES LIMITS AND DERIVATIVES The intuitive definition of a limit given in Section 2.2 is inadequate for some purposes.! This is because such phrases as x is close to 2 and f(x) gets
More information4.1 Distance and Length
Chapter Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at vectors
More informationVectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its.
Fry Texas A&M University Math 150 Chapter 9 Fall 2014 1 Chapter 9 -- Vectors Remember that is the set of real numbers, often represented by the number line, 2 is the notation for the 2-dimensional plane.
More informationGEOMETRY AND VECTORS
GEOMETRY AND VECTORS Distinguishing Between Points in Space One Approach Names: ( Fred, Steve, Alice...) Problem: distance & direction must be defined point-by-point More elegant take advantage of geometry
More informationFamilies of point-touching squares k-touching
Geombinatorics 12(2003), 167 174 Families of point-touching squares Branko Grünbaum Department of Mathematics, Box 354350 University of Washington Seattle, WA 98195-4350 e-mail: grunbaum@math.washington.edu
More informationConsider the equation different values of x we shall find the values of y and the tabulate t the values in the following table
Consider the equation y = 2 x + 3 for different values of x we shall find the values of y and the tabulate t the values in the following table x 0 1 2 1 2 y 3 5 7 1 1 When the points are plotted on the
More information