Projection of geometric models
|
|
- Corey Randall
- 5 years ago
- Views:
Transcription
1 ojecion of geomeic moels YZU Opimal Design Laboao. All ighs eseve. Las upae: Yeh-Liang Hsu (-9-). Noe: his is he couse maeial fo ME55 Geomeic moeling an compue gaphics, Yuan Ze Univesi. a of his maeial is aape fom CAD/CAM heo an acice, b Ibahim Zei, McGaw-Hill, 99. his maeial is be use sicl fo eaching an leaning of his couse. ojecion of geomeic moels. Homogeneous cooinae Man gaphics applicaions involve sequences of geomeic ansfomaions. An animaion, fo eample, migh equie an objec o be anslae an oae a each incemen of he moion. Hee we consie how he mai epesenaions can be efomulae so ha such ansfomaion sequences can be efficienl pocesse. anslaion, scaling, mioing, an oaion of a poin ae epesene b he following equaions especivel: () S M () (3) R Z (4) While he las hee equaions ae in he fom of mai muliplicaion, anslaion akes he fom of veco aiion. his makes i inconvenien o concaenae ansfomaions involving anslaion. I is esiable, heefoe, o epess all geomeic ansfomaions in he fom of mai muliplicaions onl. Repesening poins b hei homogeneous cooinaes povies an effecive wa o unif he escipion of geomeic ansfomaions as mai muliplicaions. hp://esigne.mech.u.eu.w/
2 ojecion of geomeic moels hp://esigne.mech.u.eu.w/ In hee-imensional space, a poin wih Caesian cooinaes (,, ) has he homogeneous cooinaes (,,, h), whee h is an scala faco. he wo pes of cooinaes ae elae o each ohe b he following equaions: h h h (5) Fo eample, (,, 3, ) an (, 4, 6, ) ae he same poin. anslaion can hen be epesene b mai muliplicaion using homogeneous cooinaes. (6) D (7) An he ansfomaion maices in Equaion (), (3), (4) become s s s S (8) M (9) R () Epessing posiions in homogeneous cooinaes allows us o epesen all geomeic ansfomaion equaions as mai muliplicaions. Fo eample,
3 ojecion of geomeic moels hp://esigne.mech.u.eu.w/ 3 D R D Z () whee D () cos sin sin cos Z R (3) D (4) cos sin cos sin sin cos sin cos R D D (5) o in a geneal fom, (6) he 3 3 submai [ ] pouces scaling, eflecion, o oaion. he 3 column mai [ ] geneaes anslaion. he 3 ow mai [ 3 ] pouces pespecive pojecion, which will be iscusse lae.
4 ojecion of geomeic moels Assignmen Ceae a ecangle in - plane in ou CAD sofwae; specif he cooinaes of he 4 cone poins. Assume some paamees an appl he following wo-imensional ansfomaions, anslaion, oaion, anslaion, o he objec in ou CAD sofwae. Daw he objec afe hese ansfomaions. Use homogenous cooinaes o epesen he ke poins of he objec ou geneae. Wie a pogam in Malab an concaenae he 3 ansfomaions ino one ansfomaion mai as shown in Equaion ()~(5). Calculae he posiions of he ke poins using his ansfomaion mai. Regeneae he objec b connecing he ke poins afe ansfomaion. Does he objec mach he objec ou ew in ou CAD sofwae? Assignmen Use wo-imensional oaion maices o show ha oaion is no commuaive, ha is, R R R R. Homogeneous cooinaes was inouce b Augus Feinan Möbius in 87. he have he avanage ha he cooinaes of a poin, even hose a infini, can be epesene using finie cooinaes. Ofen fomulas involving homogeneous cooinaes ae simple an moe smmeic han hei Caesian counepas. Homogeneous cooinaes have a ange of applicaions, incluing compue gaphics an 3D compue vision, whee he allow affine ansfomaions an, in geneal, pojecive ansfomaions o be easil epesene b a mai.. ojecions of geomeic moels. especive pojecion an paallel pojecion Viewing a hee-imensional moel is a ahe comple pocess ue o he fac ha ispla evices can onl ispla gaphics on wo-imensional sceens. o efine a pojecion, a cene of pojecion an a pojecion plane mus be efine. hee ae wo iffeen pes of pojecions base on he locaion of he cene of pojecion (o pojecion efeence poin) elaive o he pojecion plane, as shown in Figue. If he cene is a a finie isance fom he plane, pespecive pojecion 4 hp://esigne.mech.u.eu.w/
5 ojecion of geomeic moels esuls an all he pojecos mee a he cene. If, on he ohe han, he cene is a an infinie isance, all he pojecos become paallel (mee a infini) an paallel pojecion esuls. ojecos + Cene of pojecion ojecos Cene of pojecion a infini (a) especive pojecion (b) aallel pojecion Figue. pes of pojecions especive pojecion oes no peseve paallelism, ha is, no wo lines ae paallel. especive pojecion ceaes an aisic effec ha as some ealism o pespecive views. Sie of an eni is invesel popoional o is isance fom he cene of pojecion; ha is, he close he eni o he cene, he lage is sie is. aallel pojecion peseves acual imensions an shapes of objecs. I also peseves paallelism. Angles ae peseve onl on faces of he objec ha ae paallel o he pojecion plane. Assignmen 3 Does ou CAD sofwae suppo pespecive pojecion? If so, buil a 3D objec o emonsae he iffeence beween pespecive pojecion an paallel pojecion. In ou CAD sofwae, how o ou moif he paamees o change he viewpoin of he pespecive pojecion? hee ae wo pes of paallel pojecions base on he elaion beween he iecion of pojecion an he pojecion plane. If his iecion is nomal o he pojecion plane, ohogaphic pojecion an views esul. If he iecion is no nomal o he plane, oblique pojecion occus. hee ae wo pes of ohogaphic pojecions. he mos common pe is he one ha uses pojecion planes ha ae pepenicula o he pincipal aes of he MCS of he moel; ha is, he iecion of pojecion coincies wih one of hese aes. he muli-view 5 hp://esigne.mech.u.eu.w/
6 ojecion of geomeic moels pojecion -- fon, op, an igh views ha ae use cusomail in engineeing awings belong o his pe. he ohe pe of ohogaphic pojecion uses pojecion planes ha ae no nomal o a pincipal ais an heefoe show seveal faces of a moel a once. his pe is calle aonomeic pojecions. he peseve paallelism of lines bu no angles. Aonomeic pojecions ae fuhe ivie ino imeic, imeic an isomeic pojecions. he isomeic pojecion is he mos common aonomeic pojecion. he isomeic pojecion has he useful pope ha all hee pincipal aes ae equall foeshoene. heefoe measuemens along he aes can be mae wih he same scalehus he name: iso fo equal, meic fo measue. In aiion, he nomal o he pojecion plane makes equal angles wih each pincipal ais an he pincipal aes make equal angles ( each) wih one anohe when pojece ono he pojecion plane. he oblique pojecion places he pincipal face of he objec paallel o he plane of he pape, an is ofen use in feehan skeching. he avanage is ha eails on he fon face of he objec eain hei ue shape. he isavanage is ha oblique pojecion oes no appea ealisic. Assignmen 4 Daw a famil ee fo he pes of pojecions iscusse above. Use ou CAD sofwae o geneae all pojecions in he famil ee using he 3D objec in Assignmen 3.. Mappings of geomeic moels o he viewing cooinae ssem Mapping of a poin (o a se of poins) belonging o an objec fom one cooinae ssem o anohe is efine as changing he escipion of he poin (o he se of poins) fom he fis cooinae ssem o he secon one. his is equivalen o ansfoming one cooinae ssem o anohe. Given he cooinaes of a poin measue in a given cooinae ssem, fin he cooinaes of he poin measue in anohe cooinae ssem, sa such ha =f(, ansfomaion paamees) (7) he mapping paamees escibe he elaionship beween he wo ssems an consis of he posiion of he oigin an oienaion of he ssem elaive o he ssem. 6 hp://esigne.mech.u.eu.w/
7 ojecion of geomeic moels R (8) whee [R] an ae he oaional an anslaional mapping pas of [] especivel. heefoe, Equaion (8) gives he posiion veco of he oigin of he ssem as an is oienaion as [R] boh measue in he ssem. he columns of [R] ae he componens of he uni vecos of he ssem (along is aes) measue in he ssem. A view has a viewing cooinae ssem (VCS). I is a hee-imensional ssem wih he X v ais hoional poining o he igh an he Y v ais veical poining upwa, as shown in Figue. he Z v ais efines he viewing iecion. he posiive Z v ais has an opposie sense o he viewing iecion o keep he VCS a igh-hane cooinae ssem, even hough a lef-hane ssem ma be moe esiable hee since is posiive Z v ais is in he iecion of he lines of sigh emiing fom he viewing ee. Viewing plane Y v Viewpo (view winow) View oigin X v Z v Viewing iecion Viewing ee o Figue. View efiniion. o obain views of a moel, he viewing plane, he X v Y v plane, is mae coincien wih he - plane of he MCS such ha he VCS oigin is he same as ha of he MCS. Moel views now become a mae of oaing he moel wih espec o he VCS aes unil he esie moel plane coincies wih he viewing plane followe b pojecing he moel 7 hp://esigne.mech.u.eu.w/
8 ojecion of geomeic moels ono ha plane. hus, a view of a moel is geneae in wo seps: oae he moel popel an hen pojec i..3 Ohogaphic pojecions Figue 3 shows he elaionship beween MCS an VCS. An ohogaphic pojecion (view) of a moel is obaine b seing o eo he cooinae value coesponing o he MCS ais ha coincies wih he iecion of pojecion (o viewing) afe he moel oaion. o obain he fon view, we onl (no oaion is neee) nee o se = fo all he ke poins of he moel. hus, Equaion (8) becomes An (9) v () whee v is he poin epesse in he VCS. Fo he fon view, Equaion () gives v = an v =. 8 hp://esigne.mech.u.eu.w/
9 ojecion of geomeic moels Fon op Righ X Y v Z (a) Moel views elaive o is MCS Y Y v op X v,x Fon op Righ X v Z Z Z v X Y v,y Y v,y Fon Righ Fon view X v,x Z Righ view X v (b) MCS an VCS elaionship Figue 3. Relaionship beween MCS an VCS Fo he op view, he moel an is MCS ae oae b 9 abou he X v ais followe b seing he cooinae of he esuling poins o eo. he cooinae is he one o se o eo because he Y ais of he MCS coincies wih he pojecion iecion. In his case, [] becomes () An Equaion () gives v = an v =-. If we use he above equaion o ansfom he MCS iself, he X ais (==) ansfoms o v = an he Y ais (==) ansfoms o v =-. 9 hp://esigne.mech.u.eu.w/
10 ojecion of geomeic moels he igh view shown in Figue 3 can be obaine b oaing he moel an is MCS abou he Y v ais b -9 an seing he cooinae o eo. hus, () which gives v =- an v =. Eamining Equaion (), (), an () shows ha [] is a singula mai wih a column of eos which coespons o he MCS ais ha coincies wih he pojecion o viewing iecion. o obain he isomeic pojecion o view, he moel an is MCS ae cusomail oae an angle 45 abou he Y v ais followe b a oaion abou he X v ais. In pacice, he angle is aken as iangles in manual consucion of isomeic views. 3 o enable he afing (plasic) cos sin cos sin v (3) sin cos sin cos Assignmen 5 Ceae a 3D block an assign cooinaes o all 8 cone poins of he block Wie a pogam in Malab an use Equaion (3) o ansfom he 8 ke poins of he block. Seing cooinaes of he ke poins o eo, hen aw he block again. Do ou obain an isomeic view of he block? Show ou Malab pogam oo..4 especive pojecions One common wa o obain a pespecive view is o place he cene of pojecion along he Z v ais of he VCS an pojec ono he v = o he X v Y v plane. A new cooinae ssem calle he ee cooinae ssem (ECS) is inouce elaive o he line of sigh. he ECS has an oigin locae a he same posiion as he viewing ee. Is X e an Y e aes ae paallel o he X v an Y v aes of he VCS. Howeve, i is a lef-hane ssem. hp://esigne.mech.u.eu.w/
11 ojecion of geomeic moels hp://esigne.mech.u.eu.w/ he ansfomaion mai of cooinaes of poins fom he VCS o he ECS o vice vesa can be wien as (4) his mai simpl inves he sign of he cooinae. In he ohogaphic views, he ECS is locae a infini. I is obvious ha he ECS can be eplace b he VCS. In his case, poins wih smalle values ae inepee as being fuhe fom he viewing ee. One-poin pojecion is shown in Figue 4. Fom simila iangles, we can he following equaions. v / (5) v / (6) / v (7) If his equaion is epane i gives. / v
12 ojecion of geomeic moels Figue 4. especive pojecion along he Z v ais Assignmen 6 Use he same 3D block in Assignmen 5. Wie a pogam in Malab o appl Equaion (7) o is 8 ke poins, hen ceae he pespecive pojecion of he block. Change seveal values of, how oes he pespecive pojecion change? Show ou Malab pogam oo. hp://esigne.mech.u.eu.w/
Projection of geometric models
ojecion of geomeic moels Eie: Yeh-Liang Hsu (998-9-2); ecommene: Yeh-Liang Hsu (2-9-26); las upae: Yeh-Liang Hsu (29--3). Noe: This is he couse maeial fo ME55 Geomeic moeling an compue gaphics, Yuan Ze
More informationKINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES In igid body kinemaics, we use he elaionships govening he displacemen, velociy and acceleaion, bu mus also accoun fo he oaional moion of he body. Descipion of he moion of igid
More informationENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]
ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,
More informationOrthotropic Materials
Kapiel 2 Ohoopic Maeials 2. Elasic Sain maix Elasic sains ae elaed o sesses by Hooke's law, as saed below. The sesssain elaionship is in each maeial poin fomulaed in he local caesian coodinae sysem. ε
More informationComputer Propagation Analysis Tools
Compue Popagaion Analysis Tools. Compue Popagaion Analysis Tools Inoducion By now you ae pobably geing he idea ha pedicing eceived signal sengh is a eally impoan as in he design of a wieless communicaion
More informationCAMERA GEOMETRY. Ajit Rajwade, CS 763, Winter 2017, IITB, CSE department
CAMERA GEOMETRY Aji Rajwade, CS 763, Wine 7, IITB, CSE depamen Conens Tansfomaions of poins/vecos in D Tansfomaions of poins/vecos in 3D Image Fomaion: geome of pojecion of 3D poins ono D image plane Vanishing
More informationToday - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations
Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy
More information, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t
Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission
More informationSpecial Subject SC434L Digital Video Coding and Compression. ASSIGNMENT 1-Solutions Due Date: Friday 30 August 2002
SC434L DVCC Assignmen Special Sujec SC434L Digial Vieo Coing an Compession ASSINMENT -Soluions Due Dae: Fiay 30 Augus 2002 This assignmen consiss of wo pages incluing wo compulsoy quesions woh of 0% of
More information( ) exp i ω b ( ) [ III-1 ] exp( i ω ab. exp( i ω ba
THE INTEACTION OF ADIATION AND MATTE: SEMICLASSICAL THEOY PAGE 26 III. EVIEW OF BASIC QUANTUM MECHANICS : TWO -LEVEL QUANTUM SYSTEMS : The lieaue of quanum opics and lase specoscop abounds wih discussions
More informationLecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation
Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion
More informationGeneral Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security
1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,
More informationSections 3.1 and 3.4 Exponential Functions (Growth and Decay)
Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens
More informationThe sudden release of a large amount of energy E into a background fluid of density
10 Poin explosion The sudden elease of a lage amoun of enegy E ino a backgound fluid of densiy ceaes a song explosion, chaaceized by a song shock wave (a blas wave ) emanaing fom he poin whee he enegy
More informationLecture 22 Electromagnetic Waves
Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should
More informationRisk tolerance and optimal portfolio choice
Risk oleance and opimal pofolio choice Maek Musiela BNP Paibas London Copoae and Invesmen Join wok wih T. Zaiphopoulou (UT usin) Invesmens and fowad uiliies Pepin 6 Backwad and fowad dynamic uiliies and
More information2-d Motion: Constant Acceleration
-d Moion: Consan Acceleaion Kinemaic Equaions o Moion (eco Fom Acceleaion eco (consan eloci eco (uncion o Posiion eco (uncion o The eloci eco and posiion eco ae a uncion o he ime. eloci eco a ime. Posiion
More informationLecture 5. Chapter 3. Electromagnetic Theory, Photons, and Light
Lecue 5 Chape 3 lecomagneic Theo, Phoons, and Ligh Gauss s Gauss s Faada s Ampèe- Mawell s + Loen foce: S C ds ds S C F dl dl q Mawell equaions d d qv A q A J ds ds In mae fields ae defined hough ineacion
More informationControl Volume Derivation
School of eospace Engineeing Conol Volume -1 Copyigh 1 by Jey M. Seizman. ll ighs esee. Conol Volume Deiaion How o cone ou elaionships fo a close sysem (conol mass) o an open sysem (conol olume) Fo mass
More informationEN221 - Fall HW # 7 Solutions
EN221 - Fall2008 - HW # 7 Soluions Pof. Vivek Shenoy 1.) Show ha he fomulae φ v ( φ + φ L)v (1) u v ( u + u L)v (2) can be pu ino he alenaive foms φ φ v v + φv na (3) u u v v + u(v n)a (4) (a) Using v
More informationMECHANICS OF MATERIALS Poisson s Ratio
Fouh diion MCHANICS OF MATRIALS Poisson s Raio Bee Johnson DeWolf Fo a slende ba subjeced o aial loading: 0 The elongaion in he -diecion is accompanied b a conacion in he ohe diecions. Assuming ha he maeial
More informationMATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH
Fundamenal Jounal of Mahemaical Phsics Vol 3 Issue 013 Pages 55-6 Published online a hp://wwwfdincom/ MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Univesias
More informationThe Production of Polarization
Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview
More informationSharif University of Technology - CEDRA By: Professor Ali Meghdari
Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai Pupose: o exen he Enegy appoach in eiving euaions of oion i.e. Lagange s Meho fo Mechanical Syses. opics: Genealize Cooinaes Lagangian Euaion
More informationRelative and Circular Motion
Relaie and Cicula Moion a) Relaie moion b) Cenipeal acceleaion Mechanics Lecue 3 Slide 1 Mechanics Lecue 3 Slide 2 Time on Video Pelecue Looks like mosly eeyone hee has iewed enie pelecue GOOD! Thank you
More informationTime-Space Model of Business Fluctuations
Time-Sace Moel of Business Flucuaions Aleei Kouglov*, Mahemaical Cene 9 Cown Hill Place, Suie 3, Eobicoke, Onaio M8Y 4C5, Canaa Email: Aleei.Kouglov@SiconVieo.com * This aicle eesens he esonal view of
More informationLecture 17: Kinetics of Phase Growth in a Two-component System:
Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien
More informationCamera Models class 8
Camea Models class 8 Mulile View Geomey Com 29-89 Mac ollefeys Mulile View Geomey couse schedule (subjec o change) Jan. 7, 9 Ino & moivaion ojecive 2D Geomey Jan. 4, 6 (no class) ojecive 2D Geomey Jan.
More informationThe Effect of the Metal Oxidation on the Vacuum Chamber Impedance
SL-FL 9-5 The ffec of he Meal Oiaion on he Vacuum Chambe Impeance nani Tsaanian ambug Univesiy Main Dohlus, Igo agoonov Deusches leconen-sinchoon(dsy bsac The oiaion of he meallic vacuum chambe inenal
More informationCircular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can.
1 Cicula Moion Radians One evoluion is equivalen o 360 0 which is also equivalen o 2π adians. Theefoe we can say ha 360 = 2π adians, 180 = π adians, 90 = π 2 adians. Hence 1 adian = 360 2π Convesions Rule
More informationP h y s i c s F a c t s h e e t
P h y s i c s F a c s h e e Sepembe 2001 Numbe 20 Simple Hamonic Moion Basic Conceps This Facshee will:! eplain wha is mean by simple hamonic moion! eplain how o use he equaions fo simple hamonic moion!
More informationCS 188: Artificial Intelligence Fall Probabilistic Models
CS 188: Aificial Inelligence Fall 2007 Lecue 15: Bayes Nes 10/18/2007 Dan Klein UC Bekeley Pobabilisic Models A pobabilisic model is a join disibuion ove a se of vaiables Given a join disibuion, we can
More informationÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s
MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN
More informationChapter 7. Interference
Chape 7 Inefeence Pa I Geneal Consideaions Pinciple of Supeposiion Pinciple of Supeposiion When wo o moe opical waves mee in he same locaion, hey follow supeposiion pinciple Mos opical sensos deec opical
More informationRepresenting Knowledge. CS 188: Artificial Intelligence Fall Properties of BNs. Independence? Reachability (the Bayes Ball) Example
C 188: Aificial Inelligence Fall 2007 epesening Knowledge ecue 17: ayes Nes III 10/25/2007 an Klein UC ekeley Popeies of Ns Independence? ayes nes: pecify complex join disibuions using simple local condiional
More informationMEEN 617 Handout #11 MODAL ANALYSIS OF MDOF Systems with VISCOUS DAMPING
MEEN 67 Handou # MODAL ANALYSIS OF MDOF Sysems wih VISCOS DAMPING ^ Symmeic Moion of a n-dof linea sysem is descibed by he second ode diffeenial equaions M+C+K=F whee () and F () ae n ows vecos of displacemens
More information156 There are 9 books stacked on a shelf. The thickness of each book is either 1 inch or 2
156 Thee ae 9 books sacked on a shelf. The hickness of each book is eihe 1 inch o 2 F inches. The heigh of he sack of 9 books is 14 inches. Which sysem of equaions can be used o deemine x, he numbe of
More informationSTUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE WEIBULL DISTRIBUTION
Inenaional Jounal of Science, Technology & Managemen Volume No 04, Special Issue No. 0, Mach 205 ISSN (online): 2394-537 STUDY OF THE STRESS-STRENGTH RELIABILITY AMONG THE PARAMETERS OF GENERALIZED INVERSE
More informationVS203B Lecture Notes Spring, 2011 Topic: Diffraction
Diffracion Diffracion escribes he enency for ligh o ben aroun corners. Huygens principle All poins on a wavefron can be consiere as poin sources for he proucion of seconary waveles, an a a laer ime he
More information1. Kinematics I: Position and Velocity
1. Kinemaics I: Posiion and Velociy Inroducion The purpose of his eperimen is o undersand and describe moion. We describe he moion of an objec by specifying is posiion, velociy, and acceleraion. In his
More informationPhysics 2001/2051 Moments of Inertia Experiment 1
Physics 001/051 Momens o Ineia Expeimen 1 Pelab 1 Read he ollowing backgound/seup and ensue you ae amilia wih he heoy equied o he expeimen. Please also ill in he missing equaions 5, 7 and 9. Backgound/Seup
More informationInventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions
Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.
More informationWORK POWER AND ENERGY Consevaive foce a) A foce is said o be consevaive if he wok done by i is independen of pah followed by he body b) Wok done by a consevaive foce fo a closed pah is zeo c) Wok done
More informationCombinatorial Approach to M/M/1 Queues. Using Hypergeometric Functions
Inenaional Mahemaical Foum, Vol 8, 03, no 0, 463-47 HIKARI Ld, wwwm-hikaicom Combinaoial Appoach o M/M/ Queues Using Hypegeomeic Funcions Jagdish Saan and Kamal Nain Depamen of Saisics, Univesiy of Delhi,
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationViewing in 3D. Viewing in 3D. Planar Geometric Projections. Taxonomy of Projections. How to specify which part of the 3D world is to be viewed?
Viewig i 3D Viewig i 3D How o speci which pa o he 3D wo is o e viewe? 3D viewig voume How o asom 3D wo cooiaes o D ispa cooiae? Pojecios Cocepua viewig pipeie: Xom o ee coos 3D cippig Pojec Xom o viewpo
More informationPHYS PRACTICE EXAM 2
PHYS 1800 PRACTICE EXAM Pa I Muliple Choice Quesions [ ps each] Diecions: Cicle he one alenaive ha bes complees he saemen o answes he quesion. Unless ohewise saed, assume ideal condiions (no ai esisance,
More informationA note on characterization related to distributional properties of random translation, contraction and dilation of generalized order statistics
PobSa Foum, Volume 6, July 213, Pages 35 41 ISSN 974-3235 PobSa Foum is an e-jounal. Fo eails please visi www.pobsa.og.in A noe on chaaceizaion elae o isibuional popeies of anom anslaion, conacion an ilaion
More informationMuch that has already been said about changes of variable relates to transformations between different coordinate systems.
MULTIPLE INTEGRLS I P Calculus Cooinate Sstems Much that has alea been sai about changes of vaiable elates to tansfomations between iffeent cooinate sstems. The main cooinate sstems use in the solution
More information( ) ( ) ( ) ( u) ( u) = are shown in Figure =, it is reasonable to speculate that. = cos u ) and the inside function ( ( t) du
Porlan Communiy College MTH 51 Lab Manual The Chain Rule Aciviy 38 The funcions f ( = sin ( an k( sin( 3 38.1. Since f ( cos( k ( = cos( 3. Bu his woul imply ha k ( f ( = are shown in Figure =, i is reasonable
More information() t. () t r () t or v. ( t) () () ( ) = ( ) or ( ) () () () t or dv () () Section 10.4 Motion in Space: Velocity and Acceleration
Secion 1.4 Moion in Spce: Velociy nd Acceleion We e going o dive lile deepe ino somehing we ve ledy inoduced, nmely () nd (). Discuss wih you neighbo he elionships beween posiion, velociy nd cceleion you
More informationExponential and Logarithmic Equations and Properties of Logarithms. Properties. Properties. log. Exponential. Logarithmic.
Eponenial and Logaihmic Equaions and Popeies of Logaihms Popeies Eponenial a a s = a +s a /a s = a -s (a ) s = a s a b = (ab) Logaihmic log s = log + logs log/s = log - logs log s = s log log a b = loga
More informationChapter 2 The Derivative Applied Calculus 107. We ll need a rule for finding the derivative of a product so we don t have to multiply everything out.
Chaper The Derivaive Applie Calculus 107 Secion 4: Prouc an Quoien Rules The basic rules will le us ackle simple funcions. Bu wha happens if we nee he erivaive of a combinaion of hese funcions? Eample
More informationChapter Finite Difference Method for Ordinary Differential Equations
Chape 8.7 Finie Diffeence Mehod fo Odinay Diffeenial Eqaions Afe eading his chape, yo shold be able o. Undesand wha he finie diffeence mehod is and how o se i o solve poblems. Wha is he finie diffeence
More informationDual-Quaternions. From Classical Mechanics to Computer Graphics and Beyond. Ben Kenwright
Dual-Quaenions: Fom Classical Mechanics o Compue Gaphics an eyon Dual-Quaenions Fom Classical Mechanics o Compue Gaphics an eyon en Kenwigh www.xbev.ne bkenwigh@xbev.ne bsac This pape pesens an oveview
More informationOn The Estimation of Two Missing Values in Randomized Complete Block Designs
Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.
More informationME 141. Engineering Mechanics
ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics
More informationMATHEMATICS PAPER 121/2 K.C.S.E QUESTIONS SECTION 1 ( 52 MARKS) 3. Simplify as far as possible, leaving your answer in the form of surd
f MATHEMATICS PAPER 2/2 K.C.S.E. 998 QUESTIONS CTION ( 52 MARKS) Answe he enie queion in his cion /5. U logaihms o evaluae 55.9 (262.77) e F 2. Simplify he epeson - 2 + 3 Hence solve he equaion - - 2 +
More informationSPHERICAL WINDS SPHERICAL ACCRETION
SPHERICAL WINDS SPHERICAL ACCRETION Spheical wins. Many sas ae known o loose mass. The sola win caies away abou 10 14 M y 1 of vey ho plasma. This ae is insignifican. In fac, sola aiaion caies away 4 10
More informationHomework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
More informationLecture 4 Kinetics of a particle Part 3: Impulse and Momentum
MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an
More informationReinforcement learning
Lecue 3 Reinfocemen leaning Milos Hauskech milos@cs.pi.edu 539 Senno Squae Reinfocemen leaning We wan o lean he conol policy: : X A We see examples of x (bu oupus a ae no given) Insead of a we ge a feedback
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationPhysics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
More informationFig. 1S. The antenna construction: (a) main geometrical parameters, (b) the wire support pillar and (c) the console link between wire and coaxial
a b c Fig. S. The anenna consucion: (a) ain geoeical paaees, (b) he wie suppo pilla and (c) he console link beween wie and coaial pobe. Fig. S. The anenna coss-secion in he y-z plane. Accoding o [], he
More informationAn Automatic Door Sensor Using Image Processing
An Auomaic Doo Senso Using Image Pocessing Depamen o Elecical and Eleconic Engineeing Faculy o Engineeing Tooi Univesiy MENDEL 2004 -Insiue o Auomaion and Compue Science- in BRNO CZECH REPUBLIC 1. Inoducion
More informationChapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
More informationChapter 15 Oscillatory Motion I
Chaper 15 Oscillaory Moion I Level : AP Physics Insrucor : Kim Inroducion A very special kind of moion occurs when he force acing on a body is proporional o he displacemen of he body from some equilibrium
More informationChapter Three Systems of Linear Differential Equations
Chaper Three Sysems of Linear Differenial Equaions In his chaper we are going o consier sysems of firs orer orinary ifferenial equaions. These are sysems of he form x a x a x a n x n x a x a x a n x n
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationOnline Completion of Ill-conditioned Low-Rank Matrices
Online Compleion of Ill-condiioned Low-Rank Maices Ryan Kennedy and Camillo J. Taylo Compue and Infomaion Science Univesiy of Pennsylvania Philadelphia, PA, USA keny, cjaylo}@cis.upenn.edu Laua Balzano
More informationCSE590B Lecture 4 More about P 1
SE590 Lece 4 Moe abo P 1 Tansfoming Tansfomaions James. linn Jimlinn.om h://coses.cs.washingon.ed/coses/cse590b/13a/ Peviosly On SE590b Tansfomaions M M w w w w w The ncion w w w w w w 0 w w 0 w 0 w The
More informationChapter 3 Kinematics in Two Dimensions
Chaper 3 KINEMATICS IN TWO DIMENSIONS PREVIEW Two-dimensional moion includes objecs which are moing in wo direcions a he same ime, such as a projecile, which has boh horizonal and erical moion. These wo
More informationOverview. Overview Page 1 of 8
COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIORNIA DECEMBER 2001 COMPOSITE BEAM DESIGN AISC-LRD93 Tecnical Noe Compac and Noncompac Requiemens Tis Tecnical Noe descibes o e pogam cecks e AISC-LRD93 specificaion
More informationME 210 Applied Mathematics for Mechanical Engineers
Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the
More information2. v = 3 4 c. 3. v = 4c. 5. v = 2 3 c. 6. v = 9. v = 4 3 c
Vesion 074 Exam Final Daf swinney (55185) 1 This pin-ou should have 30 quesions. Muliple-choice quesions may coninue on he nex column o page find all choices befoe answeing. 001 10.0 poins AballofmassM
More informationME 3560 Fluid Mechanics
ME3560 Flid Mechanics Fall 08 ME 3560 Flid Mechanics Analsis of Flid Flo Analsis of Flid Flo ME3560 Flid Mechanics Fall 08 6. Flid Elemen Kinemaics In geneal a flid paicle can ndego anslaion, linea defomaion
More information7 Wave Equation in Higher Dimensions
7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,
More informationKalman Filter: an instance of Bayes Filter. Kalman Filter: an instance of Bayes Filter. Kalman Filter. Linear dynamics with Gaussian noise
COM47 Inoducion o Roboics and Inelligen ysems he alman File alman File: an insance of Bayes File alman File: an insance of Bayes File Linea dynamics wih Gaussian noise alman File Linea dynamics wih Gaussian
More informationAuthors name Giuliano Bettini* Alberto Bicci** Title Equivalent waveguide representation for Dirac plane waves
Auhos name Giuliano Beini* Albeo Bicci** Tile Equivalen waveguide epesenaion fo Diac plane waves Absac Ideas abou he elecon as a so of a bound elecomagneic wave and/o he elecon as elecomagneic field apped
More informationLinear Motion I Physics
Linear Moion I Physics Objecives Describe he ifference beween isplacemen an isance Unersan he relaionship beween isance, velociy, an ime Describe he ifference beween velociy an spee Be able o inerpre a
More informationThe Arcsine Distribution
The Arcsine Disribuion Chris H. Rycrof Ocober 6, 006 A common heme of he class has been ha he saisics of single walker are ofen very differen from hose of an ensemble of walkers. On he firs homework, we
More informationThe Fundamental Theorems of Calculus
FunamenalTheorems.nb 1 The Funamenal Theorems of Calculus You have now been inrouce o he wo main branches of calculus: ifferenial calculus (which we inrouce wih he angen line problem) an inegral calculus
More informationChapter 3: Vectors and Two-Dimensional Motion
Chape 3: Vecos and Two-Dmensonal Moon Vecos: magnude and decon Negae o a eco: eese s decon Mulplng o ddng a eco b a scala Vecos n he same decon (eaed lke numbes) Geneal Veco Addon: Tangle mehod o addon
More informationThe fundamental mass balance equation is ( 1 ) where: I = inputs P = production O = outputs L = losses A = accumulation
Hea (iffusion) Equaion erivaion of iffusion Equaion The fundamenal mass balance equaion is I P O L A ( 1 ) where: I inpus P producion O oupus L losses A accumulaion Assume ha no chemical is produced or
More informationLAB # 2 - Equilibrium (static)
AB # - Equilibrium (saic) Inroducion Isaac Newon's conribuion o physics was o recognize ha despie he seeming compleiy of he Unierse, he moion of is pars is guided by surprisingly simple aws. Newon's inspiraion
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationOne-Dimensional Kinematics
One-Dimensional Kinemaics One dimensional kinemaics refers o moion along a sraigh line. Een hough we lie in a 3-dimension world, moion can ofen be absraced o a single dimension. We can also describe moion
More informationNEWTON S SECOND LAW OF MOTION
Course and Secion Dae Names NEWTON S SECOND LAW OF MOTION The acceleraion of an objec is defined as he rae of change of elociy. If he elociy changes by an amoun in a ime, hen he aerage acceleraion during
More informationThis is an example to show you how SMath can calculate the movement of kinematic mechanisms.
Dec :5:6 - Kinemaics model of Simple Arm.sm This file is provided for educaional purposes as guidance for he use of he sofware ool. I is no guaraeed o be free from errors or ommissions. The mehods and
More informationPhysics Notes - Ch. 2 Motion in One Dimension
Physics Noes - Ch. Moion in One Dimension I. The naure o physical quaniies: scalars and ecors A. Scalar quaniy ha describes only magniude (how much), NOT including direcion; e. mass, emperaure, ime, olume,
More informationKinematics in two dimensions
Lecure 5 Phsics I 9.18.13 Kinemaics in wo dimensions Course websie: hp://facul.uml.edu/andri_danlo/teaching/phsicsi Lecure Capure: hp://echo36.uml.edu/danlo13/phsics1fall.hml 95.141, Fall 13, Lecure 5
More informationLecture-V Stochastic Processes and the Basic Term-Structure Equation 1 Stochastic Processes Any variable whose value changes over time in an uncertain
Lecue-V Sochasic Pocesses and he Basic Tem-Sucue Equaion 1 Sochasic Pocesses Any vaiable whose value changes ove ime in an unceain way is called a Sochasic Pocess. Sochasic Pocesses can be classied as
More informationA GENERAL METHOD TO STUDY THE MOTION IN A NON-INERTIAL REFERENCE FRAME
The Inenaional onfeence on opuaional Mechanics an Viual Engineeing OME 9 9 OTOBER 9, Basov, Roania A GENERAL METHOD TO STUDY THE MOTION IN A NON-INERTIAL REFERENE FRAME Daniel onuache, Vlaii Mainusi Technical
More informationWeek 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
More informationOPTIMIZATION OF TOW-PLACED, TAILORED COMPOSITE LAMINATES
6 H INERNAIONAL CONFERENCE ON COMPOSIE MAERIALS OPIMIZAION OF OW-PLACED AILORED COMPOSIE LAMINAES Adiana W. Blom* Mosafa M. Abdalla* Zafe Güdal* *Delf Univesi of echnolog he Nehelands Kewods: vaiable siffness
More informationAn Open cycle and Closed cycle Gas Turbine Engines. Methods to improve the performance of simple gas turbine plants
An Open cycle and losed cycle Gas ubine Engines Mehods o impove he pefomance of simple gas ubine plans I egeneaive Gas ubine ycle: he empeaue of he exhaus gases in a simple gas ubine is highe han he empeaue
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationNUMERICAL SIMULATION FOR NONLINEAR STATIC & DYNAMIC STRUCTURAL ANALYSIS
Join Inenaional Confeence on Compuing and Decision Making in Civil and Building Engineeing June 14-16, 26 - Monéal, Canada NUMERICAL SIMULATION FOR NONLINEAR STATIC & DYNAMIC STRUCTURAL ANALYSIS ABSTRACT
More informationServomechanism Design
Sevomechanism Design Sevomechanism (sevo-sysem) is a conol sysem in which he efeence () (age, Se poin) changes as ime passes. Design mehods PID Conol u () Ke P () + K I ed () + KDe () Sae Feedback u()
More information