Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3


 Bertina Fox
 1 years ago
 Views:
Transcription
1 Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each point in the pictue. The pupose of this lectue is to explain how to epesent colo and intensity numeically and then to develop illumination models that allow us to compute the colo and intensity at each point once we know the colo, location, and intensity of the light souces and the physical chaacteistics of the objects in the scene. 2. The RGB Colo Model Colo foms a 3dimensional vecto space. The thee pimay colos  ed, geen, and blue  ae a basis fo this vecto space: evey colo can be epesented as a linea combination of ed, geen, and blue. The unit cube is often used to epesent colo space (see Figue 1). Black is located at one cone of the cube usually associated with the oigin of the coodinate system, and the thee pimay colos ae placed along the thee othogonal axes. Intensity vaies along the edges of the cube, with the full intensity of each pimay colo coesponding to a unit distance along the associated edge. Evey combination of colo and intensity is then epesented by some linea combination of ed, geen, and blue, whee the coefficients of each pimay lie between zeo and one. Thus fo each colo c thee is a unique set of coodinates (,g,b) inside the unit cube that epesents the colo c. The numeical values (,g,b) epesent intensities: the highe the value of o g o b, the geate the contibution of ed o geen o blue to the colo c. This model is known as the RGB colo model and is one of the most common colo models in Compute Gaphics. blue cyan geen magenta white yellow black ed Figue 1: The RGB colo cube. Evey colo and intensity is epesented by a linea combination of the ed, geen, and blue, whee the coefficients of each pimay colo lie between zeo and one.
2 Pais of pimay colos combine to fom colos complementay to the missing pimay. Thus ed+geen=yellow, which is the colo complementay to blue. Similaly, blue+geen=cyan, which is the colo complementay to ed, and blue+ed=magenta, which is the colo complementay to geen. Combining all thee pimaies at full intensity yields white=ed+geen+blue, which lies at the cone of the cube diagonally opposite to black. Shades of gay ae epesented along the diagonal of the cube joining black to white. In the emainde of this lectue we shall develop illumination models. These illumination models allow us to ende a scene by computing the (,g,b) colo intensities fo each suface point once we know the light souces and the physical chaacteistics of the objects in the scene. We shall conside thee illumination models: ambient light, diffuse eflections, and specula highlights. 3. Ambient Light Ambient light is light that is eflected into a scene off outside sufaces. Fo example, sunlight that entes a oom though a window by bouncing off neaby buildings is ambient light. Ambient light softens hash shadows geneated by point light souces. Thus ambient light helps to make scenes endeed by Compute Gaphics appea moe natual. Typically we assume that the intensity I a of the ambient light in a scene is a constant. What we need to compute is the intensity I of the ambient light eflected to a viewe fom each suface point in the scene. The fomula fo this intensity I is simply I = k a I a, (3.1) whee popety of the colo and mateial of the suface, which can be detemined expeimentally o simply set by the pogamme. Notice that the intensity I is independent of the position of the viewe. set 0 k a 1 is the ambient eflection coefficient. The ambient eflection coefficient k a is a Equation (3.1) is eally thee equations, one fo each pimay colo ed, geen, and blue. If we I a = (I a,ia g,ia b ) and k a = (k a,ka,ka ), whee (I a,ia,ia) ae the ambient intensities fo ed, geen and blue, and (k a,ka,ka ) ae the ambient eflection coefficients fo ed, geen, and blue, then Equation (3.1) becomes g g b b I = (,g,b) = (k aia,ka Ia,kaIa). (3.2) Notice that if the ambient light is white, then I a = (1,1,1), so the colo of the light eflected fom a suface is the colo of the suface. Howeve, if the colo of the ambient light is blue and the colo of the suface is ed, then I a = (0,0,1) and k a = (k a,0,0), so I = (,g,b) = (0,0,0) and the colo peceived by the viewe is black. 2
3 4. Diffuse Reflection Diffuse light is light eflected off dull sufaces like cloth. Light dispesed fom dull sufaces is eflected fom each point by the same amount in all diections. Thus the intensity of the eflected light is independent of the position of the viewe. Hee we shall compute the diffuse light eflected off a dull suface fom a point light souce. Let I p be the intensity of the point light souce. We need to compute the intensity I of the diffuse light eflected to a viewe fom each point on a dull suface. Let L be the unit vecto fom the point on the suface to the point light souce, and let N be the outwad pointing unit nomal at the point on the suface (see Figue 2). Then the fomula fo the diffuse intensity I is simply I = k d (L N)I p, (4.1) whee 0 k d 1 is the diffuse eflection coefficient. The diffuse eflection coefficient k d, like the ambient eflection coefficient k a, is a popety of the colo and mateial of the suface, which can be detemined expeimentally o simply set by the pogamme. L N Suface Figue 2: The unit nomal vecto N to a point on the suface, and a unit vecto L pointing in the diection of a point light souce. Equation (4.1) is a consequence of Lambet s Law. Conside a point light souce fa away fom a small suface facet (see Figue 3). Let I facet denote the intensity of light on the facet, and let I souce denote the intensity of the light souce. Then Light Beam Coss Section I facet = = Unit Aea Facet I Aea souce. But we can see fom Figue 3 that Beam Coss Section Facet Aea = cos(), whee is the angle between the nomal to the facet and the vecto to the light souce. Thus we aive at the following esult: 3
4 Lambet s Law: I facet = cos() I souce. (4.2) N Light Souce Beam Coss Section Figue 3: Lambet s Law: I facet = cos() I souce. In Equation (4.1), L and N ae unit vectos. Hence L N = cos(), whee is the angle between the nomal to the suface (N) and the diection to the light souce (L). Thus Equation (4.1) is simply Equation (4.2) attenuated by the facto k d. Equation (4.1) is eally thee equations, one fo each pimay colo ed, geen, and blue. If we set I p = (I p,i g p,i b p) and g k d = (k d,kd,kd b ), whee (I p,ip,ip) ae the intensities of the point light souce fo ed, geen and blue, and (k d,kd,kd ) ae the diffuse eflection coefficients fo ed, geen, and blue, then Equation (4.1) becomes I = (,g,b) = (k di g g b b p,kd Ip,kd Ip). (4.3) When thee ae many light souces, we simply add the contibutions fom each light souce. As with ambient light, the colo peceived by the viewe depends both on the colo of the light souce and the colo of the suface. If the light souce is white, then I p = (1,1,1), so the colo of the eflected light is the colo of the suface. Howeve, if the colo of the light souce is blue and the colo of the suface is ed, then I p = (0,0,1) and k d = (k d,0,0), so I = (,g,b) = (0,0,0) and once again the colo peceived by the viewe is black. 5. Specula Reflection Specula eflections ae highlights eflected off shiny sufaces. Unlike ambient and diffuse eflections, specula highlights ae sensitive to the position of the viewe. Hee we shall compute the specula highlights eflected off a shiny suface fom a point light souce. Let I p be the intensity of the point light souce. We need to compute the intensity I of the 4
5 specula highlight eflected to a viewe fom each point on a shiny suface. Let L be the unit vecto fom the point on the suface to the point light souce, and let N be the outwad pointing unit nomal at the point on the suface. Let R be the image of the vecto L as the light fom the point light souce in the diection L bounces off the suface  that is, R is the vecto defined by setting the angle of incidence of L equal to the angle of eflection of R. Finally let V be a unit vecto fom the point on the suface to the viewe, and let α be the angle between V and R (see Figue 4). Then the fomula fo the intensity I of the specula highlight seen by a viewe in the diection V is given by I = k s cos n (α)i p = k s (R V ) n I p, (5.1) whee 0 k d 1 is the specula eflection coefficient and n 0 is the specula exponent, a constant that contols the concentation of the specula highlight. The specula eflection coefficient k s, like the ambient eflection coefficient k a and the diffuse eflection coefficient k d, is a popety of the colo and mateial of the suface, which can be detemined expeimentally o simply set by the pogamme. L N R V α Suface Figue 4: The unit nomal vecto N to a point on the suface, and a unit vecto L pointing in the diection of a point light souce. The unit vecto V points to the viewe, and the unit vecto R is the image of the light vecto L eflected off the suface. The angle is the angle between the vectos L and N, and the angle α is the angle between the vectos R and V. Notice that unlike ambient and diffuse eflections, specula highlights depend on the position of the viewe. The intensity of the specula highlight falls off apidly as the angle α between the eflected vecto R and the vecto to the viewe V inceases, since R V = cos(α) 0 as α 90 o. Moeove, inceasing the value of the specula exponent n futhe concentates the specula highlight aound the diection R, since high powes of cos(α) appoach zeo moe apidly than low powes of cos(α). As n, the suface appoaches a mio; only when α = 0 is any effect visible to the viewe. Convesely, when n 0, neithe the position of the viewe no the position of the light souce mattes, and the effect is much the same as fo ambient light. In Equation (5.1), the vectos L,V,N ae typically known, since the position of the light souce, the location of the viewe, and the shape of the suface ae contolled by the pogamme. The 5
6 paametes I p and k s  the intensity of the light souce and the specula eflection coefficient  ae also contolled by the pogamme. The eflection vecto R, howeve, must be computed. To find R, let L and L denote the components of L paallel and pependicula to N (see Figue 5). Then L = L + L and R = L L. Now ecall that L = (L N)N L = L L = L (L N)N. Theefoe R = 2(L N)N L. L N L L L R Suface Figue 5: Computation of the vecto R fom the vectos N and L: L = L + L and R = L L. In the calculation of specula eflection, we shall often assume that the light souces ae located at infinity. Notice that if the light souce is fa away and if the suface is plana, then L,N,R ae constants, and only R V needs to be ecalculated fo each point on the suface. Thus these assumptions help to speed up the calculation of specula eflections. Equation (5.1), like Equations (3.1) and (4.1), is eally thee equations, one fo each pimay colo ed, geen, and blue. If we set I p = (I p,i g p,i b p) and k s = (k s,ks,ks ), whee (I p,ip,ip) ae the intensities of the point light souce fo ed, geen and blue, and (k s,ks,ks ) ae the specula eflection coefficients fo ed, geen, and blue, then Equation (5.1) becomes I = (,g,b) = (k di g g b b p,kd Ip,kd Ip). (5.2) When thee ae many light souces, we simply add the contibutions fom each light souce. Specula highlights come in two types, depending on whethe the suface mateial is 6
7 homogeneous o inhomogeneous. When the suface mateial is homogeneous, like a pue metal, then the specula highlight takes on the colo of the suface much like diffuse eflections. Thus if the light souce is white, then I p = (1,1,1), so the colo of the specula highlight is the colo of the suface. Howeve, if the colo of the light souce is blue and the colo of the suface is ed, then I p = (0,0,1) and viewe. k d = (k d,0,0), so I = (,g,b) = (0,0,0) and no specula highlight is visible to the When the suface mateial is inhomogeneous, like a plastic, then unlike ambient light and diffuse eflections, the colo of the specula highlights peceived by the viewe depends only on the colo of the light souce and not on the colo of the suface. Fo example, if the colo of the light souce is blue and the colo of the suface is ed, then the colo of the specula highlight is blue not ed o black. 6. Total Intensity To find the total intensity at each point, we must add the ambient, diffuse, and specula components. Thus Total Intensity = Ambient Intensity + Diffuse Intensity + Specula Intensity. Fo each scene thee is only one ambient intensity, but fo the diffuse and specula intensities we need to add the contibutions fom each light souce. Theefoe Total Intensity = k a I a + p k d (L N)I p + pk s (R V ) n I p, (6.1) whee the sums ae taken ove all point light souces. As usual, Equation (6.1) is eally thee equations, one fo each pimay colo. Since in the RGB colo model evey colo and intensity is epesented by a point inside the unit cube, when thee ae many light souces it may be necessay to nomalize the total intensities to lie between zeo and one by dividing the intensity at each point by the maximum intensity in the scene. 7. Summay In the RGB colo model, each colo and intensity is epesented by thee coodinates, (,g,b), in the unit cube, each coodinate epesenting the contibution of the associated pimay colo ed, geen, o blue, to the given colo. Thee ae thee illumination models fo computing the colo and intensity at each suface point: ambient intensity, diffuse eflection, and specula highlights. The fomulas fo these illumination models ae summaized below: 7
8 Illumination Models Ambient Intensity = k a I a Diffuse Reflection = k d (L N)I p Specula Reflection = k s (R V) n I p Total Intensity = k a I a + p k d (L N)I p + k s (R V ) n p I p whee L is the unit vecto fom the point on the suface to the point light souce N is the outwad pointing unit nomal at the point on the suface R is the image of the vecto L as the light fom the point light souce in the diection L bounces off the suface (see Figue 5) V is a unit vecto fom the point on the suface to the viewe I a is the ambient intensity I p is the intensity of the light souce located at the point p k a is the ambient eflection coefficient k d is the diffuse eflection coefficient k s is the specula eflection coefficient n is the specula eflection exponent and the sums ae taken ove all the point light souces. The distinguishing popeties of these thee illumination models ae summaized in Table 1. Colo Viewe Location Ambient Intensity Suface Independent Diffuse Reflection Suface Independent Specula Highlights Suface Dependent (Homogeneous) Specula Highlights Light Souce Dependent (Inhomogeneous) Table 1: How the colo and intensity at a point depends on the position of the viewe as well as the colo of the suface and the colo of the light souce fo ambient, diffuse, and specula eflections. 8
9 Execises: 1. Suppose that the colo of the light souce is yellow and the colo of the suface is cyan. What is the colo peceived by the viewe fo a. diffuse eflection b. specula eflection (homogenous mateial) c. specula eflection (inhomogeneous mateial) 9
Nuno Vasconcelos UCSD
Radiomety Nuno Vasconcelos UCSD Image fomation two components: geomety and adiomety geomety: pinhole camea point x,y,z in 3D scene pojected into image pixel of coodinates x, y y accoding to the pespective
More informationChapter 26  Electric Field. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapte 6 lectic Field A PowePoint Pesentation by Paul. Tippens, Pofesso of Physics Southen Polytechnic State Univesity 7 Objectives: Afte finishing this unit you should be able to: Define the electic field
More informationTALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS
4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length planoconvex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, planoconvex long focal length lens, monochomatic
More informationGauss's Law. EAcos (for E = constant, surface flat ) 1 of 11
1 of 11 Gauss's Law Gauss's Law is one of the 4 funmental laws of electicity and magnetism called Maxwell's quations. Gauss's law elates chages and electic fields in a subtle and poweful way, but befoe
More informationTANGENTS IN POLAR COORDINATES
TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a Pole P and a halfline called the pola
More informationVector 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 informationGauss s Law. Gauss s law and electric flux. Chapter 24. Electric flux. Electric flux. Electric flux. Electric flux
Gauss s law and electic flux Gauss s Law Chapte 4 Gauss s law is based on the concept of flux: You can think of the flux though some suface as a measue of the numbe of field lines which pass though that
More informationAnnex F. Interaction of Multiple Planar Flaws: Alignment and Combination Rules
(0 May 006 FITNET MK7 Annex F Inteaction of Multiple Plana Flaws: Alignment and Combination Rules F Inteaction of Multiple Flaws: Alignment and Combination Rules F Inteaction of Multiple Flaws: Alignment
More informationAverage: 59 Standard Deviation: 19 Solutions will be on the web today
Couse Updates Midtem statistics Aveage: 59 Standad Deviation: 19 Solutions will be on the web today Assignment 2 Theoy is due today by midnight Pogamming is due a week fom today (if you haven t stated
More informationChapter 24: Gauss Law
. Kass P1 Sp Ch 1 Chapte : Gauss Law Gauss Law elates the net amount of electic chage enclosed by a suface to the electic field on that suface. Fo cetain situations Gauss Law povides an easie way to calculate
More informationSources of the Magnetic Field. Physics 231 Lecture 81
Souces of the Magnetic Field Physics 31 Lectue 81 Magnetic Field of a Point Chage Given a point chage, q, we know that it geneates an electic field egadless of whethe it is moving o not f the chage is
More informationReview Topics Lawrence B. Rees You may make a single copy of this document for personal use without written permission.
Review Topics Lawence. Rees 2006. You ma make a single cop of this document fo pesonal use without witten pemission. R.1 Vectos I assume that ou have alead studied vectos in pevious phsics couses. If ou
More informationTHREE DIMENSIONAL GEOMETRY. The moving power of mathematical invention is not reasoning but imagination. A. DEMORGAN
THREE D IMENSIONAL G EOMETRY 463 The moving powe of mathematical invention is not easoning but imagination. A. DEMORGAN. Intoduction In Class XI, while studying Analytical Geomety in two dimensions, and
More informationChapter 23: Gauss s Law
Chapte 3: Gauss s Law Homewok: Read Chapte 3 Questions, 5, 1 Poblems 1, 5, 3 Gauss s Law Gauss s Law is the fist of the fou Maxwell Equations which summaize all of electomagnetic theoy. Gauss s Law gives
More informationLINES AND TANGENTS IN POLAR COORDINATES
LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Polacoodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and
More informationFor a supplemental understanding of antennas  a quick overview. See:
Antenna Fundamentals Definitions Antenna fundamentals ae pesented in Lesson 1 though 6. If upon eading Chapte 4 of the textbook, you find that you ae still lacking an undestanding, then study the CD. If
More informationPhysics 417G : Solutions for Problem set 3
Poblem Physics 47G : Solutions fo Poblem set 3 Due : Febuay 5 6 Please show all the details of you computations including intemediate steps A thick spheical shell of inne adius a and oute adius b caies
More information3 The Electric Field Due to one or more Point Charges
Chapte 3 The lectic Field Due to one o moe Point Chages 3 The lectic Field Due to one o moe Point Chages A chaged paticle (a.k.a. a point chage, a.k.a. a souce chage) causes an electic field to exist in
More information2 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 informationGauss Law. Physics 231 Lecture 21
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 informationMagnetic Forces. Physics 231 Lecture 71
Magnetic Foces Physics 231 Lectue 71 Magnetic Foces Chaged paticles expeience an electic foce when in an electic field egadless of whethe they ae moving o not moving Thee is anothe foce that chaged paticles
More informationEpisode 401: Newton s law of universal gravitation
Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce
More informationCS559: Computer Graphics
CS559: Compute Gaphics Lectue 7: Textue Mapping Li Zhang Sping 008 Many slides fom Ravi Ramamoothi, Columbia Univ, Geg Humpheys, UVA and Rosalee Wolfe, DePaul tutoial teaching textue mapping visually,
More informationFluids Lecture 15 Notes
Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2D, this velocit
More informationThe statement of the problem of factoring integer is as follows: Given an integer N, find prime numbers p i and integers e i such that.
CS 2942 Sho s Factoing Algoithm 0/5/04 Fall 2004 Lectue 9 Intoduction Now that we have talked about uantum Fouie Tansfoms and discussed some of thei popeties, let us see an application aea fo these ideas.
More informationWaves and Superposition (Keating Chapter 21) Light as a Transverse Wave.
Waves and Supeposition (Keating Chapte 1) The ay model fo light (i.e. light tavels in staight lines) can be used to explain a lot of phenomena (like basic object and image fomation and even abeations)
More informationMoment and couple. In 3D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r
Moment and couple In 3D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita
More informationGravitational Field and its Potential
Lectue 19 Monday  Octobe 17, 2005 Witten o last updated: Octobe 17, 2005 P441 Analytical Mechanics  I Gavitational Field and its Potential c Alex. Dzieba Isaac Newton What Isaac Newton achieved was tuly
More informationVectors in three dimensions
Vectos in thee dimensions The concept of a vecto in thee dimensions is not mateially diffeent fom that of a vecto in two dimensions. It is still a quantity with magnitude and diection, except now thee
More informationHomework #4  Answers. The ISLM Model Due Mar 18
Winte Tem 2004 Alan Deadoff Homewok #4  Answes Page 1 of 12 Homewok #4  Answes The  Model Due Ma 18 1. Fun with the Keynesian Coss: a. Use the geomety of the Keynesian Coss diagam shown at the ight
More informationLecture 19: Effective Potential, and Gravity
Lectue 19: Effective Potential, and Gavity The expession fo the enegy of centalfoce motion was: 1 ( ) l E = µ + U + µ We can teat this as a onedimensional poblem if we define an effective potential:
More informationCoordinate 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 informationPrelab Quiz/PHYS 224 Earth s Magnetic Field. Your name Lab section
Pelab Quiz/PHYS 224 Eath s Magnetic Field You name Lab section. What do you investigate in this lab? 2. Fo a pai of Helmholtz coils descibed in this manual and shown in Figue 2, =.5 m, N = 3, I =.4 A,
More informationChapter 24. The Electric Field
Chapte 4. The lectic Field Physics, 6 th dition Chapte 4. The lectic Field The lectic Field Intensity 41. A chage of + C placed at a point P in an electic field epeiences a downwad foce of 8 14 N. What
More informationTransformations in Homogeneous Coordinates
Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) YanBin 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 informationComputer Graphics.  Camera Transformations  Hendrik Lensch. Computer Graphics WS07/08 Camera Transformations
Compute Gaphics  Camea Tansfomations  Hendik Lensch Compute Gaphics WS7/8 Camea Tansfomations Oveview Last lectue: Tansfomations Today: Geneating 2D image fom 3D wold Coodinate Spaces Camea Specification
More information19.1 Potential Energy
Chapte 19: Electic Potential Enegy & Electic Potential Why electic field contains enegy? Is thee an altenative way to undestand electic field? Concepts: Wok done by consevative foce Electic potential enegy
More informationElectric & Potential Fields
Electic & Potential Fields Pupose An electic field suounds any assemblage of chaged objects. To detemine the stength and diection of these fields, it is most convenient to fist map the electic potential
More informationSpirotechnics! 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 informationModel Building For ARIMA time series. Consists of three steps 1. Identification 2. Estimation 3. Diagnostic checking
Model Building Fo ARIMA time seies Consists of thee steps. Identification. Estimation 3. Diagnostic checking ARIMA Model building Identification Detemination of p, d and q To identify an ARIMA(p,d,q) we
More informationChapter 2 CHARACTERISTICS OF SYNCHROTRON RADIATION
SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION Chapte CHARACTERISTICS OF SYNCHROTRON RADIATION. Intoduction The adiation is chaacteized in geneal b the following tems: spectal ange, photon flu, photon
More informationThe Effects of Moons on Saturn s Ring System
The Effects of Moons on Satun s Ring System Kisten Lason Physics Depatment, The College of Wooste, Wooste, Ohio 44691, USA (Dated: May 10, 007) The ing system of Satun is a complex inteaction between numeous
More informationThere are two kinds of charges, namely, positive (+) charge and negative ( ) charge. Like charges repel
Unit 4 Electic Foces, Fields and Cicuits 4 Electic chage 4 Coulomb s law 43 Shell theoems fo electostatics 44 Electic field 45 Electic field lines 46 Shielding and chaging by induction 47 Electic Cicuits
More informationMapping Electric and Potential Fields
Mapping Electic and Potential Fields Pupose We detemined the magnitude and diection of the electic fields suounding thee sets of chaged electodes. Fo each electode set, we measued the electic potential
More informationCONSIDER the variations in the human race.
Exploing Genetics CONSIDE the vaiations in the human ace. Thee is a wide ange of skin colos. People s eyes may be dak bown, light bown, blue, hazel, geen, and vaious othe hues. Some people ae tall, some
More informationRelativistic Theory of Black Holes
Relativistic Theoy of Black Holes Daniele Sasso * Abstact The gavitational theoy is the most accedited theoy fo explaining black holes. In this pape we pesent a new intepetation based on the elativistic
More informationPhysics 202, Lecture 4. Gauss s Law: Review
Physics 202, Lectue 4 Today s Topics Review: Gauss s Law Electic Potential (Ch. 25Pat I) Electic Potential Enegy and Electic Potential Electic Potential and Electic Field Next Tuesday: Electic Potential
More informationCHAPTER III APPLICATION OF SNELLDESCARTES LAWS TO THE STUDY OF PRISMS
18 CHPTER III PPLICTION OF SNELLDESCRTES LWS TO THE STUDY OF PRISMS pism is a tanspaent medium limited by plane efactive sufaces that ae not paallel. In this couse, we will only conside pisms made of
More informationHomework #2. Solutions. Chapter 22. The Electric Field II: Continuous Charge Distributions
Homewok #. Solutions. Chapte. The lectic Field II: Continuous Chage Distibutions 4 If the electic flux though a closed suface is zeo, must the electic field be zeo eveywhee on that suface? If not, give
More information1.4 Phase Line and Bifurcation Diag
Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes
More informationThe 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 informationReview Module: Cross Product
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of hysics 801 Fall 2009 Review Module: Coss oduct We shall now intoduce ou second vecto opeation, called the coss poduct that takes any two vectos and geneates
More informationSamples 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 informationIn the lecture on double integrals over nonrectangular domains we used to demonstrate the basic idea
Double Integals in Pola Coodinates In the lectue on double integals ove nonectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example
More informationIrradiance versus distance. Exploring the inverse square law.
Iadiance vesus distance.. OBJECTIVE CONTENTS To examine the distance distibution of the visual adiation emitted by the light souces. The iadiance vs. distance dependence is analyzed in tems of the invese
More information4.1 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 information1 Charge & Coulomb's Law
Chapte 1 Chage & Coulomb's Law 1 Chage & Coulomb's Law Chage is a popety of matte. Thee ae two kinds of chage, positive + and negative. 1 An object can have positive chage, negative chage, o no chage at
More informationPotential Flow Theory
.06 Hydodynamics Reading #4.06 Hydodynamics Pof. A.H. Techet Potential Flow Theoy When a flow is both fictionless and iotational, pleasant things happen. F.M. White, Fluid Mechanics 4th ed. We can teat
More information7. Molecular interactions 7.1 Polar molecules
7. Molecula inteactions 7.1 Pola molecules A pola molecule is a molecule with a pemanent dipole moment (stemming fom patial chages due to electonegativity and othe featues of bonding). Nonpola molecules
More informationFXA 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 informationPhysics 212 Final Sample Exam Form A
1. A point chage q is located at position A, a distance away fom a point chage Q. The chage q is moved to position B, which is also located a distance away fom the chaged paticle Q. Which of the following
More informationLecture: Spherical Black Holes
Chapte 11 Lectue: Spheical Black Holes One of the most spectacula consequences of geneal elativity is the pediction that gavitational fields can become so stong that they can effectively tap even light.
More informationIntegration in polar coordinates
Pola Coodinates Integation in pola coodinates Pola coodinates ae a diffeent wa of descibing points in the plane. The pola coodinates (, θ) ae elated to the usual ectangula coodinates (, ) b b = cos θ,
More informationProblem Set 5: Universal Law of Gravitation; Circular Planetary Orbits.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.01T Fall Tem 2004 Poblem Set 5: Univesal Law of Gavitation; Cicula Planetay Obits. Available online Octobe 1; Due: Octobe 12 at 4:00
More informationInfinitedimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.
Infinitedimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach
More informationChapter 2 Vector Spaces  An Introduction
EE448/58 Class Notes Vesion 1.0 John Stensby Chapte Vecto Spaces  An Intoduction A vecto space ove a scala field F (in ou wok, we use both the eal numbes R o the complex numbes C as scalas) is a nonempty
More informationTransformations, continued
Tansfomations, continued D Rotation x,, x, y, y,, x, y,,, x, y, So if the ows of R ae othogonal unit vectos (othonomal), they ae the axes of a new coodinate system, and matix multiplication ewites (x,y,)
More informationGAUSS S LAW FOR SPHERICAL SYMMETRY
MISN012 GAUSS S LAW FOR SPHRICAL SYMMTRY by Pete Signell GAUSS S LAW FOR SPHRICAL SYMMTRY G.S. R 1. Intoduction.............................................. 1 a. The Impotance of Gauss s Law.........................
More informationElectric Flux, Surface Integrals, and Gauss s Law
Electic Flux, Suface Integals, and Gauss s Law If we know the locations of all electic chages, we can calculate the Electic Field at any point in space though: v E v E i all i ˆ i 4πε i ρ ˆ 4πε ( ) dv
More information11.6 Directional Derivatives and the Gradient Vector
6 Diectional Deivatives and the Gadient Vecto So a we ve ound the ate o change o a unction o two o moe vaiables in the diection paallel to the ais (, we set constant, = b plane), and in the diection paallel
More informationd r ). Therefore the speed would be much greater than the
SPH3U Position, Velocity Time Gaphs LoRusso Displacement: The staightline distance between an object's stating position and finishing position, measue as a vecto (i.e. includes diection). Displacement
More informationLaws of Motion; Circular Motion
Pactice Test: This test coves Newton s Laws of Motion, foces, coefficients of fiction, feebody diagams, and centipetal foce. Pat I. Multiple Choice 3m 2m m Engine C B A 1. A locomotive engine of unknown
More informationLecture 6 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lectue 6 Notes, Electomagnetic Theoy II D. Chistophe S. Baid, faculty.uml.edu/cbaid Univesity of Massachusetts Lowell 1. Radiation Intoduction  We have leaned about the popagation of waves, now let us
More informationCHAPTER 17 MAGNETIC DIPOLE MOMENT
1 CHAPTER 17 MAGNETIC DIPOLE MOMENT 17.1 Intoduction A numbe of diffeent units fo expessing magnetic dipole moment (heeafte simply magnetic moment ) ae commonly seen in the liteatue, including, fo example,
More informationAlgebra 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 informationVectorValued Functions and Mathcad
VectoValued Functions and Mathcad D P Mostad, Univesity of Noth Dakota Ojectives of Assignment 1 To lean how to make Mathcad gaph two and theedimensional vectovalued functions To lean how to make Mathcad
More informationPhysics 2102 Lecture 15. Physics 2102
Physics 212 Jonathan Dowling Physics 212 Lectue 15 iotsavat Law Jeanaptiste iot (17741862) Felix Savat (1791 1841) What Ae We Going to Lean? A Road Map Electic chage Electic foce on othe electic chages
More informationSkills 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 informationChapter 9 Eigenvalues, Eigenvectors and Canonical Forms Under Similarity
EE448/528 Vesion.0 John Stensby Chapte 9 Eigenvalues, Eigenvectos and Canonical Foms Unde Similaity Eigenvectos and Eigenvectos play a pominent ole in many applications of numeical linea algeba and matix
More informationBrown University PHYS 0060 ELECTRIC POTENTIAL
INTRODUCTION ELECTRIC POTENTIL You have no doubt noticed that TV sets, light bulbs, and othe electic appliances opeate on 115 V, but electic ovens and clothes dyes usually need 220 V. atteies may be ated
More informationPhysics E1ax Solutions: Assignment for Feb. 3 Feb. 10 Homework #1: Electric Potential, Coulomb s Law, Equipotentials
Physics Eax Solutions: Assignment fo Feb. 3 Feb. 0 Homewok #: Electic Potential, Coulomb s Law, Equipotentials Afte completing this homewok assignment, you should be able to Undestand the diffeence between
More informationPolarization of Dielectrics
Polaization of Dielectics Dielectics ae mateials which have no fee chages; all electons ae bound and associated with the neaest atoms. An extenal electic field causes a small sepaation of the centes of
More informationMultiItem EOQ Model with Varying Holding Cost: A Geometric Programming Approach
Intenational Mathematical Foum, Vol. 6, 0, no. 3, 3544 MultiItem EOQ Model with Vaying Holding Cost: A Geometic Pogamming Appoach K. A. M. Kotb () and Hala A. Fegany () () Depatment of Mathematics and
More informationModeling the viscous torque acting on a rotating object
Modeling the viscous toque acting on a otating object Manon E. Gugel Physics Depatment, The College of Wooste, Wooste, Ohio 6 (Apil 0, ) By dawing an analogy between linea and otational dynamics, an equation
More informationGeostrophic balance. John Marshall, Alan Plumb and Lodovica Illari. March 4, 2003
Geostophic balance John Mashall, Alan Plumb and Lodovica Illai Mach 4, 2003 Abstact We descibe the theoy of Geostophic Balance, deive key equations and discuss associated physical balances. 1 1 Geostophic
More information2.2. 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 information1 Spherical multipole moments
Jackson notes 7 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the
More informationFaraday's Law ds B B r r Φ B B S d dφ ε B = dt
ds Faaday's Law Φ ε ds dφ = Faaday s Law of Induction Recall the definition of magnetic flux is Φ = da Faaday s Law is the induced EMF in a closed loop equal the negative of the time deivative of magnetic
More informationVideo Montage  Video Abstraction
Video Montage  Video Abstaction ITing Fang 1 Intoduction The amount of video has apidly inceased in ecent yeas and theefoe how to extact useful video content and etieve video fom huge database is a cucial
More informationTRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles
TRIGONOMETRY REVIEW The Cosines and Sines of the Standad Angles P θ = ( cos θ, sin θ ) . ANGLES AND THEIR MEASURE In ode to define the tigonometic functions so that they can be used not only fo tiangula
More informationE g n i g n i e n e e r e in i g n g M e M c e h c a h n a i n c i s c : s D yn y a n m a i m c i s 151
Engineeing Mechanics: Dynamics Intoduction Kinematics of igid bodies: elations between time and the positions, velocities, and acceleations of the paticles foming a igid body. Classification of igid body
More informationInfluence of electrical steel sheet textures on their magnetization curves *
ARCHIVES OF ELECTRICAL ENGINEERING VOL. 6(3), pp. 45437 (013) DOI 10.478/aee0130034 Influence of electical steel sheet textues on thei magnetization cuves * WITOLD MAZGAJ, ADAM WARZECHA Institute of
More informationMaths for Graphics programming. Rotations. Vectors recap. Sin, Cos and Tan (basic trig) What is the value of sin θ? Sin, Cos and Tan (basic trig)
5//6 Maths fo Gaphics pogamming Vectos ecap Can use unit vectos to define positions instead of points Adding/subtacting vectos gives ou elative positions Coss poducts can find ne diections Can tansfom
More informationCourse Updates. 2) This week: Finish Chap 27 (magnetic fields and forces)
Couse Updates http://www.phys.hawaii.edu/~vane/phys272sp10/physics272.html Notes fo today: 1) Assignment #7 due Monday 2) This week: Finish Chap 27 (magnetic fields and foces) 3) Next week Chap 28 (Souces
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More informationEAS 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 TimeDawdown Data In the past lectue we discussed how to calculate dawdown if we know the hydologic
More information1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2
Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the
More informationChapter 22 The Electric Field II: Continuous Charge Distributions
Chapte The lectic Field II: Continuous Chage Distibutions 1 [M] A unifom line chage that has a linea chage density l equal to.5 nc/m is on the x axis between x and x 5. m. (a) What is its total chage?
More information