How many times have you seen something like this?

Size: px
Start display at page:

Download "How many times have you seen something like this?"

Transcription

1 VOL. 77, NO. 4, OTOR Whee the amea Was KTHRN McL. YRS JMS M. HNL Smith ollege Nothampton, M How many times have you seen something like this? Then Now Souces: outesy of The ostonian Society/Ol State House Simon lay/hysalis mages On the left is a pictue of the Massachusetts Statehouse in oston, taken about On the ight is a pictue taken in They appea in oston Then an Now [3] an ae meant to show us how the builing an its setting have change, but the effect is iminishe because the camea was not in the same place fo both photogaphs. How ha is it to etemine the exact location of the photogaphe fom infomation in a photogaph? The poblem of unestaning the elative positions of image an object is actively stuie by compute scientists. n Kanatani [2], it is pat of computational pojective geomety. The specific task of locating the camea fom the photogaph is calle camea calibation. n Kanatani s book the pocess is quite involve an technical. n a mathematical pape publishe late, gga [1] tackles the same poblem. He poves that the task can be one, but the technique is similaly complex an the pape oes not eive a pactical metho o fomula. n this pape, we pesent a metho an a fomula fo locating the position of the photogaphe. Ou basic esult is the following: PROPOSTON. f a pictue of a ectangula soli taken by a vetically-hel pinhole camea has measuements (on the photogaph) of a, b, c,, an e, a b c e

2 252 MTHMTS MZN then the camea was positione c (b c) + e(b a) to the left of in the iection fom to an ae (b c) + e(b a) in font of point,whee an ae on-site measuements. The poof is base on high-school plane geomety an the basic pinciples of pojective geomety taught in a beginning awing class. ackgoun Ou assumption is that the camea is a pinhole camea with the film in a vetical plane (plane pepenicula to the goun). Une these cicumstances, the image on the film is the same as if we pojecte the thee-imensional wol onto a plane, what we ll call the, using staight lines to the viewe s. film = pinhole The only iffeence is that with a pinhole camea, the image appeas on the film upsie own. We ll nee a few elementay facts about this pojection: () The images of lines that ae paallel to the goun an to one anothe, but not paallel to the, meet at a single point in the. This point is calle the vanishing point of the collection of paallel lines.

3 VOL. 77, NO. 4, OTOR magine a collection of planes, each passing though the an one of the paallel lines. Then the planes intesect in a line that meets the at the vanishing point. ll such vanishing points lie on a single hoizontal line calle the hoizon line. () Lines in the eal wol that ae paallel to each othe an also paallel to the ae paallel when pojecte onto the. o Sale om this it follows that eal hoizontal lines ae pojecte to hoizontal lines. () lso, atios along lines paallel to the ae peseve when pojecte to the. n the iagam below, this means that X/Y = x/y. X Y o Sale SOLD! x y inally, (D) Lines on the goun connecting an object to the photogaphe appea as vetical lines on the. gain, imagine a plane containing the of the photogaphe an the line to the photogaphe.

4 254 MTHMTS MZN photogaphe That plane is vetical an intesects the in a vetical line. conveseof(d) is also tue: lines in the goun plane whose images ae vetical connect to the photogaphe. Ou metho iven the tools above, we pesent a simple metho fo etemining the location of the photogaphe. We stat with a photogaph of John M. eene Hall at Smith ollege, taken aoun 1935 by ga Scott. Since the builing is a complex soli, we pick a ectangula soli on it whose cones ae easy to locate. Souce: Histoic Nothampton, Nothampton, Massachusetts We ll call this outline the schematic pictue.

5 VOL. 77, NO. 4, OTOR a b c e The schematic coespons to the aeial view below, whee is the font of the builing an P is the location of the photogaphe. D J P Ou goal is to compute the istances an J. We ll compute the computation of J can be one symmetically. Ou poceue is to expess in tems of the five measuements a, b, c,, ane in the. ssuming we can measue on site, we can then multiply this times the atio to fin. To make the poof easie to view, we will show ou wok on a schematic with shape angles: a b c e We begin by extening an in the schematic pictue to etemine the location of the left vanishing point, V.

6 256 MTHMTS MZN Next, notice that P in the aeial view is paallel to, hence by act (), inthe schematic pictue it passes though V. lso, since it is a line to the photogaphe, by act (D) it is vetical in the schematic pictue. Thus point is the intesection of this vetical with the extension of. V Now we a a hoizontal line though paallel to the an exten P an D to meet it. n the aeial view, it looks like: D K L J P y act (), this line is also hoizontal in the schematic. The aeial view line L is paallel to an P, so it too passes though V. V K L om K L in the aeial view we have = K L. om act (), this popotion is equal to the atio of istances /s.

7 VOL. 77, NO. 4, OTOR V K s L To fin /s, we a two moe hoizontal lines, N an the hoizon line VH, then focus on the lowe half of the esulting figue. V N K a' b' c' H J e s L om VLK VN we have + s b = + e c, fom which we can eive: om VJ VH we have s = c b + b e c. =, fom which we can eive: = b b a b a. These togethe give us s = c b b a b b + b b a e c = b b a c b + b e ea c. We pomise to expess this atio in tems of a, b, c,,ane. We can accomplish that by one moe application of simila tiangles: We have x a a' b' b e c c' y a b = x x + = a b, an c b = y y + e = c b,

8 258 MTHMTS MZN an so giving us = s = a a = b b = c c b b c b b b + b b b e e b a b b c b = c (b c) + e(b a). The coesponing fomula fo J/ can be foun symmetically: J = ae (b c) + e(b a). This completes the poof of the poposition. The last step in locating the position of the camea is fining its height. This is accomplishe in a pimitive way by noting whee the hoizon line cuts acoss the pictue. The height of the camea is the height of this line as it appeas against the builing in the pictue. Souce: Histoic Nothampton, Nothampton, Massachusetts onclusion The close ageement of the two pictues illustates the poposition. Then Now Souce: Histoic Nothampton, Nothampton, Massachusetts

9 VOL. 77, NO. 4, OTOR Thee ae poblems, though, in applying the poposition. t may be ifficult to fin an appopiate pat of a builing to analyze. t can be ifficult to measue the builing. t can be ifficult to measue the photogaph. inally, locating the spot compute by the poposition, is not easy without equipment. onsieing these poblems, the close ageement of the pictues of John M. eene Hall might be consiee goo luck. We use a high-esolution scan on the achive photogaph b was measue at 470 pixels. ven so, if b wee measue just one pixel less, the compute location of the photogaphe changes by almost two feet (because of the stategic location of b in the enominato of the fomula). RRNS 1. M. H. gga, Pinhole cameas, pespective, an pojective geomety, me. Math. Monthly 105:7 (1998), Kenichi Kanatani, eometic omputation fo Machine Vision, laenon Pess, Oxfo, lizabeth McNulty, oston Then an Now, Thune ay Pess, Poof Without Wos: xtema of the unction a cos t + b sin t y (cos t, sin t) 1 ax + by = 0 x 1 a cos t + b sin t / a 2 + b 2 1 a 2 + b 2 a cos t + b sin t a 2 + b 2 M. HSSN,M.YT, ND H. TMOOR NSTTUT OR DVND STUDS N S SNS, P. O. OX , V ZN,ZNJN 45195, RN

CLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT

CLOSE RANGE PHOTOGRAMMETRY WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT CLOSE RANGE PHOTOGRAMMETR WITH CCD CAMERAS AND MATCHING METHODS - APPLIED TO THE FRACTURE SURFACE OF AN IRON BOLT Tim Suthau, John Moé, Albet Wieemann an Jens Fanzen Technical Univesit of Belin, Depatment

More information

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6 Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

More information

Physics 111 Fall 2007 Electrostatic Forces and the Electric Field - Solutions

Physics 111 Fall 2007 Electrostatic Forces and the Electric Field - Solutions Physics 111 Fall 007 Electostatic Foces an the Electic Fiel - Solutions 1. Two point chages, 5 µc an -8 µc ae 1. m apat. Whee shoul a thi chage, equal to 5 µc, be place to make the electic fiel at the

More information

Analytical Proof of Newton's Force Laws

Analytical Proof of Newton's Force Laws Analytical Poof of Newton s Foce Laws Page 1 1 Intouction Analytical Poof of Newton's Foce Laws Many stuents intuitively assume that Newton's inetial an gavitational foce laws, F = ma an Mm F = G, ae tue

More information

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algebra and Trig. I. A point is a location or position that has no size or dimension. Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

More information

PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK

PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK PRICING MODEL FOR COMPETING ONLINE AND RETAIL CHANNEL WITH ONLINE BUYING RISK Vaanya Vaanyuwatana Chutikan Anunyavanit Manoat Pinthong Puthapon Jaupash Aussaavut Dumongsii Siinhon Intenational Institute

More information

New proofs for the perimeter and area of a circle

New proofs for the perimeter and area of a circle New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Poynting Vector and Energy Flow in a Capacitor Challenge Problem Solutions

Poynting Vector and Energy Flow in a Capacitor Challenge Problem Solutions Poynting Vecto an Enegy Flow in a Capacito Challenge Poblem Solutions Poblem 1: A paallel-plate capacito consists of two cicula plates, each with aius R, sepaate by a istance. A steay cuent I is flowing

More information

L19 Geomagnetic Field Part I

L19 Geomagnetic Field Part I Intoduction to Geophysics L19-1 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o

More information

Deflection of Electrons by Electric and Magnetic Fields

Deflection of Electrons by Electric and Magnetic Fields Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

More information

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of Oiginall TRIGONOMETRY was that banch of mathematics concened with solving tiangles using tigonometic atios which wee seen as popeties of tiangles athe than of angles. The wod Tigonomet comes fom the Geek

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90 . Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

More information

LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

More information

Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M

Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuent-caying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to

More information

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

More information

Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number.

Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number. Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 1-5 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE

More information

Thomas J. Osler, Mathematics Department and Tirupathi R. Chandrupatla, Mechanical Engineering. Rowan University Glassboro, NJ

Thomas J. Osler, Mathematics Department and Tirupathi R. Chandrupatla, Mechanical Engineering. Rowan University Glassboro, NJ /9/04 SOME UNUSUAL EXPRESSIONS FOR THE INRADIUS OF A TRIANGLE Tomas J. Osle, Matematics Depatment and Tiupati R. Candupatla, Mecanical Engineeing Rowan Univesity Glassboo, NJ 0808 Osle@owan.edu Candupatla@owan.edu

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

More information

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle

More information

Valuation of Floating Rate Bonds 1

Valuation of Floating Rate Bonds 1 Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

Review Module: Dot Product

Review Module: Dot Product MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and

More information

Seventh Edition DYNAMICS 15Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University

Seventh Edition DYNAMICS 15Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University Seenth 15Fedinand P. ee E. Russell Johnston, J. Kinematics of Lectue Notes: J. Walt Ole Texas Tech Uniesity Rigid odies CHPTER VECTOR MECHNICS FOR ENGINEERS: YNMICS 003 The McGaw-Hill Companies, Inc. ll

More information

92.131 Calculus 1 Optimization Problems

92.131 Calculus 1 Optimization Problems 9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient:

Psychology 282 Lecture #2 Outline. Review of Pearson correlation coefficient: Psychology 282 Lectue #2 Outline Review of Peason coelation coefficient: z z ( n 1) Measue of linea elationship. Magnitude Stength Sign Diection Bounded by +1.0 and -1.0. Independent of scales of measuement.

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

More information

The Detection of Obstacles Using Features by the Horizon View Camera

The Detection of Obstacles Using Features by the Horizon View Camera The Detection of Obstacles Using Featues b the Hoizon View Camea Aami Iwata, Kunihito Kato, Kazuhiko Yamamoto Depatment of Infomation Science, Facult of Engineeing, Gifu Univesit aa@am.info.gifu-u.ac.jp

More information

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

Butterfly Network Analysis and The Beneˇ s Network

Butterfly Network Analysis and The Beneˇ s Network 6.895 Theoy of Paallel Systems Lectue 17 Buttefly Netwok Analysis and The Beneˇ s Netwok Lectue: Chales Leiseson Lectue Summay 1. Netwok with N Nodes This section poves pat of the lowe bound on expected

More information

Hip Hop solutions of the 2N Body problem

Hip Hop solutions of the 2N Body problem Hip Hop solutions of the N Boy poblem Esthe Baabés baabes@ima.ug.es Depatament Infomàtica i Matemàtica Aplicaa, Univesitat e Giona. Josep Maia Cos cos@eupm.upc.es Depatament e Matemàtica Aplicaa III, Univesitat

More information

Review of Coordinate Systems

Review of Coordinate Systems Review o Coodinate Sstems good undestanding o coodinate sstems can be ve helpul in solving poblems elated to Mawell s Equations. The thee most common coodinate sstems ae ectangula (,, ), clindical (,,

More information

Problem Set 6: Solutions

Problem Set 6: Solutions UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

More information

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to . Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

Lab #7: Energy Conservation

Lab #7: Energy Conservation Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

More information

Product reviews by third parties are growing in popularity. This paper examines when and how a manufacturing

Product reviews by third parties are growing in popularity. This paper examines when and how a manufacturing Vol. 24, No. 2, Sping 2005, pp. 218 240 issn 0732-2399 eissn 1526-548X 05 2402 0218 infoms oi 10.1287/mksc.1040.0089 2005 INFORMS Thi-Paty Pouct Review an Fim Maketing Stategy Yubo Chen Elle College of

More information

Introduction to Electric Potential

Introduction to Electric Potential Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

Experiment MF Magnetic Force

Experiment MF Magnetic Force Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating

More information

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

More information

Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electric Fields Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

More information

Voltage ( = Electric Potential )

Voltage ( = Electric Potential ) V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

More information

Magnetism: a new force!

Magnetism: a new force! -1 Magnetism: a new foce! o fa, we'e leaned about two foces: gaity and the electic field foce. F E = E, FE = E Definition of E-field kq E-fields ae ceated by chages: E = 2 E-field exets a foce on othe

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

More information

CHAT Pre-Calculus Section 10.7. Polar Coordinates

CHAT Pre-Calculus Section 10.7. Polar Coordinates CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to

More information

CHAPTER 4 POSITION, VELOCITY AND ACCELERATION ANALYSES FOR PLANAR MECHANISMS USING COMPLEX NUMBER METHOD

CHAPTER 4 POSITION, VELOCITY AND ACCELERATION ANALYSES FOR PLANAR MECHANISMS USING COMPLEX NUMBER METHOD CHPTER POSITION, VELOCITY ND CCELERTION NLYSES FOR PLNR MECHNISMS USING COMPLEX NUMER METHOD Vecto nalysis: Fo the position vectos shown below, the positive angle is measued counte-clock wise (ccw) fom

More information

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

More information

!( r) =!( r)e i(m" + kz)!!!!. (30.1)

!( r) =!( r)e i(m + kz)!!!!. (30.1) 3 EXAMPLES OF THE APPLICATION OF THE ENERGY PRINCIPLE TO CYLINDRICAL EQUILIBRIA We now use the Enegy Pinciple to analyze the stability popeties of the cylinical! -pinch, the Z-pinch, an the Geneal Scew

More information

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth

Ch. 8 Universal Gravitation. Part 1: Kepler s Laws. Johannes Kepler. Tycho Brahe. Brahe. Objectives: Section 8.1 Motion in the Heavens and on Earth Ch. 8 Univesal Gavitation Pat 1: Keple s Laws Objectives: Section 8.1 Motion in the Heavens and on Eath Objectives Relate Keple s laws of planetay motion to Newton s law of univesal gavitation. Calculate

More information

Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w

Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w 1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a

More information

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27 Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

More information

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

More information

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0. Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads

More information

Ilona V. Tregub, ScD., Professor

Ilona V. Tregub, ScD., Professor Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

Road tunnel. Road tunnel information sheet. Think about. Using the information

Road tunnel. Road tunnel information sheet. Think about. Using the information Road tunnel This activity is about using a gaphical o algebaic method to solve poblems in eal contets that can be modelled using quadatic epessions. The fist poblem is about a oad tunnel. The infomation

More information

MAT 080-Algebra II. Literal Equation

MAT 080-Algebra II. Literal Equation MAT 080-Algeba II Lieal Equaions Objecives a Solve (linea) lieal equaions which o no equie facoing b Solve (linea) lieal equaions which equie facoing a Solving lieal equaions which o no equie facoing A

More information

Continuous Compounding and Annualization

Continuous Compounding and Annualization Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

Section 5-3 Angles and Their Measure

Section 5-3 Angles and Their Measure 5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

More information

Transformations in Homogeneous Coordinates

Transformations in Homogeneous Coordinates Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +

More information

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

More information

Forces & Magnetic Dipoles. r r τ = μ B r

Forces & Magnetic Dipoles. r r τ = μ B r Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

More information

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years. 9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

More information

An Introduction to Omega

An Introduction to Omega An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

More information

Notes on Electric Fields of Continuous Charge Distributions

Notes on Electric Fields of Continuous Charge Distributions Notes on Electic Fields of Continuous Chage Distibutions Fo discete point-like electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution

More information

Sequence Impedances of Transmission Lines R. E. Fehr, May 2004

Sequence Impedances of Transmission Lines R. E. Fehr, May 2004 Sequence Impeances o Tansmission Lines R. E. Feh, May 004 In oe to analye unbalance conitions on tansmission lines, we nee to apply the metho o symmetical components, as escibe by Chales Fotescue in his

More information

ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION

ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION ON NEW CHALLENGES FOR CFD SIMULATION IN FILTRATION Michael Dedeing, Wolfgang Stausbeg, IBS Filtan, Industiestasse 19 D-51597 Mosbach-Lichtenbeg, Gemany. Oleg Iliev(*), Zaha Lakdawala, Faunhofe Institut

More information

PY1052 Problem Set 3 Autumn 2004 Solutions

PY1052 Problem Set 3 Autumn 2004 Solutions PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the

More information

4.1 Cylindrical and Polar Coordinates

4.1 Cylindrical and Polar Coordinates 4.1 Cylindical and Pola Coodinates 4.1.1 Geometical Axisymmety A lage numbe of pactical engineeing poblems involve geometical featues which have a natual axis of symmety, such as the solid cylinde, shown

More information

Thank you for participating in Teach It First!

Thank you for participating in Teach It First! Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident

More information

The transport performance evaluation system building of logistics enterprises

The transport performance evaluation system building of logistics enterprises Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics

More information

Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109

Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109 Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

More information

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

Nontrivial lower bounds for the least common multiple of some finite sequences of integers J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

More information

N V V L. R a L I. Transformer Equation Notes

N V V L. R a L I. Transformer Equation Notes Tnsfome Eqution otes This file conts moe etile eivtion of the tnsfome equtions thn the notes o the expeiment 3 wite-up. t will help you to unestn wht ssumptions wee neee while eivg the iel tnsfome equtions

More information

EAS Groundwater Hydrology Lecture 13: Well Hydraulics 2 Dr. Pengfei Zhang

EAS Groundwater Hydrology Lecture 13: Well Hydraulics 2 Dr. Pengfei Zhang EAS 44600 Goundwate Hydology Lectue 3: Well Hydaulics D. Pengfei Zhang Detemining Aquife Paametes fom Time-Dawdown Data In the past lectue we discussed how to calculate dawdown if we know the hydologic

More information

Physics 107 HOMEWORK ASSIGNMENT #14

Physics 107 HOMEWORK ASSIGNMENT #14 Physics 107 HOMEWORK ASSIGNMENT #14 Cutnell & Johnson, 7 th edition Chapte 17: Poblem 44, 60 Chapte 18: Poblems 14, 18, 8 **44 A tube, open at only one end, is cut into two shote (nonequal) lengths. The

More information

AP Physics Electromagnetic Wrap Up

AP Physics Electromagnetic Wrap Up AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle

More information

The Capital Asset Pricing Model. Chapter 9

The Capital Asset Pricing Model. Chapter 9 The Capital Asset Picing odel Chapte 9 Capital Asset Picing odel CAP centepiece of moden finance gives the elationship that should be obseved between isk and etun of an asset it allows fo the evaluation

More information

MATHEMATICAL SIMULATION OF MASS SPECTRUM

MATHEMATICAL SIMULATION OF MASS SPECTRUM MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec

More information

The Binomial Distribution

The Binomial Distribution The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

More information

9.5 Volume of Pyramids

9.5 Volume of Pyramids Page of 7 9.5 Volume of Pyamids and Cones Goal Find the volumes of pyamids and cones. Key Wods pyamid p. 49 cone p. 49 volume p. 500 In the puzzle below, you can see that the squae pism can be made using

More information

PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013 PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

More information

PDHonline Course L121 (4 PDH) Vertical Curves. Instructor: Jan Van Sickle, P.L.S. PDH Online PDH Center

PDHonline Course L121 (4 PDH) Vertical Curves. Instructor: Jan Van Sickle, P.L.S. PDH Online PDH Center PDHonline Couse L11 (4 PDH) Vetical Cuves Instucto: Jan Van Sickle, PLS 01 PDH Online PDH Cente 57 Meadow Estates Dive Faifa, VA 030-6658 Phone & Fa: 703-988-0088 wwwpdhonlineo wwwpdhcentecom An Appoved

More information

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

More information

VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

International Monetary Economics Note 1

International Monetary Economics Note 1 36-632 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated

More information

The Prime Coe cient Formulas for Rotation of Conics

The Prime Coe cient Formulas for Rotation of Conics The Pime Coe cient Fomulas fo Rotation of Conics It is known that the gah of a conic section in the x; y -coodinate lane has a second degee equation in the vaiables x and y: I.e., Ax + xy +Cy +Dx+Ey +F

More information

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1) Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

More information