CSE 5365 Computer Graphics. Take Home Test #1
|
|
- Doreen Barnett
- 6 years ago
- Views:
Transcription
1 CSE 5365 Comper Graphics Take Home Tes #1 Fall/1996 Tae-Hoon Kim
2 roblem #1) A bi-cbic parameric srface is defined by Hermie geomery in he direcion of parameer. In he direcion, he geomery ecor is defined by a a a angen and a second order angen a) Find he geomery mari [G] for he gien srface. Solion) arameric cbic cre for one coordinae in direcion for a gien of he srface (,) is epressed as follows. (,) [G 1 () G () G 3 () G 4 ()]MV (1) where V is a colmn ecor and a ranspose of a row ecor [ 3 1]. i, j (,) Assme, ij (,){i,j}, {i,j}, i, j (,) and {i,j}, ec.---- Since he geomery ecor for a gien in direcion is defined by a a a angen and a second order angen we can epress G 1 (), G (), G 3 (), and G 4 () as follows. G 1 (),0 ; G (),0.5 ; G 3 (), 1 G 4 (), Since G 1 (), G (), G 3 (), and G 4 () are epressed in Hermie geomery in direcions, hey can be epressed as follows. 1 ; G 1 (),0 0,0 1,0 0,0 1,0 MH U 0,0 1,0 H 0,0 1,0 U M () where U is a row ecor [ 3 1]. Similarly, oher erms are epressed as follows.
3 G (),0.5 U MH 0,0.5 1,0.5 0,0.5 1,0.5 (3) G 3 (), 1 U MH 0,1 1,1 (4) G 4 (), 1 U MH 3 3 (5) Enering (3), (4), and (5) ino (1), we ge he following geomery mari. G 0,0 0,0.5 0,1 0,1 1,0 1,0.5 1,1 1,1 0,0 0,0.5 0,1 3 0,1 1,0 1,0.5 1,1 3 1,1 (6) b) If he eqaion of his srface is gien by (,) ; y(,) ; z(,) ; Find he nmerical ales for he [G ] and [G y ] for his srface. Solion) Le s firs find he differenial epressions for and y coordinaes ; ; ; ;
4 ; (7) y ; y ; y 3 y ; y ; ; (8) From he se of eqaions in (7) and (8) and sing he epression for G in (6), we obain G and G y marices wih nmerical ales as follows. G (0,0) (1,0) (0,0) (1,0) (0,0.5) (1,0.5) (0,0.5) (1,0.5) (0,1) (0,1) (,) 11 (,) 11 3 (0,1) (0,1) 3 (,) 11 (,) Similarly, G y roblem #) Viewing olme in a perspecie projecion is defined as View plane: -0; Fron plane: -80; Back plane: -180;
5 Side planes: y-80, -y0; Top plane: -4z+40; Boom plane: z-30; Assming a righ handed coordinae sysem, find he seqence of ransformaions which will ransform his iewing olme ino a sandard perspecie iew olme. Solion) From he inersecion of wo side planes and a op plane we obain he R before ransformaion as (8,8,3). We also ge he inersecion poins of he back plane and he wo side planes, a op plane, and a boom plane as follows. 1 and are locaed a he op plane and 3 and 4 are locaed a he boom plane. 1 (18,8,5.5); (18,18,5.5); 3 (18,18,3); 4 (18,8,3); (1) From he iew plane eqaion and assming ha VN is direced from R o he back plane, we ge he VN as [ ] in homogeneos coordinae sysem. From he inersecion poins of planes, sing he direcion from he boom plane poin (18,8,3) o op plane poin (18,8,5.5) ha are on he same side plane, we ge VU as [ ]. From R, we ge he firs ransformaion ranslaion mari as follows. T () Since VN coincides wih ais, we don hae o roae wih respec o ais. From VN, we ge he second ransformaion roaion mari wih respec o y ais as follows. The roaion angle β here is 70 coner clockwise or 90 clockwise. Ths, R cos β 0 sin β sin β 0 cos β (3) Applying R roaion mari o VU, we ge he new VU as follows. VU R[ ] [ ] (4) From his VU, we obain he hird ransformaion roaion mari wih respec o z ais as follows. The roaion angle γ is 70. Ths,
6 cosγ sinγ 0 0 sinγ cosγ 0 0 R (5) Afer he series of ransformaions T1, R, and R3, he poins 1,, 3, and 4 in (1) are ransformed o be he following poins 1,, 3, and 4 respeciely. 1 [ ]; [ ]; 3 [ ]; 4 [ ] Now we ge he cener of aboe poins 1,, 3, and 4 as CW [ ] From CW we can ge he 4 h ransformaion shear mari as follows; Sh / / (6) From 1 and 3 which are diagonally locaed poins in he ransformed back plane, we ge he following seqence of scale ransformaions. Sc5 is o render he iew olme heigh o be eqal o z ais coordinae of he back plane. Sc6 is o scale he iew olme niformly in all hree aes direcions sch ha he back clipping plane becomes he z 1 plane. In he following eqaions 1 represens componen of 1, 1y represens y componen of 1, ec. Sc5 1z z y - 3y /( 5. / ) (7)
7 1/ 1z / / 1z 0 0 Now, Sc6 0 1/ (8) 0 0 1/ 1z / If I combine all he ransformaion marices ino one ransformaion mari, he following form of combined ransformaion mari resls T Sc6 Sc5 Sh4 R3 R T
8 roblem #3) A cred srface is cbic in he direcion and qadric in he direcion. The parameric eqaions of he cres corresponding o 0,., and 1 are gien as; () ; y() ; z() ; when 0 (1) () ; y() ; z() ; when. () () ; y() ; z() ; when 1 ; (3) a) Find he coefficien mari C, C y, and C z for he gien srface. Solion) Using C, we can epress as follows; c11 c1 c13 c1 c c3 [ 3 1] c31 c3 c33 1 c41 c4 c43 (4) Eqaions in (1), (), and (3) proides 1 eqaions for 1 nknowns c11 c1 -- c43 in (4). Coefficiens for from eqaions in (1), (), and (3) for 0,., and 1 and (4) can be organized as follows o sole for 1 nknowns in (4) c11 c1 c13 [ c1 c c ] [ 3 1] c31 c3 c c41 c4 c43 (5) From (5) we can sole for C as follows. c11 c1 c13 c1 c c3 C c31 c3 c33 c41 c4 c Similarly, we can ge Cy and Cz as follows.
9 cy11 cy1 cy13 cy1 cy cy3 Cy cy31 cy3 cy33 cy41 cy4 cy43 cz11 cz1 cz13 cz1 cz cz3 Cz cz31 cz3 cz33 cz41 cz4 cz b) Find he geomery mari of his srface if he direcion is assmed o hae Hermie geomery and he geomery ecor in he direcion is defined by and a angen ecor o he Solion) Following he similar seps in rob #1), we ge he geomery mari as follows; arameric qadric cre in direcion for a gien for a srface (,) is gien as he following form. (,) [G 1 () G () G 3 ()]MV (6) where V is a colmn ecor and is a ranspose of a row ecor [ 1]. i, j (,) Assme, ij (,){i,j}, {i,j}, i, j (,) and {i,j}, ec.---- Since he geomery ecor for a gien in direcion is defined by a a and a angen we can epress G 1 (), G (), and G 3 () as follows. G 1 (),0 ; G (),1 ;, G 3 () 1 (7) Le s find he mari M ha ransforms he geomery ecor o qadric cre coefficiens of V. We can ge M by sing hree consrains on (,) when 0 and 1 as in (7). Ths,
10 ,0,1, [G 1() G () G 3 ()]M (8) From (8), we ge M as Since G 1 (), G (), and G 3 () are epressed in Hermie geomery in direcions, hey can be epressed as follows. G 1 (),0 G (),1 G 3 (), 0,0 1,0 H 0,0 1,0 U M U MH 1 U MH (9) (10) (11) Enering (9), (10), and (11) ino (8), we ge he following geomery mari. G 0,0 0,1 1,0 1,1 0,0 0,1 1,0 1,1 0,1 1,1 0,1 1,1 (1) Now we know ha he parameric srface can be epressed as follows;
11 (,) [ 3 1] M H GM[ 1] (13) From aboe, we know ha G (M H ) -1 CM -1 (14) From (14), we ge he following nmerically aled geomery marices G, G y, and G z. G C Gy (M H ) -1 C y M Gz (M H ) -1 C z M c) Find he normal o his Solion) We can ge he angen ecors in he direcions of increasing and as follows. (,) [3 1 0] C [ 1] [3 1 0] Cy [ 1] {1,.5} [3 1 0] Cz [ 1] (,) 3 [ 1] C [ 1 0] 3 [ 1] Cy [ 1 0] 3 [ 1] Cz [ 1 0] {1,.5} 5 1 Firs sep o ge he srface normal is o obain he cross prodc of aboe wo ecors. The ecor from he cross prodc is;
12 N1 (,) (,) Normalized ersion of N1 is he srface normal. The resl is; N
CSE-4303/CSE-5365 Computer Graphics Fall 1996 Take home Test
Comper Graphics roblem #1) A bi-cbic parameric srface is defined by Hermie geomery in he direcion of parameer. In he direcion, he geomery ecor is defined by a poin @0, a poin @0.5, a angen ecor @1 and
More information3D Coordinate Systems. 3D Geometric Transformation Chapt. 5 in FVD, Chapt. 11 in Hearn & Baker. Right-handed coordinate system:
3D Geomeric ransformaion Chap. 5 in FVD, Chap. in Hearn & Baker 3D Coordinae Ssems Righ-handed coordinae ssem: Lef-handed coordinae ssem: 2 Reminder: Vecor rodc U V UV VU sin ˆ V nu V U V U ˆ ˆ ˆ 3D oin
More informationElements of Computer Graphics
CS580: Compuer Graphics Min H. Kim KAIST School of Compuing Elemens of Compuer Graphics Geomery Maerial model Ligh Rendering Virual phoography 2 Foundaions of Compuer Graphics A PINHOLE CAMERA IN 3D 3
More informationMethod of Moment Area Equations
Noe proided b JRR Page-1 Noe proided b JRR Page- Inrodcion ehod of omen rea qaions Perform deformaion analsis of flere-dominaed srcres eams Frames asic ssmpions (on.) No aial deformaion (aiall rigid members)
More informationTHE DARBOUX TRIHEDRONS OF REGULAR CURVES ON A REGULAR TIME-LIKE SURFACE. Emin Özyilmaz
Mahemaical and Compaional Applicaions, Vol. 9, o., pp. 7-8, 04 THE DARBOUX TRIHEDROS OF REULAR CURVES O A REULAR TIME-LIKE SURFACE Emin Özyilmaz Deparmen of Mahemaics, Facly of Science, Ee Uniersiy, TR-500
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
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 informationMat 267 Engineering Calculus III Updated on 04/30/ x 4y 4z 8x 16y / 4 0. x y z x y. 4x 4y 4z 24x 16y 8z.
Ma 67 Engineering Calcls III Updaed on 04/0/0 r. Firoz Tes solion:. a) Find he cener and radis of he sphere 4 4 4z 8 6 0 z ( ) ( ) z / 4 The cener is a (, -, 0), and radis b) Find he cener and radis of
More informationIntegration of the equation of motion with respect to time rather than displacement leads to the equations of impulse and momentum.
Inegraion of he equaion of moion wih respec o ime raher han displacemen leads o he equaions of impulse and momenum. These equaions greal faciliae he soluion of man problems in which he applied forces ac
More informationHYPOTHESIS TESTING. four steps. 1. State the hypothesis. 2. Set the criterion for rejecting. 3. Compute the test statistics. 4. Interpret the results.
Inrodcion o Saisics in Psychology PSY Professor Greg Francis Lecre 23 Hypohesis esing for correlaions Is here a correlaion beween homework and exam grades? for seps. Sae he hypohesis. 2. Se he crierion
More informationSurfaces in the space E
3 Srfaces in he sace E Le ecor fncion in wo ariables be efine on region R = x y y whose scalar coorinae fncions x y y are a leas once iffereniable on region. Hoograh of ecor fncion is a iece-wise smooh
More informationME 425: Aerodynamics
ME 45: Aerodnamics Dr. A.B.M. Toiqe Hasan Proessor Deparmen o Mechanical Engineering Bangladesh Uniersi o Engineering & Technolog BUET, Dhaka Lecre-7 Fndamenals so Aerodnamics oiqehasan.be.ac.bd oiqehasan@me.be.ac.bd
More informationThe motions of the celt on a horizontal plane with viscous friction
The h Join Inernaional Conference on Mulibody Sysem Dynamics June 8, 18, Lisboa, Porugal The moions of he cel on a horizonal plane wih viscous fricion Maria A. Munisyna 1 1 Moscow Insiue of Physics and
More informationHYPOTHESIS TESTING. four steps. 1. State the hypothesis and the criterion. 2. Compute the test statistic. 3. Compute the p-value. 4.
Inrodcion o Saisics in Psychology PSY Professor Greg Francis Lecre 24 Hypohesis esing for correlaions Is here a correlaion beween homework and exam grades? for seps. Sae he hypohesis and he crierion 2.
More informationScalar Conservation Laws
MATH-459 Nmerical Mehods for Conservaion Laws by Prof. Jan S. Heshaven Solion se : Scalar Conservaion Laws Eercise. The inegral form of he scalar conservaion law + f ) = is given in Eq. below. ˆ 2, 2 )
More information3, so θ = arccos
Mahemaics 210 Professor Alan H Sein Monday, Ocober 1, 2007 SOLUTIONS This problem se is worh 50 poins 1 Find he angle beween he vecors (2, 7, 3) and (5, 2, 4) Soluion: Le θ be he angle (2, 7, 3) (5, 2,
More informationAP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationEarthquake, Volcano and Tsunami
A. Merapi Volcano Erpion Earhqake, Volcano and Tsnami Qesion Answer Marks A. Using Black s Principle he eqilibrim emperare can be obained Ths,.5 A. For ideal gas, pv e e RTe, hs.3 A.3 The relaive velociy
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationBasilio Bona ROBOTICA 03CFIOR 1
Indusrial Robos Kinemaics 1 Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables
More informationDifferential Geometry: Revisiting Curvatures
Differenial Geomery: Reisiing Curaures Curaure and Graphs Recall: hus, up o a roaion in he x-y plane, we hae: f 1 ( x, y) x y he alues 1 and are he principal curaures a p and he corresponding direcions
More information4.2 Continuous-Time Systems and Processes Problem Definition Let the state variable representation of a linear system be
4 COVARIANCE ROAGAION 41 Inrodcion Now ha we have compleed or review of linear sysems and random processes, we wan o eamine he performance of linear sysems ecied by random processes he sandard approach
More informationy z P 3 P T P1 P 2. Werner Purgathofer. b a
Einführung in Viual Compuing Einführung in Viual Compuing 86.822 in co T P 3 P co in T P P 2 co in Geomeric Tranformaion Geomeric Tranformaion W P h f Werner Purgahofer b a Tranformaion in he Rendering
More information1 First Order Partial Differential Equations
Firs Order Parial Differenial Eqaions The profond sdy of nare is he mos ferile sorce of mahemaical discoveries. - Joseph Forier (768-830). Inrodcion We begin or sdy of parial differenial eqaions wih firs
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
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 informationUnit 1 Test Review Physics Basics, Movement, and Vectors Chapters 1-3
A.P. Physics B Uni 1 Tes Reiew Physics Basics, Moemen, and Vecors Chapers 1-3 * In sudying for your es, make sure o sudy his reiew shee along wih your quizzes and homework assignmens. Muliple Choice Reiew:
More informationHW6: MRI Imaging Pulse Sequences (7 Problems for 100 pts)
HW6: MRI Imaging Pulse Sequences (7 Problems for 100 ps) GOAL The overall goal of HW6 is o beer undersand pulse sequences for MRI image reconsrucion. OBJECTIVES 1) Design a spin echo pulse sequence o image
More informationDISPLACEMENT ESTIMATION FOR IMAGE PREDICTIVE CODING AND FRAME MOTION-ADAPTIVE INTERPOLATION
DSPLACEMENT ESTMATON FOR MAGE PREDCTVE CODNG AND FRAME MOTON-ADAPTVE NTERPOLATON Georges TZRTAS Laboraoire des Signa e Ssèmes (C.N.R.S.), Ecole Spériere d'elecricié, Plaea d Molon 9119 GF-sr-YVETTE, FRANCE
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 information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More information, u denotes uxt (,) and u. mean first partial derivatives of u with respect to x and t, respectively. Equation (1.1) can be simply written as
Proceedings of he rd IMT-GT Regional Conference on Mahemaics Saisics and Applicaions Universii Sains Malaysia ANALYSIS ON () + () () = G( ( ) ()) Jessada Tanhanch School of Mahemaics Insie of Science Sranaree
More informationVelocity is a relative quantity
Veloci is a relaie quani Disenangling Coordinaes PHY2053, Fall 2013, Lecure 6 Newon s Laws 2 PHY2053, Fall 2013, Lecure 6 Newon s Laws 3 R. Field 9/6/2012 Uniersi of Florida PHY 2053 Page 8 Reference Frames
More informationDifferential Geometry: Numerical Integration and Surface Flow
Differenial Geomery: Numerical Inegraion and Surface Flow [Implici Fairing of Irregular Meshes using Diffusion and Curaure Flow. Desbrun e al., 1999] Energy Minimizaion Recall: We hae been considering
More informationPhys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole
Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1 Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationConcourse Math Spring 2012 Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations
Concourse Mah 80 Spring 0 Worked Examples: Marix Mehods for Solving Sysems of s Order Linear Differenial Equaions The Main Idea: Given a sysem of s order linear differenial equaions d x d Ax wih iniial
More informationSolutions for homework 12
y Soluions for homework Secion Nonlinear sysems: The linearizaion of a nonlinear sysem Consider he sysem y y y y y (i) Skech he nullclines Use a disincive marking for each nullcline so hey can be disinguished
More informationRoller-Coaster Coordinate System
Winer 200 MECH 220: Mechanics 2 Roller-Coaser Coordinae Sysem Imagine you are riding on a roller-coaer in which he rack goes up and down, wiss and urns. Your velociy and acceleraion will change (quie abruply),
More informationAsymptotic Solution of the Anti-Plane Problem for a Two-Dimensional Lattice
Asympoic Solion of he Ani-Plane Problem for a Two-Dimensional Laice N.I. Aleksandrova N.A. Chinakal Insie of Mining, Siberian Branch, Rssian Academy of Sciences, Krasnyi pr. 91, Novosibirsk, 6391 Rssia,
More informationTopology of the Intersection of Two Parameterized Surfaces, Using Computations in 4D Space
Topology of he Inersecion of Two Parameerized Srfaces, Using Compaions in 4D Space Séphane Cha, André Galligo To cie his ersion: Séphane Cha, André Galligo. Topology of he Inersecion of Two Parameerized
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 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 informationCoE4TN3 Image Processing
CoE4T3 Imae Processin Imae Reisraion Imae Reisraion Throuh Transform A B f Imae reisraion provides ransformaion of a source imae space o he are imae space. The are imae ma be of differen modaliies from
More informationMOMENTUM CONSERVATION LAW
1 AAST/AEDT AP PHYSICS B: Impulse and Momenum Le us run an experimen: The ball is moving wih a velociy of V o and a force of F is applied on i for he ime inerval of. As he resul he ball s velociy changes
More informationSolution of Integro-Differential Equations by Using ELzaki Transform
Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp://www.irphouse.com Soluion of Inegro-Differenial Equaions by Using ELzaki Transform
More informationCheck in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2
7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions
More informationa. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.
Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A.
More informationKinematics and kinematic functions
Kinemaics and kinemaic funcions Kinemaics deals wih he sudy of four funcions (called kinemaic funcions or KFs) ha mahemaically ransform join variables ino caresian variables and vice versa Direc Posiion
More informationAnswers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
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 informationSolutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
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 informationNumerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
More informationExponential and Logarithmic Functions -- ANSWERS -- Logarithms Practice Diploma ANSWERS 1
Eponenial and Logarihmic Funcions -- ANSWERS -- Logarihms racice Diploma ANSWERS www.puremah.com Logarihms Diploma Syle racice Eam Answers. C. D 9. A 7. C. A. C. B 8. D. D. C NR. 8 9. C 4. C NR. NR 6.
More informationKinematics of Wheeled Robots
1 Kinemaics of Wheeled Robos hps://www.ouube.com/wach?=gis41ujlbu 2 Wheeled Mobile Robos robo can hae one or more wheels ha can proide seering direcional conrol power eer a force agains he ground an ideal
More informationDetecting Movement SINA 07/08
Deecing Moemen How do we perceie moemen? This is no a simple qesion becase we are neer saionar obserers (ees and head moe An imporan isse is how we discriminae he moion of he eernal world from he moion
More informationLogarithms Practice Exam - ANSWERS
Logarihms racice Eam - ANSWERS Answers. C. D 9. A 9. D. A. C. B. B. D. C. B. B. C NR.. C. B. B. B. B 6. D. C NR. 9. NR. NR... C 7. B. C. B. C 6. C 6. C NR.. 7. B 7. D 9. A. D. C Each muliple choice & numeric
More informationLecture 10: Wave equation, solution by spherical means
Lecure : Wave equaion, soluion by spherical means Physical modeling eample: Elasodynamics u (; ) displacemen vecor in elasic body occupying a domain U R n, U, The posiion of he maerial poin siing a U in
More informationKinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.
Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages
More informationPH2130 Mathematical Methods Lab 3. z x
PH130 Mahemaical Mehods Lab 3 This scrip shold keep yo bsy for he ne wo weeks. Yo shold aim o creae a idy and well-srcred Mahemaica Noebook. Do inclde plenifl annoaions o show ha yo know wha yo are doing,
More informationA Direct Method for Solving Nonlinear PDEs and. New Exact Solutions for Some Examples
In. J. Conemp. Mah. Sciences, Vol. 6, 011, no. 46, 83-90 A Direc Mehod for Solving Nonlinear PDEs and New Eac Solions for Some Eamples Ameina S. Nseir Jordan Universiy of Science and Technology Deparmen
More informationA Mathematical model to Solve Reaction Diffusion Equation using Differential Transformation Method
Inernaional Jornal of Mahemaics Trends and Technology- Volme Isse- A Mahemaical model o Solve Reacion Diffsion Eqaion sing Differenial Transformaion Mehod Rahl Bhadaria # A.K. Singh * D.P Singh # #Deparmen
More informationKinematics: Motion in One Dimension
Kinemaics: Moion in One Dimension When yo drie, yo are spposed o follow he hree-second ailgaing rle. When he car in fron of yo passes a sign a he side of he road, yor car shold be far enogh behind i ha
More information10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have
More informationCLASS XI SET A PHYSICS. 1. If and Let. The correct order of % error in. (a) (b) x = y > z (c) x < z < y (d) x > z < y
PHYSICS 1. If and Le. The correc order of % error in (a) (b) x = y > z x < z < y x > z < y. A hollow verical cylinder of radius r and heigh h has a smooh inernal surface. A small paricle is placed in conac
More informationMotion along a Straight Line
chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)
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 informationfirst-order circuit Complete response can be regarded as the superposition of zero-input response and zero-state response.
Experimen 4:he Sdies of ransiional processes of 1. Prpose firs-order circi a) Use he oscilloscope o observe he ransiional processes of firs-order circi. b) Use he oscilloscope o measre he ime consan of
More informationFinite Element Analysis of Structures
KAIT OE5 Finie Elemen Analysis of rucures Mid-erm Exam, Fall 9 (p) m. As shown in Fig., we model a russ srucure of uniform area (lengh, Area Am ) subjeced o a uniform body force ( f B e x N / m ) using
More informationand v y . The changes occur, respectively, because of the acceleration components a x and a y
Week 3 Reciaion: Chaper3 : Problems: 1, 16, 9, 37, 41, 71. 1. A spacecraf is raveling wih a veloci of v0 = 5480 m/s along he + direcion. Two engines are urned on for a ime of 84 s. One engine gives he
More informationThe average rate of change between two points on a function is d t
SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope
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 informationEquations of motion for constant acceleration
Lecure 3 Chaper 2 Physics I 01.29.2014 Equaions of moion for consan acceleraion Course websie: hp://faculy.uml.edu/andriy_danylo/teaching/physicsi Lecure Capure: hp://echo360.uml.edu/danylo2013/physics1spring.hml
More informationMath 315: Linear Algebra Solutions to Assignment 6
Mah 35: Linear Algebra s o Assignmen 6 # Which of he following ses of vecors are bases for R 2? {2,, 3, }, {4,, 7, 8}, {,,, 3}, {3, 9, 4, 2}. Explain your answer. To generae he whole R 2, wo linearly independen
More informationThe Paradox of Twins Described in a Three-dimensional Space-time Frame
The Paradox of Twins Described in a Three-dimensional Space-ime Frame Tower Chen E_mail: chen@uguam.uog.edu Division of Mahemaical Sciences Universiy of Guam, USA Zeon Chen E_mail: zeon_chen@yahoo.com
More informationSuggested Practice Problems (set #2) for the Physics Placement Test
Deparmen of Physics College of Ars and Sciences American Universiy of Sharjah (AUS) Fall 014 Suggesed Pracice Problems (se #) for he Physics Placemen Tes This documen conains 5 suggesed problems ha are
More informationDistance Between Two Ellipses in 3D
Disance Beween Two Ellipses in 3D David Eberly Magic Sofware 6006 Meadow Run Cour Chapel Hill, NC 27516 eberly@magic-sofware.com 1 Inroducion An ellipse in 3D is represened by a cener C, uni lengh axes
More informationLearning from a Golf Ball
Session 1566 Learning from a Golf Ball Alireza Mohammadzadeh Padnos School of Engineering Grand Valley Sae Uniersiy Oeriew Projecile moion of objecs, in he absence of air fricion, is sdied in dynamics
More informationECE 2100 Circuit Analysis
ECE 1 Circui Analysis Lesson 35 Chaper 8: Second Order Circuis Daniel M. Liynski, Ph.D. ECE 1 Circui Analysis Lesson 3-34 Chaper 7: Firs Order Circuis (Naural response RC & RL circuis, Singulariy funcions,
More informationContent-Based Shape Retrieval Using Different Shape Descriptors: A Comparative Study Dengsheng Zhang and Guojun Lu
Conen-Based Shape Rerieval Using Differen Shape Descripors: A Comparaive Sudy Dengsheng Zhang and Guojun Lu Gippsland School of Compuing and Informaion Technology Monash Universiy Churchill, Vicoria 3842
More informationEECS20n, Solution to Midterm 2, 11/17/00
EECS20n, Soluion o Miderm 2, /7/00. 0 poins Wrie he following in Caresian coordinaes (i.e. in he form x + jy) (a) 2 poins j 3 j 2 + j += j ++j +=2 (b) 2 poins ( j)/( + j) = j (c) 2 poins cos π/4+jsin π/4
More informationThen. 1 The eigenvalues of A are inside R = n i=1 R i. 2 Union of any k circles not intersecting the other (n k)
Ger sgorin Circle Chaper 9 Approimaing Eigenvalues Per-Olof Persson persson@berkeley.edu Deparmen of Mahemaics Universiy of California, Berkeley Mah 128B Numerical Analysis (Ger sgorin Circle) Le A be
More informationSequences Arising From Prudent Self-Avoiding Walks Shanzhen Gao, Heinrich Niederhausen Florida Atlantic University, Boca Raton, Florida 33431
Seqences Arising From Prden Self-Avoiding Walks Shanzhen Gao, Heinrich Niederhasen Florida Alanic Universiy, Boca Raon, Florida 33431 Absrac A self-avoiding walk (SAW) is a seqence of moves on a laice
More informatione 2t u(t) e 2t u(t) =?
EE : Signals, Sysems, and Transforms Fall 7. Skech he convoluion of he following wo signals. Tes No noes, closed book. f() Show your work. Simplify your answers. g(). Using he convoluion inegral, find
More informationAQA Maths M2. Topic Questions from Papers. Differential Equations. Answers
AQA Mahs M Topic Quesions from Papers Differenial Equaions Answers PhysicsAndMahsTuor.com Q Soluion Marks Toal Commens M 600 0 = A Applying Newonís second law wih 0 and. Correc equaion = 0 dm Separaing
More informationLecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.
Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in
More information4 3 a b (C) (a 2b) (D) (2a 3b)
* A balloon is moving verically pwards wih a velociy of 9 m/s. A sone is dropped from i and i reaches he grond in 10 sec. The heigh of he balloon when he sone was dropped is (ake g = 9.8 ms - ) (a) 100
More informationChapter 12: Velocity, acceleration, and forces
To Feel a Force Chaper Spring, Chaper : A. Saes of moion For moion on or near he surface of he earh, i is naural o measure moion wih respec o objecs fixed o he earh. The 4 hr. roaion of he earh has a measurable
More informationOutline of Topics. Analysis of ODE models with MATLAB. What will we learn from this lecture. Aim of analysis: Why such analysis matters?
of Topics wih MATLAB Shan He School for Compuaional Science Universi of Birmingham Module 6-3836: Compuaional Modelling wih MATLAB Wha will we learn from his lecure Aim of analsis: Aim of analsis. Some
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationMiscellanea Miscellanea
Miscellanea Miscellanea Miscellanea Miscellanea Miscellanea CENRAL EUROPEAN REVIEW OF ECONOMICS & FINANCE Vol., No. (4) pp. -6 bigniew Śleszński USING BORDERED MARICES FOR DURBIN WASON D SAISIC EVALUAION
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationThe role of the error function in three-dimensional singularly perturbed convection-diffusion problems with discontinuous data
The role of he error funcion in hree-dimensional singularl perurbed convecion-diffusion problems wih disconinuous daa José Luis López García, Eser Pérez Sinusía Depo. de Maemáica e Informáica, U. Pública
More informationAP CALCULUS BC 2016 SCORING GUIDELINES
6 SCORING GUIDELINES Quesion A ime, he posiion of a paricle moving in he xy-plane is given by he parameric funcions ( x ( ), y ( )), where = + sin ( ). The graph of y, consising of hree line segmens, is
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More informationBEng (Hons) Telecommunications. Examinations for / Semester 2
BEng (Hons) Telecommunicaions Cohor: BTEL/14/FT Examinaions for 2015-2016 / Semeser 2 MODULE: ELECTROMAGNETIC THEORY MODULE CODE: ASE2103 Duraion: 2 ½ Hours Insrucions o Candidaes: 1. Answer ALL 4 (FOUR)
More information