Rigid Body Transformations

Size: px
Start display at page:

Download "Rigid Body Transformations"

Transcription

1 igid od Kinemtics igid od Trnsformtions Vij Kumr

2 igid od Kinemtics emrk out Nottion Vectors,,, u, v, p, q, Potentil for Confusion! Mtrices,, C, g, h,

3 igid od Kinemtics The vector nd its skew smmetric mtri counterprt For n vector +

4 igid od Displcement igid od Kinemtics igid od Trnsformtion g : O t O igid od Motion g () t : O 4

5 igid od Kinemtics Coordinte Trnsformtions nd Displcements Trnsformtions of points Trnsformtion (g) of points induces n ction (g * ) on vectors p v q g (p) g * (v) g (q) Wht re rigid od trnsformtions? Displcements? g preserves lengths g * preserves cross products 5

6 igid od Kinemtics igid od Trnsformtions in Cn show tht the most generl coordinte trnsformtion from {} to {} hs the following form {} ' r P P ' O r P r O O' ' position vector of P in {} is trnsformed to position vector of P in {} description of {} s seen from n oserver in {} P P O r r + r ottion of {} with respect to {} Trnsltion of the origin of {} with respect to origin of {} 6

7 igid od Kinemtics ottionl trnsformtions in Properties of rottion mtrices Trnspose is the inverse Determinnt is + ottions preserve cross products u v (u v) ottion of skew smmetric mtrices For n rottion mtri : w T ( w) 7

8 igid od Kinemtics Emple: ottion ottion out the -is through θ ot (, θ) cosθ sin θ sin θ cosθ θ Displcement 8

9 igid od Kinemtics Emple: ottion ottion out the -is through θ ottion out the -is through θ ot (, θ) cosθ sin θ sin θ cosθ ot (, θ) cosθ sin θ sin θ cosθ ' θ ' ' θ 9 '

10 ' igid od Kinemtics igid Motion in ' {} {} r P P ' {} r P r P P ' O r P r O ' O' P P O r r + r Coordinte trnsformtion from {} to {} r P P r O O ' O' P P O r r + r Displcement of od-fied frme from {} to {} The sme eqution cn hve two interprettions: It trnsforms the position vector of n point in {} to the position vector in {} It trnsforms the position vector of n point in the first position/orienttion (descried {}) to its new position vector in the second position orienttion (descried {}).

11 Moile oots igid od Kinemtics θ W W g W cosθ sin θ sin θ cosθ

12 igid od Kinemtics The Lie group SE() () I r r T T SE,,,

13 igid od Kinemtics SE() is Lie group SE() stisfies the four ioms tht must e stisfied the elements of n lgeric group: The set is closed under the inr opertion. In other words, if nd re n two mtrices in SE(), SE(). The inr opertion is ssocitive. In other words, if,, nd C re n three mtrices SE(), then () C (C). For ever element SE(), there is n identit element given the 4 4 identit mtri, I SE(), such tht I. For ever element SE(), there is n identit inverse, - SE(), such tht - I. SE() is continuous group. the inr opertion ove is continuous opertion the product of n two elements in SE() is continuous function of the two elements the inverse of n element in SE() is continuous function of tht element. In other words, SE() is differentile mnifold. group tht is differentile mnifold is clled Lie group[ Sophus Lie (84-899)].

14 Displcement from {} to {} Displcement from {} to {C} Displcement from {} to {C} C C igid od Kinemtics Composition of Displcements C C r r r r O' O O O,, O O C r + C r POSITION r O {} O ' ' {} O' '' POSITION POSITION O'' ' {C} Note X Y descries the displcement of the od-fied frme from {X} to {Y} in reference frme {X} '' '' 4

15 igid od Kinemtics Composition (continued) Composition of displcements Displcements re generll descried in od-fied frme Emple: C is the displcement of rigid od from to C reltive to the es of the first frme. Composition of trnsformtions Sme sic ide POSITION {} O ' ' {} O' POSITION O'' ' {C} '' C C Note tht our description of trnsformtions (e.g., C ) is reltive to the first frme (, the frme with the leding superscript). '' POSITION Note X Y descries the displcement of the od-fied frme from {X} to {Y} in reference frme {X} '' 5

16 igid od Kinemtics Sugroup Nottion Definition Significnce The group of rottions in three dimensions SO() SO The set of ll proper orthogonl mtrices. T T ( ) {, I} ll sphericl displcements. Or the set of ll displcements tht cn e generted sphericl joint (S-pir). Sugroups of SE() Specil Eucliden group in two dimensions The group of rottions in two dimensions SE() SO() The set of ll mtrices with the structure: cosθ sinθ r sinθ cosθ r where θ, r, nd r re rel numers. The set of ll proper orthogonl mtrices. The hve the structure: cosθ sinθ sinθ cosθ, ll plnr displcements. Or the set of displcements tht cn e generted plnr pir (E-pir). ll rottions in the plne, or the set of ll displcements tht cn e generted single revolute joint (-pir). where θ is rel numer. The group of trnsltions in n dimensions. The group of trnsltions in one dimension. T(n) T() The group of SO() T() clindricl displcements The group of screw displcements The set of ll n rel vectors with vector ddition s the inr opertion. The set of ll rel numers with ddition s the inr opertion. The Crtesin product of SO() nd T() ll trnsltions in n dimensions. n indictes plnr, n indictes sptil displcements. ll trnsltions prllel to one is, or the set of ll displcements tht cn e generted single prismtic joint (P-pir). ll rottions in the plne nd trnsltions long n is perpendiculr to the plne, or the set of ll displcements tht cn e generted clindricl joint (C-pir). H() one-prmeter sugroup of SE() ll displcements tht cn e generted helicl joint (H-pir). 6

MA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork

MA1008. Calculus and Linear Algebra for Engineers. Course Notes for Section B. Stephen Wills. Department of Mathematics. University College Cork MA1008 Clculus nd Liner Algebr for Engineers Course Notes for Section B Stephen Wills Deprtment of Mthemtics University College Cork s.wills@ucc.ie http://euclid.ucc.ie/pges/stff/wills/teching/m1008/ma1008.html

More information

AML710 CAD LECTURE 16 SURFACES. 1. Analytical Surfaces 2. Synthetic Surfaces

AML710 CAD LECTURE 16 SURFACES. 1. Analytical Surfaces 2. Synthetic Surfaces AML7 CAD LECTURE 6 SURFACES. Anlticl Surfces. Snthetic Surfces Surfce Representtion From CAD/CAM point of view surfces re s importnt s curves nd solids. We need to hve n ide of curves for surfce cretion.

More information

Topic 3: 2D Transformations 9/10/2016. Today s Topics. Transformations. Lets start out simple. Points as Homogeneous 2D Point Coords

Topic 3: 2D Transformations 9/10/2016. Today s Topics. Transformations. Lets start out simple. Points as Homogeneous 2D Point Coords Tody s Topics 3. Trnsformtions in 2D 4. Coordinte-free geometry 5. (curves & surfces) Topic 3: 2D Trnsformtions 6. Trnsformtions in 3D Simple Trnsformtions Homogeneous coordintes Homogeneous 2D trnsformtions

More information

Topics in Analytic Geometry

Topics in Analytic Geometry Nme Chpter 10 Topics in Anltic Geometr Section 10.1 Lines Objective: In this lesson ou lerned how to find the inclintion of line, the ngle between two lines, nd the distnce between point nd line. Importnt

More information

Lecture 5: Spatial Analysis Algorithms

Lecture 5: Spatial Analysis Algorithms Lecture 5: Sptil Algorithms GEOG 49: Advnced GIS Sptil Anlsis Algorithms Bsis of much of GIS nlsis tod Mnipultion of mp coordintes Bsed on Eucliden coordinte geometr http://stronom.swin.edu.u/~pbourke/geometr/

More information

Lecture 4 Single View Metrology

Lecture 4 Single View Metrology Lecture 4 Single View Metrology Professor Silvio Svrese Computtionl Vision nd Geometry Lb Silvio Svrese Lecture 4-4-Jn-5 Lecture 4 Single View Metrology Review clibrtion nd 2D trnsformtions Vnishing points

More information

CS380: Computer Graphics Modeling Transformations. Sung-Eui Yoon ( 윤성의 ) Course URL:

CS380: Computer Graphics Modeling Transformations. Sung-Eui Yoon ( 윤성의 ) Course URL: CS38: Computer Grphics Modeling Trnsformtions Sung-Eui Yoon ( 윤성의 ) Course URL: http://sgl.kist.c.kr/~sungeui/cg/ Clss Ojectives (Ch. 3.5) Know the clssic dt processing steps, rendering pipeline, for rendering

More information

Viewing and Projection

Viewing and Projection 15-462 Computer Grphics I Lecture 5 Viewing nd Projection Sher Trnsformtion Cmer Positioning Simple Prllel Projections Simple Perspective Projections [Angel, Ch. 5.2-5.4] Jnury 30, 2003 [Red s Drem, Pixr,

More information

4-1 NAME DATE PERIOD. Study Guide. Parallel Lines and Planes P Q, O Q. Sample answers: A J, A F, and D E

4-1 NAME DATE PERIOD. Study Guide. Parallel Lines and Planes P Q, O Q. Sample answers: A J, A F, and D E 4-1 NAME DATE PERIOD Pges 142 147 Prllel Lines nd Plnes When plnes do not intersect, they re sid to e prllel. Also, when lines in the sme plne do not intersect, they re prllel. But when lines re not in

More information

Matrices and Systems of Equations

Matrices and Systems of Equations Mtrices Mtrices nd Sstems of Equtions A mtri is rectngulr rr of rel numbers. CHAT Pre-Clculus Section 8. m m m............ n n n mn We will use the double subscript nottion for ech element of the mtri.

More information

Geometric transformations

Geometric transformations Geometric trnsformtions Computer Grphics Some slides re bsed on Shy Shlom slides from TAU mn n n m m T A,,,,,, 2 1 2 22 12 1 21 11 Rows become columns nd columns become rows nm n n m m A,,,,,, 1 1 2 22

More information

Introduction Transformation formulae Polar graphs Standard curves Polar equations Test GRAPHS INU0114/514 (MATHS 1)

Introduction Transformation formulae Polar graphs Standard curves Polar equations Test GRAPHS INU0114/514 (MATHS 1) POLAR EQUATIONS AND GRAPHS GEOMETRY INU4/54 (MATHS ) Dr Adrin Jnnett MIMA CMth FRAS Polr equtions nd grphs / 6 Adrin Jnnett Objectives The purpose of this presenttion is to cover the following topics:

More information

8.2 Areas in the Plane

8.2 Areas in the Plane 39 Chpter 8 Applictions of Definite Integrls 8. Ares in the Plne Wht ou will lern out... Are Between Curves Are Enclosed Intersecting Curves Boundries with Chnging Functions Integrting with Respect to

More information

5/9/17. Lesson 51 - FTC PART 2. Review FTC, PART 1. statement as the Integral Evaluation Theorem as it tells us HOW to evaluate the definite integral

5/9/17. Lesson 51 - FTC PART 2. Review FTC, PART 1. statement as the Integral Evaluation Theorem as it tells us HOW to evaluate the definite integral Lesson - FTC PART 2 Review! We hve seen definition/formul for definite integrl s n b A() = lim f ( i )Δ = f ()d = F() = F(b) F() n i=! where F () = f() (or F() is the ntiderivtive of f() b! And hve seen

More information

Math 17 - Review. Review for Chapter 12

Math 17 - Review. Review for Chapter 12 Mth 17 - eview Ying Wu eview for hpter 12 1. Given prmetric plnr curve x = f(t), y = g(t), where t b, how to eliminte the prmeter? (Use substitutions, or use trigonometry identities, etc). How to prmeterize

More information

Ray surface intersections

Ray surface intersections Ry surfce intersections Some primitives Finite primitives: polygons spheres, cylinders, cones prts of generl qudrics Infinite primitives: plnes infinite cylinders nd cones generl qudrics A finite primitive

More information

B. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a

B. Definition: The volume of a solid of known integrable cross-section area A(x) from x = a Mth 176 Clculus Sec. 6.: Volume I. Volume By Slicing A. Introduction We will e trying to find the volume of solid shped using the sum of cross section res times width. We will e driving towrd developing

More information

Stained Glass Design. Teaching Goals:

Stained Glass Design. Teaching Goals: Stined Glss Design Time required 45-90 minutes Teching Gols: 1. Students pply grphic methods to design vrious shpes on the plne.. Students pply geometric trnsformtions of grphs of functions in order to

More information

2D Projective transformation

2D Projective transformation 2D Projective trnsformtion The mpping of points from n N-D spce to n M-D subspce (M < N) w' w' w m m m 2 m m m 2 m m m 2 2 22 w' w' w m m m 2 m m m 2 m m 2 2 Boqun Chen 22 2D Projective trnsformtion w'

More information

15. 3D-Reconstruction from Vanishing Points

15. 3D-Reconstruction from Vanishing Points 15. 3D-Reconstruction from Vnishing Points Christin B.U. Perwss 1 nd Jon Lseny 2 1 Cvendish Lortory, Cmridge 2 C. U. Engineering Deprtment, Cmridge 15.1 Introduction 3D-reconstruction is currently n ctive

More information

Date: 9.1. Conics: Parabolas

Date: 9.1. Conics: Parabolas Dte: 9. Conics: Prols Preclculus H. Notes: Unit 9 Conics Conic Sections: curves tht re formed y the intersection of plne nd doulenpped cone Syllus Ojectives:. The student will grph reltions or functions,

More information

Basic Geometry and Topology

Basic Geometry and Topology Bsic Geometry nd Topology Stephn Stolz Septemer 7, 2015 Contents 1 Pointset Topology 1 1.1 Metric spces................................... 1 1.2 Topologicl spces................................ 5 1.3 Constructions

More information

Algebra II Notes Unit Ten: Conic Sections

Algebra II Notes Unit Ten: Conic Sections Sllus Ojective: 0. The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting the

More information

6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it.

6.3 Volumes. Just as area is always positive, so is volume and our attitudes towards finding it. 6.3 Volumes Just s re is lwys positive, so is volume nd our ttitudes towrds finding it. Let s review how to find the volume of regulr geometric prism, tht is, 3-dimensionl oject with two regulr fces seprted

More information

1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES)

1. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) Numbers nd Opertions, Algebr, nd Functions 45. SEQUENCES INVOLVING EXPONENTIAL GROWTH (GEOMETRIC SEQUENCES) In sequence of terms involving eponentil growth, which the testing service lso clls geometric

More information

Objective: Students will understand what it means to describe, graph and write the equation of a parabola. Parabolas

Objective: Students will understand what it means to describe, graph and write the equation of a parabola. Parabolas Pge 1 of 8 Ojective: Students will understnd wht it mens to descrie, grph nd write the eqution of prol. Prols Prol: collection of ll points P in plne tht re the sme distnce from fixed point, the focus

More information

A dual of the rectangle-segmentation problem for binary matrices

A dual of the rectangle-segmentation problem for binary matrices A dul of the rectngle-segmenttion prolem for inry mtrices Thoms Klinowski Astrct We consider the prolem to decompose inry mtrix into smll numer of inry mtrices whose -entries form rectngle. We show tht

More information

50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula:

50 AMC LECTURES Lecture 2 Analytic Geometry Distance and Lines. can be calculated by the following formula: 5 AMC LECTURES Lecture Anlytic Geometry Distnce nd Lines BASIC KNOWLEDGE. Distnce formul The distnce (d) between two points P ( x, y) nd P ( x, y) cn be clculted by the following formul: d ( x y () x )

More information

Class-XI Mathematics Conic Sections Chapter-11 Chapter Notes Key Concepts

Class-XI Mathematics Conic Sections Chapter-11 Chapter Notes Key Concepts Clss-XI Mthemtics Conic Sections Chpter-11 Chpter Notes Key Concepts 1. Let be fixed verticl line nd m be nother line intersecting it t fixed point V nd inclined to it t nd ngle On rotting the line m round

More information

EXPONENTIAL & POWER GRAPHS

EXPONENTIAL & POWER GRAPHS Eponentil & Power Grphs EXPONENTIAL & POWER GRAPHS www.mthletics.com.u Eponentil EXPONENTIAL & Power & Grphs POWER GRAPHS These re grphs which result from equtions tht re not liner or qudrtic. The eponentil

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

1.1. Interval Notation and Set Notation Essential Question When is it convenient to use set-builder notation to represent a set of numbers?

1.1. Interval Notation and Set Notation Essential Question When is it convenient to use set-builder notation to represent a set of numbers? 1.1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS Prepring for 2A.6.K, 2A.7.I Intervl Nottion nd Set Nottion Essentil Question When is it convenient to use set-uilder nottion to represent set of numers? A collection

More information

Modeling and Simulation of Short Range 3D Triangulation-Based Laser Scanning System

Modeling and Simulation of Short Range 3D Triangulation-Based Laser Scanning System Modeling nd Simultion of Short Rnge 3D Tringultion-Bsed Lser Scnning System Theodor Borngiu Anmri Dogr Alexndru Dumitrche April 14, 2008 Abstrct In this pper, simultion environment for short rnge 3D lser

More information

It is recommended to change the limits of integration while doing a substitution.

It is recommended to change the limits of integration while doing a substitution. MAT 21 eptember 7, 216 Review Indrjit Jn. Generl Tips It is recommended to chnge the limits of integrtion while doing substitution. First write the min formul (eg. centroid, moment of inerti, mss, work

More information

Orientation & Quaternions. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2017

Orientation & Quaternions. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2017 Orienttion & Quternions CSE69: Computer Animtion Instructor: Steve Rotenberg UCSD, Winter 7 Orienttion Orienttion We will define orienttion to men n object s instntneous rottionl configurtion Think of

More information

called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola.

called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola. Review of conic sections Conic sections re grphs of the form REVIEW OF CONIC SECTIONS prols ellipses hperols P(, ) F(, p) O p =_p REVIEW OF CONIC SECTIONS In this section we give geometric definitions

More information

a(e, x) = x. Diagrammatically, this is encoded as the following commutative diagrams / X

a(e, x) = x. Diagrammatically, this is encoded as the following commutative diagrams / X 4. Mon, Sept. 30 Lst time, we defined the quotient topology coming from continuous surjection q : X! Y. Recll tht q is quotient mp (nd Y hs the quotient topology) if V Y is open precisely when q (V ) X

More information

Graphing Conic Sections

Graphing Conic Sections Grphing Conic Sections Definition of Circle Set of ll points in plne tht re n equl distnce, clled the rdius, from fixed point in tht plne, clled the center. Grphing Circle (x h) 2 + (y k) 2 = r 2 where

More information

Math 35 Review Sheet, Spring 2014

Math 35 Review Sheet, Spring 2014 Mth 35 Review heet, pring 2014 For the finl exm, do ny 12 of the 15 questions in 3 hours. They re worth 8 points ech, mking 96, with 4 more points for netness! Put ll your work nd nswers in the provided

More information

1.1 Lines AP Calculus

1.1 Lines AP Calculus . Lines AP Clculus. LINES Notecrds from Section.: Rules for Rounding Round or Truncte ll finl nswers to 3 deciml plces. Do NOT round before ou rech our finl nswer. Much of Clculus focuses on the concept

More information

INTRODUCTION TO SIMPLICIAL COMPLEXES

INTRODUCTION TO SIMPLICIAL COMPLEXES INTRODUCTION TO SIMPLICIAL COMPLEXES CASEY KELLEHER AND ALESSANDRA PANTANO 0.1. Introduction. In this ctivity set we re going to introduce notion from Algebric Topology clled simplicil homology. The min

More information

Tilt-Sensing with Kionix MEMS Accelerometers

Tilt-Sensing with Kionix MEMS Accelerometers Tilt-Sensing with Kionix MEMS Accelerometers Introduction Tilt/Inclintion sensing is common ppliction for low-g ccelerometers. This ppliction note describes how to use Kionix MEMS low-g ccelerometers to

More information

Constrained Optimization. February 29

Constrained Optimization. February 29 Constrined Optimiztion Februry 9 Generl Problem min f( ) ( NLP) s.. t g ( ) i E i g ( ) i I i Modeling nd Constrints Adding constrints let s us model fr more richer set of problems. For our purpose we

More information

APPLICATIONS OF INTEGRATION

APPLICATIONS OF INTEGRATION Chpter 3 DACS 1 Lok 004/05 CHAPTER 5 APPLICATIONS OF INTEGRATION 5.1 Geometricl Interprettion-Definite Integrl (pge 36) 5. Are of Region (pge 369) 5..1 Are of Region Under Grph (pge 369) Figure 5.7 shows

More information

Tree Structured Symmetrical Systems of Linear Equations and their Graphical Solution

Tree Structured Symmetrical Systems of Linear Equations and their Graphical Solution Proceedings of the World Congress on Engineering nd Computer Science 4 Vol I WCECS 4, -4 October, 4, Sn Frncisco, USA Tree Structured Symmetricl Systems of Liner Equtions nd their Grphicl Solution Jime

More information

CPSC (T1) 2nd Midterm Exam

CPSC (T1) 2nd Midterm Exam Signture: Fire Alrm Code: CPSC 44 2-2 (T) 2nd Midterm Exm Deprtment of Computer Science University of British Columbi K. Booth & R. Schrein Exm Instructions (Red Crefully). Sign the first pge of the exm

More information

The notation y = f(x) gives a way to denote specific values of a function. The value of f at a can be written as f( a ), read f of a.

The notation y = f(x) gives a way to denote specific values of a function. The value of f at a can be written as f( a ), read f of a. Chpter Prerequisites for Clculus. Functions nd Grphs Wht ou will lern out... Functions Domins nd Rnges Viewing nd Interpreting Grphs Even Functions nd Odd Functions Smmetr Functions Defined in Pieces Asolute

More information

GENERATING ORTHOIMAGES FOR CLOSE-RANGE OBJECTS BY AUTOMATICALLY DETECTING BREAKLINES

GENERATING ORTHOIMAGES FOR CLOSE-RANGE OBJECTS BY AUTOMATICALLY DETECTING BREAKLINES GENEATING OTHOIMAGES FO CLOSE-ANGE OBJECTS BY AUTOMATICALLY DETECTING BEAKLINES Efstrtios Stylinidis 1, Lzros Sechidis 1, Petros Ptis 1, Spiros Sptls 2 Aristotle University of Thessloniki 1 Deprtment of

More information

Fig.25: the Role of LEX

Fig.25: the Role of LEX The Lnguge for Specifying Lexicl Anlyzer We shll now study how to uild lexicl nlyzer from specifiction of tokens in the form of list of regulr expressions The discussion centers round the design of n existing

More information

Integration. September 28, 2017

Integration. September 28, 2017 Integrtion September 8, 7 Introduction We hve lerned in previous chpter on how to do the differentition. It is conventionl in mthemtics tht we re supposed to lern bout the integrtion s well. As you my

More information

Spatial Transformations. Prof. George Wolberg Dept. of Computer Science City College of New York

Spatial Transformations. Prof. George Wolberg Dept. of Computer Science City College of New York Sptil Trnsformtions Prof. George Wolberg Dept. of Compter Science City College of New York Objecties In this lectre we reiew sptil trnsformtions: - Forwrd nd inerse mppings - Trnsformtions Liner Affine

More information

ON THE DEHN COMPLEX OF VIRTUAL LINKS

ON THE DEHN COMPLEX OF VIRTUAL LINKS ON THE DEHN COMPLEX OF VIRTUAL LINKS RACHEL BYRD, JENS HARLANDER Astrct. A virtul link comes with vriety of link complements. This rticle is concerned with the Dehn spce, pseudo mnifold with oundry, nd

More information

Determining the spatial orientation of remote sensing sensors on the basis of incomplete coordinate systems

Determining the spatial orientation of remote sensing sensors on the basis of incomplete coordinate systems Scientific Journls of the Mritime University of Szczecin Zeszyty Nukowe Akdemii Morskiej w Szczecinie 216, 45 (117), 29 ISSN 17-867 (Printed) Received: 1.8.215 ISSN 292-78 (Online) Accepted: 18.2.216 DOI:

More information

Surfaces. Differential Geometry Lia Vas

Surfaces. Differential Geometry Lia Vas Differentil Geometry Li Vs Surfces When studying curves, we studied how the curve twisted nd turned in spce. We now turn to surfces, two-dimensionl objects in three-dimensionl spce nd exmine how the concept

More information

International Conference on Mechanics, Materials and Structural Engineering (ICMMSE 2016)

International Conference on Mechanics, Materials and Structural Engineering (ICMMSE 2016) \ Interntionl Conference on Mechnics, Mterils nd tructurl Engineering (ICMME 2016) Reserch on the Method to Clibrte tructure Prmeters of Line tructured Light Vision ensor Mingng Niu1,, Kngnin Zho1, b,

More information

Essential Question What are some of the characteristics of the graph of a rational function?

Essential Question What are some of the characteristics of the graph of a rational function? 8. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A..A A..G A..H A..K Grphing Rtionl Functions Essentil Question Wht re some of the chrcteristics of the grph of rtionl function? The prent function for rtionl functions

More information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Physics 208: Electricity and Magnetism Exam 1, Secs Feb IMPORTANT. Read these directions carefully:

Physics 208: Electricity and Magnetism Exam 1, Secs Feb IMPORTANT. Read these directions carefully: Physics 208: Electricity nd Mgnetism Exm 1, Secs. 506 510 11 Feb. 2004 Instructor: Dr. George R. Welch, 415 Engineering-Physics, 845-7737 Print your nme netly: Lst nme: First nme: Sign your nme: Plese

More information

Name Date Class. cot. tan. cos. 1 cot 2 csc 2

Name Date Class. cot. tan. cos. 1 cot 2 csc 2 Fundmentl Trigonometric Identities To prove trigonometric identit, use the fundmentl identities to mke one side of the eqution resemle the other side. Reciprocl nd Rtio Identities csc sec sin cos Negtive-Angle

More information

CS311H: Discrete Mathematics. Graph Theory IV. A Non-planar Graph. Regions of a Planar Graph. Euler s Formula. Instructor: Işıl Dillig

CS311H: Discrete Mathematics. Graph Theory IV. A Non-planar Graph. Regions of a Planar Graph. Euler s Formula. Instructor: Işıl Dillig CS311H: Discrete Mthemtics Grph Theory IV Instructor: Işıl Dillig Instructor: Işıl Dillig, CS311H: Discrete Mthemtics Grph Theory IV 1/25 A Non-plnr Grph Regions of Plnr Grph The plnr representtion of

More information

A TRIANGULAR FINITE ELEMENT FOR PLANE ELASTICITY WITH IN- PLANE ROTATION Dr. Attia Mousa 1 and Eng. Salah M. Tayeh 2

A TRIANGULAR FINITE ELEMENT FOR PLANE ELASTICITY WITH IN- PLANE ROTATION Dr. Attia Mousa 1 and Eng. Salah M. Tayeh 2 A TRIANGLAR FINITE ELEMENT FOR PLANE ELASTICITY WITH IN- PLANE ROTATION Dr. Atti Mous nd Eng. Slh M. Teh ABSTRACT In the present pper the strin-bsed pproch is pplied to develop new tringulr finite element

More information

Solution of Linear Algebraic Equations using the Gauss-Jordan Method

Solution of Linear Algebraic Equations using the Gauss-Jordan Method Solution of Liner Algebric Equtions using the Guss-Jordn Method Populr pproch for solving liner equtions The Guss Jordn method depends on two properties of liner equtions: Scling one or more of ny of the

More information

arxiv: v1 [math.mg] 27 Jan 2008

arxiv: v1 [math.mg] 27 Jan 2008 Geometric Properties of ssur Grphs rxiv:0801.4113v1 [mth.mg] 27 Jn 2008 rigitte Servtius Offer Shi Wlter Whiteley June 18, 2018 bstrct In our previous pper, we presented the combintoril theory for miniml

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes by disks: volume prt ii 6 6 Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem 6) nd the ccumultion process is to determine so-clled volumes

More information

Fundamentals of Engineering Analysis ENGR Matrix Multiplication, Types

Fundamentals of Engineering Analysis ENGR Matrix Multiplication, Types Fundmentls of Engineering Anlysis ENGR - Mtri Multiplition, Types Spring Slide Mtri Multiplition Define Conformle To multiply A * B, the mtries must e onformle. Given mtries: A m n nd B n p The numer of

More information

Cameras. Importance of camera models

Cameras. Importance of camera models pture imges mesuring devie Digitl mers mers fill in memor ith olor-smple informtion D hrge-oupled Devie insted of film film lso hs finite resolution grininess depends on speed IS 00 00 6400 sie 35mm IMAX

More information

MATH 2530: WORKSHEET 7. x 2 y dz dy dx =

MATH 2530: WORKSHEET 7. x 2 y dz dy dx = MATH 253: WORKSHT 7 () Wrm-up: () Review: polr coordintes, integrls involving polr coordintes, triple Riemnn sums, triple integrls, the pplictions of triple integrls (especilly to volume), nd cylindricl

More information

Integration. October 25, 2016

Integration. October 25, 2016 Integrtion October 5, 6 Introduction We hve lerned in previous chpter on how to do the differentition. It is conventionl in mthemtics tht we re supposed to lern bout the integrtion s well. As you my hve

More information

Lecture 7: Integration Techniques

Lecture 7: Integration Techniques Lecture 7: Integrtion Techniques Antiderivtives nd Indefinite Integrls. In differentil clculus, we were interested in the derivtive of given rel-vlued function, whether it ws lgeric, eponentil or logrithmic.

More information

Fig.1. Let a source of monochromatic light be incident on a slit of finite width a, as shown in Fig. 1.

Fig.1. Let a source of monochromatic light be incident on a slit of finite width a, as shown in Fig. 1. Answer on Question #5692, Physics, Optics Stte slient fetures of single slit Frunhofer diffrction pttern. The slit is verticl nd illuminted by point source. Also, obtin n expression for intensity distribution

More information

Orthogonal line segment intersection

Orthogonal line segment intersection Computtionl Geometry [csci 3250] Line segment intersection The prolem (wht) Computtionl Geometry [csci 3250] Orthogonl line segment intersection Applictions (why) Algorithms (how) A specil cse: Orthogonl

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY Joe McBride/Stone/Gett Imges Air resistnce prevents the velocit of skdiver from incresing indefinitel. The velocit pproches it, clled the terminl velocit. The development of clculus

More information

MTH 146 Conics Supplement

MTH 146 Conics Supplement 105- Review of Conics MTH 146 Conics Supplement In this section we review conics If ou ne more detils thn re present in the notes, r through section 105 of the ook Definition: A prol is the set of points

More information

Area & Volume. Chapter 6.1 & 6.2 September 25, y = 1! x 2. Back to Area:

Area & Volume. Chapter 6.1 & 6.2 September 25, y = 1! x 2. Back to Area: Bck to Are: Are & Volume Chpter 6. & 6. Septemer 5, 6 We cn clculte the re etween the x-xis nd continuous function f on the intervl [,] using the definite integrl:! f x = lim$ f x * i )%x n i= Where fx

More information

Ray Casting II. Courtesy of James Arvo and David Kirk. Used with permission.

Ray Casting II. Courtesy of James Arvo and David Kirk. Used with permission. y Csting II Courtesy of Jmes Arvo nd Dvid Kirk. Used with permission. MIT EECS 6.837 Frédo Durnd nd Brb Cutler Some slides courtesy of Leonrd McMilln MIT EECS 6.837, Cutler nd Durnd 1 eview of y Csting

More information

The Basic Properties of the Integral

The Basic Properties of the Integral The Bsic Properties of the Integrl When we compute the derivtive of complicted function, like + sin, we usull use differentition rules, like d [f()+g()] d f()+ d g(), to reduce the computtion d d d to

More information

Solutions to Math 41 Final Exam December 12, 2011

Solutions to Math 41 Final Exam December 12, 2011 Solutions to Mth Finl Em December,. ( points) Find ech of the following its, with justifiction. If there is n infinite it, then eplin whether it is or. ( ) / ln() () (5 points) First we compute the it:

More information

this grammar generates the following language: Because this symbol will also be used in a later step, it receives the

this grammar generates the following language: Because this symbol will also be used in a later step, it receives the LR() nlysis Drwcks of LR(). Look-hed symols s eplined efore, concerning LR(), it is possile to consult the net set to determine, in the reduction sttes, for which symols it would e possile to perform reductions.

More information

ASTs, Regex, Parsing, and Pretty Printing

ASTs, Regex, Parsing, and Pretty Printing ASTs, Regex, Prsing, nd Pretty Printing CS 2112 Fll 2016 1 Algeric Expressions To strt, consider integer rithmetic. Suppose we hve the following 1. The lphet we will use is the digits {0, 1, 2, 3, 4, 5,

More information

Order these angles from smallest to largest by wri ng 1 to 4 under each one. Put a check next to the right angle.

Order these angles from smallest to largest by wri ng 1 to 4 under each one. Put a check next to the right angle. Lines nd ngles Connect ech set of lines to the correct nme: prllel perpendiculr Order these ngles from smllest to lrgest y wri ng to 4 under ech one. Put check next to the right ngle. Complete this tle

More information

How to Design REST API? Written Date : March 23, 2015

How to Design REST API? Written Date : March 23, 2015 Visul Prdigm How Design REST API? Turil How Design REST API? Written Dte : Mrch 23, 2015 REpresenttionl Stte Trnsfer, n rchitecturl style tht cn be used in building networked pplictions, is becoming incresingly

More information

Math 4 Review for Quarter 2 Cumulative Test

Math 4 Review for Quarter 2 Cumulative Test Mth 4 Review for Qurter 2 Cumultive Test Nme: I. Right Tringle Trigonometry (3.1-3.3) Key Fcts Pythgoren Theorem - In right tringle, 2 + b 2 = c 2 where c is the hypotenuse s shown below. c b Trigonometric

More information

Recap: rigid motions. [L7] Robotics (ME671): Forward Kinematics. Recap: homogeneous transforms. Robot Kinematics Suril Shah IIT Jodhpur

Recap: rigid motions. [L7] Robotics (ME671): Forward Kinematics. Recap: homogeneous transforms. Robot Kinematics Suril Shah IIT Jodhpur --6 Rep: rgd motons [L7] Robots (ME67): Forwrd Knemts Rgd moton s ombnton of rotton nd trnslton It n be represented usng homogeneous trnsform R d H Surl Shh IIT Jodhpur Inverse trnsforms: T T R R d H Rep:

More information

Section 10.4 Hyperbolas

Section 10.4 Hyperbolas 66 Section 10.4 Hyperbols Objective : Definition of hyperbol & hyperbols centered t (0, 0). The third type of conic we will study is the hyperbol. It is defined in the sme mnner tht we defined the prbol

More information

Definition of Regular Expression

Definition of Regular Expression Definition of Regulr Expression After the definition of the string nd lnguges, we re redy to descrie regulr expressions, the nottion we shll use to define the clss of lnguges known s regulr sets. Recll

More information

ON THE KINEMATICS OF THE SCORBOT ER-VII ROBOT

ON THE KINEMATICS OF THE SCORBOT ER-VII ROBOT U... Sci. ull. Series D Vol. 77 Iss. 05 ISSN 5-58 N THE KINEMATICS F THE SCRT ER-VII RT Lurenţiu REDESCU Ion STRE Due to the relly fst dynmics of the development nd diversifiction of industril robots nd

More information

Image interpolation. A reinterpretation of low-pass filtering. Image Interpolation

Image interpolation. A reinterpretation of low-pass filtering. Image Interpolation Imge interpoltion A reinterprettion of low-pss filtering Imge Interpoltion Introduction Wht is imge interpoltion? (D-A conversion) Wh do we need it? Interpoltion Techniques 1D zero-order, first-order,

More information

HW Stereotactic Targeting

HW Stereotactic Targeting HW Stereotctic Trgeting We re bout to perform stereotctic rdiosurgery with the Gmm Knife under CT guidnce. We instrument the ptient with bse ring nd for CT scnning we ttch fiducil cge (FC). Above: bse

More information

Section 9.2 Hyperbolas

Section 9.2 Hyperbolas Section 9. Hperols 597 Section 9. Hperols In the lst section, we lerned tht plnets hve pproimtel ellipticl orits round the sun. When n oject like comet is moving quickl, it is le to escpe the grvittionl

More information

Summer Review Packet For Algebra 2 CP/Honors

Summer Review Packet For Algebra 2 CP/Honors Summer Review Pcket For Alger CP/Honors Nme Current Course Mth Techer Introduction Alger uilds on topics studied from oth Alger nd Geometr. Certin topics re sufficientl involved tht the cll for some review

More information

MATH 25 CLASS 5 NOTES, SEP

MATH 25 CLASS 5 NOTES, SEP MATH 25 CLASS 5 NOTES, SEP 30 2011 Contents 1. A brief diversion: reltively prime numbers 1 2. Lest common multiples 3 3. Finding ll solutions to x + by = c 4 Quick links to definitions/theorems Euclid

More information

Shape Representation and Indexing Based on Region Connection Calculus and Oriented Matroid Theory

Shape Representation and Indexing Based on Region Connection Calculus and Oriented Matroid Theory Shpe Representtion nd Indexing Bsed on Region Connection Clculus nd Oriented Mtroid Theory Ernesto Stffetti 1, Antoni Gru 2, Frncesc Serrtos 3, nd Alerto Snfeliu 1 1 Institute of Industril Rootics (CSIC-UPC)

More information

Planning with Reachable Distances: Fast Enforcement of Closure Constraints

Planning with Reachable Distances: Fast Enforcement of Closure Constraints Plnning with Rechle Distnces: Fst Enforcement of Closure Constrints Xinyu Tng, Shwn Thoms, nd Nncy M. Amto Astrct Motion plnning for closed-chin systems is prticulrly difficult due to dditionl closure

More information

Chapter 2 Sensitivity Analysis: Differential Calculus of Models

Chapter 2 Sensitivity Analysis: Differential Calculus of Models Chpter 2 Sensitivity Anlysis: Differentil Clculus of Models Abstrct Models in remote sensing nd in science nd engineering, in generl re, essentilly, functions of discrete model input prmeters, nd/or functionls

More information

COLOUR IMAGE MATCHING FOR DTM GENERATION AND HOUSE EXTRACTION

COLOUR IMAGE MATCHING FOR DTM GENERATION AND HOUSE EXTRACTION Hee Ju Prk OLOUR IMAGE MATHING FOR DTM GENERATION AND HOUSE EXTRATION Hee Ju PARK, Petr ZINMMERMANN * Swiss Federl Institute of Technology, Zuric Switzerlnd Institute for Geodesy nd Photogrmmetry heeju@ns.shingu-c.c.kr

More information

Introduction to Integration

Introduction to Integration Introduction to Integrtion Definite integrls of piecewise constnt functions A constnt function is function of the form Integrtion is two things t the sme time: A form of summtion. The opposite of differentition.

More information

Ma/CS 6b Class 1: Graph Recap

Ma/CS 6b Class 1: Graph Recap M/CS 6 Clss 1: Grph Recp By Adm Sheffer Course Detils Instructor: Adm Sheffer. TA: Cosmin Pohot. 1pm Mondys, Wednesdys, nd Fridys. http://mth.cltech.edu/~2015-16/2term/m006/ Min ook: Introduction to Grph

More information

ΕΠΛ323 - Θεωρία και Πρακτική Μεταγλωττιστών

ΕΠΛ323 - Θεωρία και Πρακτική Μεταγλωττιστών ΕΠΛ323 - Θωρία και Πρακτική Μταγλωττιστών Lecture 3 Lexicl Anlysis Elis Athnsopoulos elisthn@cs.ucy.c.cy Recognition of Tokens if expressions nd reltionl opertors if è if then è then else è else relop

More information

2 b. 3 Use the chain rule to find the gradient:

2 b. 3 Use the chain rule to find the gradient: Conic sections D x cos θ, y sinθ d y sinθ So tngent is y sin θ ( x cos θ) sinθ Eqution of tngent is x + y sinθ sinθ Norml grdient is sinθ So norml is y sin θ ( x cos θ) xsinθ ycos θ ( )sinθ, So eqution

More information

A GENERALIZED PROCEDURE FOR DEFINING QUOTIENT SPACES. b y HAROLD G. LAWRENCE A THESIS OREGON STATE UNIVERSITY MASTER OF ARTS

A GENERALIZED PROCEDURE FOR DEFINING QUOTIENT SPACES. b y HAROLD G. LAWRENCE A THESIS OREGON STATE UNIVERSITY MASTER OF ARTS A GENERALIZED PROCEDURE FOR DEFINING QUOTIENT SPACES b y HAROLD G. LAWRENCE A THESIS submitted to OREGON STATE UNIVERSITY in prtil fulfillment of the requirements for the degree of MASTER OF ARTS June

More information