Lecture 9: Vector Geometry: A Coordinate-Free Approach. And ye shall know the truth, and the truth shall make you free. John 8:32

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1 Lecture 9: Vector Geometry: A Coordiate-Free Approach Ad ye shall kow the truth, ad the truth shall make you free. Joh 8:32 1. Coordiate-Free Methods We are goig to asced ow from the flat world of 2-dimesios ito the real world of 3- dimesios. Whe workig i 3-dimesios, we shall isist predomiatly o coordiate-free methods. Ee i 2-dimesios we adopted coordiate-free techiques to describe shapes because it is easier to represet geometry without troublig about coordiates. For example, it is simpler to describe a bump usig a turtle program rather tha tryig to delieate the coordiates of all the ertices of the bump. Similarly, we ioke affie trasformatios -- traslatio, rotatio, scalig, ad shear -- to moe ad reshape geometry without worryig about the etries -- the coordiates -- of the correspodig matrices. Coordiates are useful for computatios, but coceptually we prefer to work at a higher leel of abstractio. Turtle programs ad affie trasformatios were our etry to coordiate-free methods i 2-dimesios. I 3-dimesios coordiate-free methods are ee more crucial. You may be comfortable with coordiate techiques i 2-dimesios, but i 3-dimesios it is much harder to coceptualize geometry usig coordiates. I additio to allowig us to work at a higher leel of abstractio, coordiate-free methods proide a more cocise otatio. Typically we will eed to deal with oly oe equatio for a poit or a ector, rather tha with three equatios for their coordiates. Also we shall see that coordiate-free methods lead aturally to geometrically meaigful expressios. With coordiates we ca perform seseless computatios that hae o itrisic geometric sigificace -- for example, we could add the coordiates of two poits. Coordiate-free methods will help us to aoid the pitfalls of such midless computatios. Fially, ad most importatly, coordiate-free techiques capture the geometric meaig behid our computatioal methods. You may kow that the dot product of two ectors u, ca be calculated from the formula u = u u u 3 3, but why would ayoe eer wat to compute the expressio iolig the rectagular coordiates o the right had side? The geometric meaig of the dot product is captured by the coordiatefree expressio u = u cos(ϑ), where u ad are the legths of the ectors u ad, ad ϑ is the agle betwee u ad. Agle ad legth (as well as projectio), these quatities are the geometric cotet of the dot product, cotet that is completely hidde i the coordiate computatio.

2 Our approach to Computer Graphics is to simplify geometry as much as possible by iokig the Mathematics most appropriate to the problem at had. I 2-dimesios we employ Turtle Graphics (LOGO) rooted i coformal trasformatios ad Affie Graphics (CODO) based o affie trasformatios. I 3-dimesios, affie trasformatios alog with a ew trasformatio, perspectie projectio, play a ee more cetral role. These trasformatios are applied to maipulate shapes i 3-dimesios, but at bottom most shapes i 3-dimesios are represeted i terms of poits ad ectors. Therefore we begi our study of 3-dimesioal Computer Graphics by itroducig a coordiate-free approach to the algebra ad geometry of poits ad ectors. 2. Vectors ad Vector Spaces Vectors ad ector spaces should be familiar to you from stadard courses o liear algebra. Vectors ca be added, subtracted, ad multiplied by scalars, ad these ector operatios all hae coordiate-free defiitios (see Figure 1). + w w w w (a) additio (b) subtractio (c) scalar multiplicatio Figure 1: Coordiate-free geometric defiitios of (a) additio, (b) subtractio, ad (c) scalar multiplicatio for ectors. w cw These ector operatios obey the usual laws of arithmetic: additio is associatie (Figure 2(a)) ad commutatie (Figure 2(b)) ad scalar multiplicatio distributes through additio (Figure 2(c)). u + u+ ( + w) = (u + ) + w u + w w u u + u = u + u + c(u + ) (a) associatie (b) commutatie (c) distributie Figure 2: The associatie, commutatie, ad distributie properties of ector additio ad scalar multiplicatio. Vector additio is (a) associatie because o matter how we group u,, w, if we place these ectors head to tail, the ector u + + w goes from the tail of u to the head of w. Vector additio is (b) commutatie because + u, u + both represet the diagoal of the parallelogram with sides u,. Fially, by similar triagles, (c) scalar multiplicatio distributes through additio. 2 u cu c

3 Neertheless, although ectors ad ector operatios are useful ad familiar, ectors are ot the primary focus of Computer Graphics. O the graphics termial we see poits, ot ectors, so it is to poits that we ext tur our attetio. 3. Poits ad Affie Spaces Poits are ot ectors. Poits hae a fixed positio, but o directio or legth; ectors hae directio ad legth, but o fixed locatio. Vectors ca be added, subtracted, ad multiplied by scalars ad the result is always a ector. Poits ca be subtracted from oe aother, but the result is a ector, ot a poit (see Figure 3), A ector ca be added to a poit, ad the result is a poit (see Figure 3), but there is o coordiate-free way to add two poits or to multiply a poit by a scalar. Q P + Q P P Figure 3: Subtractig a poit from a poit, ad addig a ector to a poit. Notice that P + (Q P) = Q, so the usual cacellatio law of additio applies. P Expressios of the form k c k k are called liear combiatios. For ectors, liear combiatios always make sese because additio ad scalar multiplicatio are always defied for ectors. Although we caot, i geeral, add two poits or multiply poits by scalars, eertheless some liear combiatios of poits also make sese. For example, the expressio P + Q = P Q 2 represets the midpoit of the lie segmet joiig the poits P ad Q, ee though oe of the expressios P + Q, P / 2, Q / 2 are well-defied. We would like to hae some aalogue of liear combiatios for poits to accout for expressios like the midpoit (P + Q) / 2. Although, i geeral, scalar multiplicatio for poits caot be defied i a coordiate-free maer, we ca defie a coordiate free ersio of scalar multiplicatio i two special cases: 1 P = P 0 P = 0 c P = udefied c 0,1 3

4 By the way the zero o the right had side of the secod equatio is the zero ector, ot the origi of the coordiate system. Now for a crucial obseratio. Notice that formally c k P k = ( c k )P 0 + c k (P k P 0 ). k=1 That is, if the usual rules of arithmetic are to apply, the the terms c k P 0 ad c k P 0 for k 1 should cacel. The expressio c k (P k P 0 ) k=1 is well-defied, sice this expressio is just a liear combiatio of ectors. Moreoer, the expressio ( c k )P 0 is well-defied wheeer the costats sum to zero or oe. Based o these obseratios, we itroduce the followig defiitios: c k P k = P 0 + c k (P k P 0 ) if c k =1 k=1 = c k (P k P 0 ) if c k = 0 k=1 = udefied if c k 0,1 Expressios of the form k c k P k, where k c k 1, are called affie combiatios. Affie combiatios are the aalogues for poits of liear combiatios for ectors. Vectors form a ector space; poits form a affie space. Vectors are closed uder liear combiatios; poits are closed uder affie combiatios. This distictio betwee poits ad ectors, betwee ector spaces ad affie spaces, is what makes the algebra uderlyig Computer Graphics just a bit differet from the stadard algebra of ector spaces that you lear i courses o Liear Algebra. 4. Vector Products There is more to ector algebra tha just additio, subtractio, ad scalar multiplicatio. For ectors, there are also seeral distict otios of multiplicatio: dot product, cross product, ad 4

5 determiat. These products are related geometrically to legth, area, ad olume, so these products show up i may geometric applicatios. Here we reiew the coordiate-free defiitios of these three products, emphasizig their major algebraic ad geometric properties. 4.1 Dot Product. The dot product of two ectors u, is the scalar defied by u = u cos(ϑ), where ϑ is the agle betwee the ectors u ad. Dot product is commutatie ad distributes through additio (see Exercise 6a). We are iterested i dot product because dot product ca be used to compute seeral importat geometric quatities. The followig properties follow easily from the defiitio of the dot product. i. Legth ii. u 2 = u u Cosie (Agle Betwee Two Vectors) cos(ϑ ) = u u iii. Orthogoality u = 0 u i. Projectios (see Figure 4) u = u u = (u ) if =1 u = u u = u u u = u (u ) if =1 The first three properties follow easily from the defiitio of the dot product. To derie the formulas for projectio, otice that the secod formula follows from the first formula because u + u = u (see Figure 4). To proe the first formula, obsere from Figure 4 that u = u cos(ϑ) = u cos(ϑ) Therefore, sice u is parallel to, u = u = u = u. = u 5

6 ϑ u u u Figure 4: Parallel ad perpedicular projectio. Thik dot product! 4.2 Cross Product. The cross product of two ectors u, is the uique ector with the followig three properties (see Figure 5): u = u si(ϑ ) u u, sg(u,,u ) > 0 The third coditio meas that the ectors u,, u hae positie orietatio -- that is, these three ectors obey the right had rule: if you curl the figers of your right had from u to, the your thumb will poit i the directio of u. Cross product distributes through additio (see Exercise 6b), but cross product is ot associatie ad cross product ati-commutatie. That is. u ( + w) = u + u w (distributie) u = u (ati-commutatie) (u ) w u ( w) (o-associatie) The ati-commutatiity follows because by the right had rule: sg(u,,u ) = sg(,u, u). The o-associatiity follows because (u ) w u,w whereas u ( w) u, w. A useful formula to remember is that two cross products ca be reduced to two dot products: ( w) u = (u )w (u w). We shall derie this idetity i Appedix 1 to this lecture. We are iterested i the cross product because, just like the dot product, cross product ca be used to compute seeral importat geometric quatities. The followig properties follow easily from the defiitio of the cross product. i. Area area(u,) = u ii. Sie (Agle Betwee Two Vectors) si(ϑ ) = u u iii. Parallelism u u = 0 6

7 u ϑ u Figure 5: The cross product of two ectors is the ector perpedicular to the two ectors with positie orietatio, ad has legth equal to the area of the associated parallelogram. 4.3 Determiat. The determiat is the scalar defied by the triple product det(u,,w) = (u ) w. The determiat iherits the properties of the dot ad cross product: determiat is multi-liear (liear i each ariable) ad skew symmetric (iterchagig the order of the two ectors chages the sig of the determiat). The mai geometric property of the determiat is the relatio betwee determiat ad olume (see Figure 6). i. Volume ol(u,, w) = det(u,,w). w u Figure 6: The determiat of three ectors is the siged olume of the associated parallelepiped because det(u,,w) = u w cos(ϑ ) = area(u,) height(u,, w) = ol(u,,w). 5. Summary Liear Algebra proides the mathematical foudatio for Computer Graphics. Howeer, the Liear Algebra uderlyig Computer Graphics differs somewhat from stadard Liear Algebra because the fudametal costituets of Computer Graphics -- what oe actually sees o a graphics termial -- are poits, ot ectors. Poits, ulike ectors, caot be combied usig arbitrary liear combiatios; poits are restricted to affie combiatios. Thus the poits form a affie space, rather tha a ector space. This distictio betwee affie spaces (spaces of poits) ad ector spaces (spaces of ectors) is the mai differece betwee the mathematical foudatio of 3- dimesioal Computer Graphics ad the stadard Liear Algebra of 3-dimesioal ector spaces. 7

8 Neertheless, ectors play a fudametal role i Computer Graphics because most of the iformatio i a affie space is stored i the ectors. Fix a sigle poit Q i affie space. The ay other poit P ca be writte as the sum of the fixed poit Q ad the ector = P Q from Q to P -- that is, P = Q + = Q + (P Q). Thus, for example, if we kow how a affie trasformatio T affects the ectors ad we wat to kow how this trasformatio behaes o the poits P, all we eed to do is to compute the trasformatio o the sigle poit Q; the rest of the trasformatio is kow from its affect o the ectors because T(P) = T(Q)+ T(P Q). To emphasize cocepts oer computatio, we adopt a coordiate-free approach to ector geometry. Additio, subtractio, scalar multiplicatio, dot product, cross product, ad determiat all hae coordiate-free, geometric iterpretatios. These geometric properties are the mai coceptual tools we will apply i our aalysis of geometry for Computer Graphics. Coordiate computatios are discussed i the ext lecture. These coordiate computatios ca be implemeted oce ad forgotte, but the geometric cocepts behid these computatios will be reused agai ad agai. Appedix 1: The No-Associatiity of the Cross Product The purpose of this appedix is to derie the hady idetities ( w) u = (u )w (u w) u ( w) = (u w) (u ) w. We begi with a importat special case from which the geeral results will follow. Lemma: (u w) u = (u u)w (u w)u Proof: To proe that two ectors are equal, we eed to check that they hae the same directio, orietatio, ad legth. a. Directio. ((u u)w (u w)u) u = (u u)(w u) (u w)(u u) = 0 ((u u)w (u w)u) (u w) = (u u)( w (u w) ) (u w)( u (u w) ) = 0. Therefore (u u)w (u w)u is perpedicular both to u ad to u w, so (u u)w (u w)u is either parallel or ati-parallel to (u w) u. b. Orietatio. ( ) = det( u, (u u)w (u w)u,u w) det u w, u, (u u)w (u w)u ( ) (u w) = u {(u u)w (u w)u} = {(u u)(u w) } (u w) ( ) = (u u) (u w) (u w) = u 2 u w 2 > 0. 8

9 Therefore the ectors u w, u, (u u)w (u w)u form a right haded system, so (u u)w (u w)u is parallel to (u w) u. c. Legth. Let θ be the agle betwee u ad w. The (u u)w (u w)u 2 = (u u)w (u w)u ( ) ((u u)w (u w)u) = (u u) 2 (w w) (u w) 2 (u u) ( ) ( ) = (u u) (u u)(w w) (u w) 2 = u 2 u 2 w 2 u 2 w 2 cos 2 θ = u 4 w 2 si 2 θ = u 2 u w 2. Sice (u u)w (u w)u ad (u w) u hae the same directio, orietatio, ad legth, it follows that (u w) u = (u u)w (u w)u. Theorem: ( w) u = (u )w (u w) Proof: If is parallel to w, the both sides are zero. Hece we ca assume that,w, w form a basis. Let us first check that this result is true whe u is a elemet of this basis. If u = or u = w, the the result follows from the Lemma. If u = w, the ( w) u = ( w) ( w) = 0 ad (u )w (u w) = (( w) )w (( w) w) = 0. Hece the result is alid whe u is a elemet of the basis,w, w. Now for a arbitrary ector u, there are costats λ, µ,ν such that u = λ + µw +ν w. Therefore the result follows i the geeral case by liearity, sice dot product ad cross product both distribute through additio. Corollary: u ( w) = (u w) (u ) w Proof: u ( w) = ( w) u = (u w) (u ) w. 9

10 Appedix 2: The Algebra of Poits ad Vectors For easy referece we collect below the mai algebraic idetities for additio, subtractio, scalar multiplicatio, dot product, cross product, ad determiat. Most of these formulas either follow easily from the defiitios or are deried i the text; the remaider are proed i the exercises at the ed of this lecture. Additio, Subtractio, ad Scalar Multiplicatio for Vectors u + ( + w) = (u + ) + w (associatie) u + = + u (commutatie) c(u + ) = c u + c (distributie) u + u + (triagular iequality) Additio, Subtractio, ad Affie Combiatios for Poits P + ( + w) = (P + ) + w (associatie) P + (Q P) = Q (R Q) + (Q P) = R P (cacellatio) c k P k = P 0 + c k (P k P 0 ) if c k =1 (affie combiatios) k=1 = c k (P k P 0 ) if c k = 0 k=1 = udefied if c k 0,1 Dimesioal Aalysis for Poits ad Vectors Poit + Poit = Udefied Poit Poit = Vector Poit ± Vector = Poit Vector±Vector = Vector Scalar Vector = Vector Scalar Vector = Vector Scalar Poit = Poit Scalar =1 =Vector Scalar = 0 = Udefied Otherwise 10

11 Dot Product u = u cos(θ) u = u u ( + w) = u + u w ( + w) u = u + w u (defiitio) (commutatie) (distributie) 2 = (legth) cos(θ) = u u (agle) u = u u = (u ) if = 1 u = u u = u u u = u (u ) if =1 u = 0 u Cross Product u = u si(θ) u u, sg(u,,u ) > 0 area(u,) = u (u ) u = 0 (u ) = 0 u u = 0 u = 0 ±u u ( + w) = u + u w ( + w) u = u + w u u = u u ( w) = (u w) (u ) w ( w) u = (u )w (u w) (parallel projectio) (perpedicular projectio) (orthogoality) (defiitio) (area) (orthogoality) (parallelism) (distributie) (ati-commutatie) (o-associatie) u 2 = u 2 2 (u ) 2 (legth) (u 1 u 2 ) ( 1 2 ) = (u 1 1 )(u 2 2 ) (u 1 2 )(u 2 1 ) (Lagrage Idetity) 11

12 Determiat det(u,,w) = (u ) w ol(u,, w) = det(u,,w) det(u,,w) > 0 sg(u,, w) > 0 det(u,,w) 0 u,,w are liearly idepedet det(u, u,w) = det(u,,u) = det(u,,) = 0 det(u,,w) = det(,w,u) = det(w,u,) det(,u,w) = det(u,,w) det(u 1 + c u 2,,w) = det(u 1,,w) + c det(u 2,,w) det(u, 1 + c 2,, w) = det(u, 1,w) + c det(u, 2,w) det(u,,w 1 + c w 2 ) = det(u,,w 1 ) + c det(u,, w 2 ) (defiitio) (olume) (orietatio) (liear idepedece) (skew symmetry) (multi-liearity) Exercises: 1. Proe that for ay poit R a. c k P k = R + c k (P k R) wheeer c k =1. b. c k P k = c k (P k R) wheeer c k = 0. Iterpret these results geometrically. (Hit: You may ot use the equality c k (P k R) = c k P k c k R because the right had side has o itrisic meaig.) 2. Proe that: u 2 = u 2 2 (u ) The purpose of this exercise is to proe the Lagrage idetity: (u 1 u 2 ) ( 1 2 ) = (u 1 1 )(u 2 2 ) (u 1 2 )(u 2 1 ). a. If u 2 is parallel to u 1, show that both sides of the Lagrage idetity are zero. b. If u 2 is ot parallel to u 1, the u 1,u 2,u 1 u 2 forms a basis for ectors i 3-space. Therefore there are costats such that 1 = c 1 u 1 + c 2 u 2 + c 3 u 1 u 2 2 = d 1 u 1 + d 2 u 2 + d 3 u 1 u 2. Now usig the distributie law ad other stadard rules for the cross product together with the result of Exercise 2, proe the Lagrage idetity. 12

13 4. Proe that: a. w = w b. w = w 5. Fix a ector w. Let u, deote the parallel projectios ad let u, deote the perpedicular projectios of u ad o w. Proe that: a. (u + ) = u + b. (u + ) = u + 6. Usig Exercises 4 ad 5, proe that dot product ad cross product both distribute through additio. That is, proe that: a. (u + ) w = u w + w b. (u + ) w = u w + w 7, By drawig the appropriate figures, show that: a. (P + ) + w = P + ( + w) (associatiity) b. (R Q) + (Q P) = R P (cacellatio) c. u + ( 1) = u (egatio) 8. Show that: a. det(u,,w) 0 u,,w are liearly idepedet b. det(u, u,w) = det(u,,u) = det(u,,) = 0 9. Show that: a. sg(u,,w) = sg(,w,u) = sg(w,u,) b. sg(,u,w) = sg(u,,w) 10. Usig Exercise 9, show that the determiat fuctio is skew symmetric. That is, show that: a. det(u,,w) = det(,w,u) = det(w,u,) b. det(,u,w) = det(u,,w) 11. Show that the determiat fuctio is multi-liear. That is, show that: a. det(u 1 + c u 2,,w) = det(u 1,,w) + c det(u 2,,w) b. det(u, 1 + c 2,w) = det(u, 1,w) + c det(u, 2,w) c. det(u,,w 1 + c w 2 ) = det(u,,w 1 ) + c det(u,, w 2 ) 13

14 12. Usig Exercises 8 ad 11, show that: a. det(u + c,,w) = det(u,, w) b. det(u, + c w,w) = det(u,, w) c. det(u + c w,,w) = det(u,, w) 13. Show that: a. (u 1 u 2 ) ( 1 2 ) = det(u 1, u 2, 2 ) 1 det(u 1, u 2, 1 ) 2 b. (u 1 u 2 ) ( 1 2 ) = det(u 1, 1, 2 )u 2 det(u 2, 1, 2 )u Show that: u ( w) + (w u) + w (u ) = 0. 14

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

Inverse Matrix. A meaning that matrix B is an inverse of matrix A. Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix