Geometric Algebra: A Computational Framework For Geometrical Applications

Transcription

1 Feture Tutoril Geometric Algebr: A Computtionl Frmework For Geometricl Applictions Prt This is the second of two-prt tutoril on geometric lgebr. In prt one, we introduced bldes, computtionl lgebric representtion of oriented subspces, which re the bsic elements of computtion in geometric lgebr. We lso looked t the geometric product nd two products derived from it, the inner nd outer products. A crucil feture of the geometric product is tht it is invertible. From tht first rticle, you should hve gthered tht every vector This second prt of our spce with n inner product hs geometric lgebr, whether or not tutoril uses geometric you choose to use it. This rticle shows how to cll on this structure lgebr to represent to define common geometricl constructs, ensuring consistent computtionl frmework. The gol is to rottions, intersections, nd show you tht this cn be done nd differentition. We show tht it is compct, directly computtionl, nd trnscends the dimensionlity of subspces. We will not how smll set of products use geometric lgebr to develop simplifies mny geometricl new lgorithms for grphics, but we hope to convince you tht you cn opertions. utomticlly tke cre some of the lower level lgorithmic spects, without tricks, eceptions, or hidden degenerte cses by using geometric lgebr s lnguge. Rottions Geometric lgebr hndles rottions of generl subspces in V m through n interesting sndwiching product using geometric products. We introduce this construction grdully. Rottions in D In the lst rticle, we sw tht the rtio of vectors Stephen Mnn University of Wterloo Leo Dorst University of Amsterdm defines rottion/diltion opertor. Let us do slightly simpler problem. In Figure, if nd b hve the sme norm, wht is the vector in the ( b) plne tht is to c s the vector b is to? Geometric lgebr phrses this s c = b nd solves it (see Eqution 3 from prt one ) s b c () Here Iφ is the ngle in the I plne from to b, so Iφ is the ngle from b to. Figure suggests tht we obtin from c by rottion, so we should pprently interpret premultiplying by e Iφ s rottion opertor in the I plne. The vector c in the I plne nticommutes with I: ci= Ic showing tht I is not merely comple number, even though I =. Using this to switch I nd c in Eqution, we obtin tht the rottion is lterntively representble s postmultipliction: e = ( ) = + Iφ b = cosφ Isinφ c ( ) = ( ) b b c Iφ c = ccosφ Icsinφ = ccosφ+ cisinφ= ce () Wht is ci? First, temporrily introduce orthonorml coordintes {e, e } in the I plne, with e long c, so tht c ce. Then, I = e e = e e. Therefore, ci = ce e e = ce it is c turned over right ngle, following the orienttion of the -blde I (here nticlockwise). So c cosφ + ci sinφ is bit of c plus bit of its nticlockwise perpendiculr, nd those mounts re precisely right to mke it equl to the rottion by φ (see Figure b). e Iφ c 58 July/August /0/$ IEEE

2 Angles s geometricl objects In Eqution, the combintion Iφ is full indiction of the ngle between the two vectors. It denotes not only the mgnitude but lso the plne in which the ngle is mesured nd even the ngle s orienttion. If you sk for the sclr mgnitude of Iφ in the plne I (the plne from b to rther thn from to b), it is φ. Therefore, the sclr vlue of the ngle utomticlly gets the right sign. The fct tht the ngle s epressed by Iφ is now geometricl quntity independent of the convention used in its definition removes mjor hedche from mny geometricl computtions involving ngles. We cll this true geometric quntity the bivector ngle. (It is just -blde, of course, nd not new kind of element, but we use it s n ngle, hence the nme). Rottions in m dimensions Eqution rottes only within the plne I. Generlly, we would like to hve rottions in spce. For vector, the outcome of rottion R Iφ should be R Iφ = + R Iφ where nd re the perpendiculr nd prllel components of reltive to the rottion plne I, respectively. We hve seen tht we cn seprte vector into such components by commuttion (s in Equtions 3 nd 3 from prt one ). As you cn verify, the following formul effects this seprtion nd rottion simultneously: rottion over Iφ: R Iφ = e Iφ/ e Iφ/ (3) The opertor e Iφ/, used in this wy, is clled rotor. In the D rottion we treted before, I= I, nd moving either rotor to the other side of retrieves Eqution if is in the I plne. Two successive rottions R nd R re equivlent to single new rottion R of which the rotor R is the geometric product of the rotors R nd R, since ( RoR ) = R( R R ) R RR RR R R = ( ) ( ) = with R = R R. Therefore, the combintion of rottions is simple consequence of the ppliction of the geometric product on rotors tht is, elements of the form e Iφ/ = cos (φ/) I sin (φ/), with I =. This is true in ny dimension greter thn (nd even in dimension, if you relize tht ny bivector there is zero, so tht rottions do not eist). Let us see how it works in 3-spce. In three dimensions, we re used to specifying rottions by rottion is rther thn by rottion plne I. Given unit vector for n is, we find the plne s the -blde complementry to it in the 3D spce with volume element I 3: I = I 3 = I 3 = I 3. A rottion over n ngle φ round n is with unit vector is therefore represented by the rotor e I3φ/. For emple, to compose rottion R round the e is of π/ with subsequent rottion R over the e is () over π/, we write out their rotors (using the shorthnd e 3 = e e 3 = e e 3 nd so on): nd R R I-plne b I3e π / = e = e 4 3 I3e π / = e = e 4 3 The totl rotor is their product, nd we rewrite it bck to the eponentil form to find the is: R RR = 3 3 e e = ( ) e e e + = 3 3 I e e e 3 e ( )( ) Iπ 3 / 3 3 Therefore, the totl rottion is over the is =(e + e e 3)/ 3, over the ngle π/3. Geometric lgebr permits strightforwrd generliztion to the rottion of higher dimensionl subspces. We cn pply rotor immeditely to n rbitrry blde through the formul generl rottion: X R X R c c I (b) I-plne Rc = e Iφ c = ce Iφ This lets you rotte plne in one opertion, for instnce, using rottion by R (s in the emple we just sw): R( e e) R = ( e e e e e e e e 4 + ) ( + + ) = There is no need to decompose the plne into its spnning vectors first. Quternions bsed on bivectors You might hve recognized the lst emple s strongly similr to quternion computtions. Quternions re indeed prt of geometric lgebr in the following strightforwrd mnner. Choose n orthonorml bsis {e i} 3 i=. Construct out c () The rottion opertor s rtio between vectors. (b) Coordinte-free specifiction of rottion. IEEE Computer Grphics nd Applictions 59

3 Feture Tutoril () I 3 b B* b B ^ B (b) B () The dul B* of bivector B nd the cross product with. (b) The sme result using the inner product of bldes (from prt one ). of tht bivector bsis {e, e 3, e 3}. Note tht these elements stisfy e =e 3=e 3=, e e 3 = e 3 (nd cyclic), nd lso e e 3 e 3 =. In fct, setting i e 3, j e 3, nd k e, we find i = j = k = i j k = nd j i = k nd cyclic. Algebriclly, these objects form bsis for quternions obeying the quternion product, commonly interpreted s some kind of 4D comple number system. To us, there is nothing comple bout quternions, but they re not vectors either they re rel - bldes in 3-spce, denoting elementry rottion plnes nd multiplying through the geometric product. Therefore, visulizing quternions is strightforwrd; ech is rottion plne with rottion ngle, nd the bivector ngle concept represents tht well. Of course, in geometric lgebr, we cn combine quternions directly with vectors nd other subspces. In tht lgebric combintion, they re not merely form of comple sclrs; quternion products re neither fully commuttive nor fully nticommuttive (for emple, i e = e i, but i e = e i). It ll depends on the reltive ttitude of the vectors nd quternions, nd these rules re precisely right to mke Eqution 3 be the rottion of vector. Liner lgebr In the clssicl wys of using vector spces, liner lgebr is n importnt tool. In geometric lgebr, this remins true; liner trnsformtions re of interest in their own right or s first order pproimtions to more complicted mppings. Indeed, liner lgebr is n integrl prt of geometric lgebr, nd it cquires much etended coordinte-free methods through this inclusion. We show some of the bsic principles in this rticle, but you cn find more in relted literture.,3 Outermorphisms When vectors re trnsformed by liner trnsformtion on the vector spce, the bldes they spn cn be viewed to trnsform s well, simply by the rule the trnsform of spn of vectors is the spn of the trnsformed vectors. This mens tht liner trnsformtion f: V m V m of vector spce hs nturl etension to the whole geometric lgebr of tht vector spce, s n outermorphism tht is, s mpping tht preserves the outer product structure: f( k) f( ) f( ) f( k) (4) B Note tht this is grde preserving k-blde trnsforms to k-blde. We supplement this by stting wht the etension does to sclrs, which is simply f(α) = α. Geometriclly, this mens tht liner trnsformtion leves the origin intct. Liner trnsformtions re outermorphisms, which eplins why we cn generlize so mny opertions from vectors to generl subspces in strightforwrd mnner. Dul representtion Liner lgebr often uses dul representtions of hyperplnes to etend the scope of vector-bsed opertions. In geometric lgebr, duliztion is lso importnt nd in fct is fleible tool to convert between viewpoints of spnning nd perpendiculrity, for rbitrry subspces. Consider n m-dimensionl spce V m nd blde A in it. The dul of the blde A in V m is its complement, definble using the contrction inner product: A = A I m where I m is the pseudosclr of V m (n m-blde giving the volume element) nd ~ is the reversion (obtined by reversing the fctors of the blde it is pplied to nd leding to grde-dependent sign chnge). The chrcteriztion of subspce by dul blde rther thn by blde enbles mnipultion of epressions involving spnning to being bout perpendiculrity nd vice vers. A fmilir emple in 3D Eucliden spce is the dul of -blde (or bivector). Using n orthonorml bsis {e i} 3 i= nd the corresponding bivector bsis, we write B = b e e 3 + b e 3 e + b 3e e. We tke the dul reltive to the spce with volume element I 3 e e e 3 (tht is, the right-hnded volume formed by using right-hnded bsis). The subspce of I 3 dul to B is then B 3 I = = ( b e e + b e e + b e e ) e e e = b e + b e + b e ( ) (5) This is vector, nd we recognize it (in this Eucliden spce) s the norml vector to the plnr subspce represented by B (see Figure b). So we hve norml vectors in geometric lgebr s the duls of -bldes, if we would wnt them but we will soon see better lterntive. We cn use either blde or its dul to represent subspce, nd it is convenient to hve some terminology. We will sy tht blde B represents subspce B if B B = 0 (6) nd tht blde B* dully represents the subspce B if B B* = 0 (7) We switch between the two stndpoints by using the distributive reltion (A B) C = A (B C) (which is Equ- 60 July/August 00

4 tion 9 from prt one ) used for vector, blde B, nd pseudosclr I: ( B) I= ( B I) (8) nd by converse (but conditionl) reltionship tht we stte without proof ( B) I= ( B I) if I= 0 (9) If is known to be in the subspce of I, we cn write these simply s ( B)* = B* nd ( B)* = B*, which mkes the equivlence of the two representtions obvious. Cross product Clssicl computtions with vectors in 3-spce often use the cross product, which produces from two vectors nd b new vector b perpendiculr to both (by the right-hnd rule), proportionl to the re they spn. We cn mke this in geometric lgebr s the dul of the -blde spnned by the vectors (see Figure b): ( ) = ( ) b b 3 I b (0) You cn verify tht computing this eplicitly using Eqution from prt one nd Eqution 5 indeed retrieves the usul epression b = ( b 3 3b )e + ( 3b b 3)e + ( b b )e 3. Eqution 0 eplicitly shows severl things tht we lwys need to remember bout the cross product: there is convention involved on hndedness (this is coded in the sign of I 3), there re metric spects becuse it is perpendiculr to plne (this is coded in the usge of the inner product ), nd the construction only works in three dimensions becuse only then is the dul of -blde vector (this is coded in the 3-grdedness of I 3). The vector product b does not depend on ny of these embedding properties yet chrcterizes the (, b) plne just s well. In geometric lgebr, we therefore hve the possibility of replcing the cross product by more elementry construction. Liner lgebr gives good reson for doing so. No norml vectors or cross products The trnsformtion of n inner product under liner mpping is more involved thn tht of the outer product in Eqution 4 becuse perpendiculrity is more complicted concept to trnsform thn spnning. Hestenes 3 gives the generl trnsformtion formul. For bldes, it becomes ( )= ( ) ( ) f A B f A f B where - f is the djoint, defined s the etension of n outermorphism of the liner mpping defined for vectors by - f() b= f(b) its mtri representtion on vectors would be the trnspose. Becuse of the compleity of this trnsformtion behvior, we should steer cler of ny constructions tht involve the inner product, especilly when chrcterizing bsic properties of geometric objects. The prctice of chrcterizing plne by its norml vector which contins the inner product in its dulity should be voided. Under liner trnsformtions, the norml vector of trnsformed plne is not the trnsform of the plne s norml vector. (This is well-known fct, but it is lwys shock to novices.) Rther, the norml vector is cross product of vectors, which trnsforms s ( ) = ( ) ( ) ( ) f b f f b det f () where det(f) is the determinnt defined by det(f)=f(i m)i m. (This is coordinte-free definition of the determinnt of the mtri representtion of f.) The right-hnd side of Eqution is usully not equl to f() f(b), so liner trnsformtion is not cross-product preserving. Therefore, it is much better to chrcterize the plne by -blde, now tht we cn. The -blde of the trnsformed plne is the trnsform of the -blde of the plne becuse liner trnsformtions re outermorphisms preserving the -blde construction. Especilly when the plnes re tngent plnes constructed by differentition, - bldes re pproprite. Under ny trnsformtion f, the construction of the tngent plne depends only on the first-order liner pproimtion mpping f of f. Using bldes for those tngent spces should enormously simplify the tretment of objects through differentil geometry, especilly in the contet of ffine trnsformtions. Intersecting subspces Geometric lgebr lso contins opertions to determine the union nd intersection of subspces. These re the Join nd Meet opertions. Severl nottions eist for these in the relted literture, cusing some confusion. In this rticle, we will use the set nottions nd to mke the formuls more esily redble. The Join of two subspces is their smllest superspce tht is, the smllest spce contining them both. Representing the spces by bldes A nd B, the Join is denoted A B. If the subspces of A nd B re disjoint, their Join is obviously proportionl to A B. A problem is tht if A nd B re not disjoint (which is precisely the cse we re interested in), then A B contins n unknown scling fctor tht is fundmentlly unresolvble due to the bldes reshpble nture (see Figure 3). (Stolfi 4 lso observed this mbiguity.) Fortuntely, in ll geometriclly relevnt entities tht we compute, it ppers tht this sclr mbiguity cncels out. The Join is more complicted product of subspces thn the outer product nd inner product. We cn give no simple formul for the grde of the result nd cnnot chrcterize it with list of lgebric computtion rules. Although computtion of the Join IEEE Computer Grphics nd Applictions 6

5 Feture Tutoril 3 The mbiguity of scle for Meet M nd Join J of two bldes A nd B. Both figures re cceptble solutions to the problem of finding blde representing the union nd intersection of the subspces of the bldes A nd B. B M M B A J A J B = e e. (We hve normlized them so tht A Ã==B B; it gives the Meet of these bldes numericl fctor tht we cn interpret geometriclly.) These re plnes in generl position in 3D spce, so their Join is proportionl to I 3. It mkes sense to tke J = I 3. For the Meet, this gives A B= = ( e e ) ( e e e ) e + e e + e = e3 (( e+ e) e3) e + e = ( e + e ) = ( 3 ) (( ) ( 3) ) (4) 4 An emple of the Meet of two plnes. e 3 A B A e e B We hve epressed the result in normlized form. The numericl fctor for the resulting blde is in fct the sine of the ngle between the rguments. Figure 4 illustrtes the nswer. As in Stolfi, 4 the sign of A B is the righthnd rule pplied to the turn required to mke A coincide with B, in the correct orienttion. Clssiclly, we usully compute the intersection of two plnes in 3-spce by first converting them to norml vectors nd then tking the cross product. This gives the sme nswer in this nondegenerte cse in 3-spce, using our previous Eqution 9, Eqution 8, nd noting tht Ĩ 3= I 3: my pper to require some optimiztion process, the smllest superspce is directly relted to the nonzero prt of highest grde in the epnsion of the geometric product nd cn therefore be done in constnt time. 5 The Meet of two subspces A nd B is their lrgest common subspce. Given the Join J A B of A nd B, we cn compute their Meet A B by the property tht its dul (with respect to the Join) is the outer product of their duls. In formul, this is ( A B) J = ( B J ) ( A J ) or ( ) = A B B A () with the dul tken with respect to the Join J. (The somewht strnge order in Eqution mens tht the Join J cn be written using the Meet M in the fctoriztion J = (AM ) M (M B), nd it corresponds to Stolfi 4 for vectors.) This leds to formul for the Meet of A nd B reltive to the chosen Join (use Eqution 8): A B B J A =( ) (3) Let us do n emple: the intersection of two plnes represented by the -bldes A= (e +e ) (e +e 3) nd ( ) ( )= (( ) ( )) A I B I A I B I I = (( B I ) ( A I )) I = ( B I3) ( A I3) I3 B I A A B ( ) = ( ) = So the clssicl result is specil cse of Eqution 3, but Eqution 3 is much more generl; it pplies to the intersection of subspces of ny grde, within spce of ny dimension. The norm of the Meet gives n impression of the intersection strength. Between normlized subspces in Eucliden spce, the mgnitude of the Meet is the sine of the ngle between them. From numericl nlysis, this is well-known mesure for the distnce between subspces in terms of their orthogonlity it is if the spces re orthogonl nd decys grcefully to 0 s the spces get more prllel, before chnging signs. This numericl significnce is useful in pplictions. Differentition Geometric lgebr hs n etended opertion of differentition, which contins the clssicl vector clculus nd much more. It is possible to differentite with respect to sclr or vector, s before, but now lso with respect to k-bldes. This enbles efficient encoding of differentil geometry, in coordinte-free mnner nd gives n lterntive look t differentil shpe 6 July/August 00

6 descriptors like the second fundmentl form. (It becomes n immedite indiction of how the tngent plne chnges when we slide long the surfce.) This would led us too fr, but we will show two emples of differentition. A rotor s sclr differentition Suppose we hve rotor R = e Iφ/ (where Iφ is function of time t) nd use it to produce rotted version X = RX 0R of some constnt blde X 0. Using the chin rule nd commuttion rules, sclr differentition with respect to t gives P() P () 5 The derivtive of the sphericl projection. d X dt d dt e Iφ/ e Iφ/ Xo d Iφ/ Iφ/ Iφ e Xoe dt Iφ/ Iφ/ d + ( e Xoe ) ( Iφ) dt d d X I I X φ φ dt dt d X Iφ dt = ( ) = ( )( ) = ( ) ( ) = ( ) (5) using the commuttor product defined in geometric lgebr s the shorthnd A B /(AB BA). This product often crops up in computtions with continuous groups such s the rottions. The simple epression tht results ssumes more fmilir form when X is vector in 3-spce nd when the ttitude of the rottion plne is fied so tht di/dt =0. We introduce sclr ngulr velocity ω dφ/dt, nd the vector dul to the plne s the ngulr velocity vector w, so w ωi Ĩ 3=ωI/I 3. Therefore, ωi = wi 3, which equls w I 3. Using the fct tht B=/(B B)= B for vector nd -blde B, we obtin d d Iφ w I3 dt dt = ( w I ) = w I w = ( ) = ( ) 3 ( ) 3 = where is the vector cross product. As before, when we treted other opertions, we find tht n eqully simple geometric lgebr epression is more generl. Here Eqution 5 describes the differentil rottion of k dimensionl subspces in n dimensionl spce, rther thn merely of vectors in 3D. Differentition of sphericl projection Suppose tht we project vector on the unit sphere by the function P()=/. We compute its derivtive in the direction, denoted s ( )P() or P (), s stndrd differentil quotient nd using Tylor series epnsion. Note how geometric lgebr permits compct epression of the result, with geometricl significnce: Homogeneous model So fr we hve been treting only homogeneous subspces of the vector spce V m tht is, subspces contining the origin. We hve spnned them, projected them, nd rotted them, but we hve not moved them out of the origin to mke more interesting geometricl structures such s lines floting in spce. We construct those now, by etending the ides behind homogeneous coordintes to geometric lgebr. It turns out tht such elements of geometry cn lso be represented by bldes, in representtionl spce with n etr dimension. The geometric lgebr of this spce gives us precisely wht we need. In this view, more complicted geometricl objects do not require new opertions or tech- ( ) lim λ 0 = lim = lim = + λ λ + λ + λ λ + λ + λ λ λ 0 λ 0 ( ) ( ) λ ( ) = ( ) We recognize the result s the rejection of by, scled ppropritely (see the Projecting subspces section from prt one ). Figure 5 confirms the outcome. You cn verify in similr mnner tht ( ) = nd interpret geometriclly. For more dvnced usge of differentition reltive to bldes, see the Dorn et l. tutoril, which introduces these differentitions using emples from physics, nd the Lsenby et l. 6 ppliction pper. Models of geometry Geometric lgebr cn help us epress severl stndrd models of geometry. The dvntge of doing so is n increse in epressive power nd structurl integrtion of seemingly d-hoc constructions into the geometry. Here we look t geometric lgebr representtions of homogeneous coordintes nd Plücker coordintes s well s model of Eucliden geometry tht nturlly hndles spheres. IEEE Computer Grphics nd Applictions 63

7 Feture Tutoril 6 Representing offset subspces of E m in (m + ) dimensionl spce. () A point P denoted by vector p with Eucliden prt p. (b) A line element is represented by the bivector formed s the outer product of two points. (c) Reshping the bivector shows the correspondence with Plücker coordintes; here, v q p. p P e p E m P Q e p q p ^ q v () (b) (c) E m e p P Q v p ^ v E m niques, merely the stndrd computtions in higher dimensionl spce. The homogeneous model is often described s ugmenting 3D vector v with coordintes [v, v, v 3] to 4-vector [v, v, v 3, ]. Tht etension mkes nonliner opertions such s trnsltions implementble s liner mppings. We give the (m + )-dimensionl homogeneous spce into which we embed our m-dimensionl Eucliden spce full geometric lgebr. Let the unit vector for the etr dimension be denoted by e. This vector must be perpendiculr to ll regulr vectors in the Eucliden spce E m, so e = 0 for ll E m. We lso need to define e e to mke our lgebr complete. This involves some dilemms tht re only fully resolvble in the double homogeneous model, which we eplin lter on. For now, we cn tke e e=. We interpret es the Eucliden point t the origin. Let us now represent the subspces of interest into this model, simply by using the structure of its geometric lgebr. Points. A point t loction p is mde by trnslting the point t the origin over the Eucliden vector p. We do this by dding p to e. This construction gives the representtion of the point P t loction p s the vector p in (m + )-dimensionl spce: p = e + p This is regulr vector in the (m + )-spce, now interpretble s Eucliden point. It is of course no more thn the usul homogeneous-coordintes method in disguise p hs coordintes [p, p, p 3, ] on the orthonorml bsis {e, e, e 3, e}. We will denote points of the m-dimensionl Eucliden spce in script, the vectors nd bldes in the corresponding vector spce in bold, nd vectors nd bldes in the (m + )-dimensionl homogeneous spce in itlic. You cn visulize this construction s in Figure 6 (necessrily drwn for m = ). We cn multiply these vectors in (m + )-dimensionl spce using the products in geometric lgebr. Let us consider in prticulr the outer product nd form bldes. Lines. To represent line, we compute the -blde spnned by the representtive vectors of two points: p q = (e + p) (e + q) = e (q p) + p q (6) We recognize the vector q p nd the re spnned by p nd q. Both re elements tht we need to describe n element of the directed line through the points P nd Q. The former is the direction vector of the directed line; the ltter is n re tht we cll the moment of the line through p nd q. It specifies the distnce to the origin, for we cn rewrite it to rectngle spnned by the direction (q p) nd the perpendiculr support vector d: p q = p (q p) = d(q p) (7) where d (p (q p))(q p) = (p q)(q p) is the rejection of p by q p. We cn therefore rewrite the sme -blde p q in vrious wys, such s p q = p (q p) = d(q p) (with d = e + d), seprting the positionl prt p or d nd the purely Eucliden directionl prt v q p (see Figure 6c). The element p q contins ll these potentil interprettions in one dt structure, which we cn construct in ll these wys. Temporrily reverting to the cross product to mke connection with clssicl representtion, we cn rewrite Eqution 6 s p q = e (q p) + p q = (p q)e + (p q)i3 (8) We recognize the si Plücker coefficients [p q; p q], chrcterizing the line by its direction vector v = q p nd its moment vector m = p q s l = ve + mi3 The si coefficients [ v, v, v 3, m, m, m 3] of the line in this representtion re the coefficients of -blde on the bivector bsis {e e, e e, e 3e, e e 3, e 3e, e e }. This integrtes the Plücker representtion fully into the homogeneous model (for which it ws historiclly designed). We will see tht the compct nd efficient Plücker intersection formuls re now strightforwrd consequences of the Meet opertion in geometric lgebr. Hyperplnes. If we hve n (m )-dimensionl hyperplne chrcterized s n = δ, this cn be written s (e + ) (n δe) = 0, so (n δe) = 0. Therefore, n δe is the dul of the blde representing the hyperplne. The dul A* of blde A in the homogeneous model is obtined reltive to the pseudosclr e I3 = ei3 of the full spce s A* = A (ei3) = A (ei3) = A(eI3), so we get (n δe)(ei3) = nei3 δi3 This hs the usul four Plücker coefficients [n, n, n 3, δ], but on the trivector bsis { e e 3e, e 3e e, e e e, e e e 3} it is clerly different from the vector bsis for points. 64 July/August 00

8 Of course, we cn lso construct the blde representing the hyperplne directly (rther thn dully), given m points on it nmely, s the outer product of the vectors representing those points. And beyond. These wys of mking offset plnr subspces etend esily. An element of the oriented plne through the points P, Q, nd R is represented by the 3-blde p q r nd so on for higher dimensionl offset subspces if the spce hs enough dimensions to ccommodte them. The bldes we construct this wy cn lwys be rewritten in the form A = da, where A is purely Eucliden blde nd d is vector of the form e + d, with d Eucliden vector. We should interpret A s the direction element, so its grde denotes the dimensionlity of the flt subspce represented by A. The vector d represents the closest point to the origin (so tht d is the perpendiculr support vector). Sclr distnces. A smll surprise is tht even 0 blde (tht is, sclr) is useful; it is the representtion of sclr distnce in the Eucliden spce (with sign but without direction), s we will see in the net section. Such distnces re of course regulr elements of geometry, so it is stisfying to find them on pr with position vectors, direction vectors, nd other elements of higher dimensionlity s just nother cse of representing blde in the homogeneous model of flt Eucliden spce. Cseless subspce interctions Hving such unified representtion for the vrious geometricl elements implies tht computtions using them re unified s well; they hve just become opertions on bldes in (m + )-spce, blissfully ignornt of wht different geometricl situtions these computtions might represent. This opens the wy to cseless computtion in geometricl lgorithms. The Meet nd Join in the homogeneous model function just s you would epect, providing the intersection of lines, plnes, nd so forth. Writing these out in their (Plücker) coordintes retrieves the fmilir compct formuls for the coefficients of the result, but they re now ccompnied by utomtic evlution of the bsis on which these coefficients should be interpreted. Tht provides immedite identifiction of the kind of intersection in mnner so well integrted tht it suggests we might continue our computtion without intermedite interprettion. This leds to cseless geometricl lgorithms in which the dimensionlity of intermedite results does not ffect the dt flow. Even though the ctul computtion is cseless Meet pplied to bldes, let us see wht is going on in detil in some typicl situtions. Following the internl computtionl combintion of the Plücker-like coefficients shows how the lgebr of the bsis elements tkes cre of the proper intersection computtion, t smll dditionl epense compred to the usul implementtion using precompiled tbles of intersection formuls. (We write the contrction of vectors s the clssicl inner product to show the correspondence clerly.) Line nd plne. The Meet of line l nd plne π* = n δe in generl position is computed s l π= π* l = (n δe) ( ve + mi 3) = (n v)e + n (mi3) δv + 0 = (n v)e + (n m)i 3 δv = (n v)e + (m n δv) This is the correct result, representing point t the loction (δv + n m)/(n v) in its homogeneous (Plücker) coordintes. Note how the orthogonlity reltionships between the bsis elements utomticlly kill the potentil term involving δ nd m. But the fct tht this term is zero is computed, nd tht is slight inefficiency reltive to the direct implementtion of the sme result from tble with Plücker formuls. It is the computtionl price we py for the membership of the full geometric lgebr. Two lines. The Meet of two lines in generl position is mesure of their signed distnce (remember tht in this model dul is mde through right-multiply by ei3, so the dul of the line v e + m I 3 is v I 3 + m e): ( ) ( + ) 3 3 The inner product is defined in terms of the Eucliden distnce d E between points: p q= d E(P,Q). Tht mens tht the representtionl spce hs rther specil metric becuse it follows tht p p = 0 for ny vector p. Only when we ssign specil point s the origin cn we define vectors denoting the reltive position of point such vector p does of course hve nonzero norm. But we cn specify ll computtions without ever introducing such n origin. The outer product constructs spheres k-blde repl l = l l = v I + m e v e m I = m v + v m retrieving the well-known compct Plücker wy of determining how lines pss ech other in spce. Three tests on the signs of such quntities representing the edges of tringle determine efficiently whether ry hits the tringle. Agin, the bsis orthogonlity reltionships hve mde terms contining m m or v v equl to zero. The directionl outcomes re ccompnied by numericl fctors (such s n v in the first emple) relting to the computtion s numericl significnce. These re n intrinsic prt of the object s computtion, not just secondry spects we need to think of seprtely (with the dnger of being d hoc) or tht we need to compute seprtely (costing time). Double homogeneous model Hestenes 7 hs recently shown tht embedding of Eucliden spce into representtionl spce of two etr dimensions nd its geometric lgebr is powerful nd simplifying. This double homogeneous model of Eucliden spce embeds the Eucliden distnce properties into the fbric of the lgebr used to compute with it. IEEE Computer Grphics nd Applictions 65

9 Feture Tutoril Correction Due to printing error, the indices in Eqution of prt one re unredble. The eqution should red ( i j) = ( i j) ( i j) = ( i j) ( i j) e e e e e e ee ee = eeee = eeee = i j i j i i j j resents Eucliden (k )-sphere. As consequence, p q is the ordered point pir (P, Q) nd p q r is the circle through P, Q, nd R. This provides Plücker coordintes for spheres. The dul of the (m + )-blde representing n m-sphere in E m is vector in the double homogeneous representtion spce, which hs coefficients tht immeditely provide the sphere s center nd rdius. Flt subspces re represented s spheres through infinity. This is possible becuse one of the two etr representtionl dimensions is vector representing the point t infinity. The sndwiching by the geometric product gives rottions nd ll conforml mppings, including trnsltion nd sphericl inversion. This is why the double homogeneous model is often clled the conforml model. The Meet of two bldes is interpretble s intersecting k-spheres in m-spce, nd its embedding gin reduces the seprte cses tht would need to be distinguished for such intersections. This model looks pproprite for mny computer grphics pplictions, nd we re currently developing it further for prcticl usge. It is truly coordinte-free model, in which we cn specify ll opertions of Eucliden geometry without ever referring to n origin. Implementtion The geometric lgebr of n m-dimensionl vector spce contins nicely liner objects (the bldes) tht cn be represented on bsis tht should contin m elements (becuse we need ( m k ) for ech k blde). The vrious products re ll liner, nd we cn implement them using mtri products of m m mtrices. Tht might be strightforwrd, but it is obviously inefficient in both spce nd time. This is even more urgent when we use the homogeneous model in which n m-dimensionl Eucliden spce requires n (m + )-dimensionl geometric lgebr, or the (m + )-dimensionl double homogeneous model. It seems lost cuse. However, recognizing tht the importnt elements re bldes nd their products suggests more efficient implementtion. When two bldes multiply by n inner or outer product, blde of unique grde results. This suggests designing dt representtion for the elements of geometric lgebr tht permits esy retrievl of their grdes nd lso utomticlly generting optimized code for the inner nd outer multipliction of bldes of specific grdes k nd l. Their geometric product genertes more generl element of mied, but still limited, set of grdes l k, l k +, k + l. Division requires inversion, but this is closely relted to the much simpler opertion of reversion simply switching the signs of certin grdes. The structurl membership of n element to the lrger geometric lgebr, with its benefit of unified reltionships between the vrious opertions, is then merely pid for by grde-dependent jump to piece of code. When processing dt in btch mode with mny similr opertions, this would not slow down things significntly. Work is underwy on n efficient implementtion tht cpitlizes on these structurl properties of geometric lgebr. 8 Our first results look promising nd re getting close to the usul efficiency of the geometricl computtions but in simpler code without eceptions or d-hoc dt structures nd nturlly integrting computtionl techniques tht clssiclly belong to different relms thn vector/mtri lgebr (such s quternions nd Plücker coordintes). Conclusion This two-prt introduction to geometric lgebr intends to lert you to the eistence of smll set of products tht ppers to generte ll geometric constructions in one consistent frmework. Using this frmework cn simplify the set of dt structures representing objects becuse it inherently encodes ll reltionships nd symmetries of the geometricl primitives in those opertors. Although there re mny interesting fcets to geometric lgebr, we would like to highlight the following: Division by subspces. Hving geometric product with n inverse lets us divide by subspces, incresing our bility to mnipulte lgebric equtions involving vectors. Subspces re bsic elements of computtion. Thus, no specil representtions re needed for subspces of dimension greter thn (for emple, tngent plnes), nd we cn mnipulte them like we mnipulte vectors. Generliztion. Epressions for opertions on subspces re often s simple s those for vectors (tht is especilly true for liner opertions) nd s esy to compute. Cseless computtion. Degenerte cses re computed utomticlly, results remin interpretble, nd the computtion lets us test the solution s numerics. Quternions. In geometric lgebr, quternions re subsumed nd become nturl prt of the lgebr, with no need to convert between representtions to perform rottions. Plücker coordintes. Geometric lgebr subsumes nd etends Plücker coordintes nd the concise epressions they give for the interctions of lines, plnes, nd so on. This rticle only covers some of wht we feel re the most importnt or useful ides of geometric lgebr s it reltes to computer grphics. We hve left out mny topics, including description of more geometries (the homogeneous model implements nd generlizes the Grssmnn spces of Goldmn 9 nd the double homogeneous model implements nd generlizes projective spces), nd we could sy lot more bout differentition 66 July/August 00

10 nd coordinte-free differentil geometry. You should be ble to glen the connections from the Further Reding sidebr, lthough n ccessible eplntion for computer grphics of such issues is still necessry. Acknowledgments The Netherlnds Orgniztion for Scientific Reserch nd the Nturl Sciences nd Engineering Reserch Council of Cnd supported this work in prt. References. L. Dorst nd S. Mnn, Geometric Algebr: A Computtionl Frmework for Geometricl Applictions (Prt ), IEEE Computer Grphics nd Applictions, vol., no. 3, My/June 00, pp C. Dorn nd A. Lsenby, Physicl Applictions of Geometric Algebr, 00, ptiiicourse/. 3. D. Hestenes, The Design of Liner Algebr nd Geometry, Act Applicnde Mthemtice, vol. 3, 99, pp J. Stolfi, Oriented Projective Geometry, Acdemic Press, Sn Diego, T.A. Boum, L. Dorst, nd H. Pijls, Geometric Algebr for Subspce Opertions, to be published in Act Applicnde Mthemtice, preprint vilble mth.la/ J. Lsenby et l., New Geometric Methods for Computer Vision, Int l J. Computer Vision, vol. 36, no. 3, 998, pp D. Hestenes, Old Wine in New Bottles, Geometric Algebr: A Geometric Approch to Computer Vision, Quntum nd Neurl Computing, Robotics, nd Engineering, E. Byro- Corrochno nd G. Sobczyk, eds., Birkhäuser, Boston, 00, pp D. Fontijne, GAIGEN: A Geometric Algebr Implementtion Genertor, 9. R. Goldmn, The Ambient Spces of Computer Grphics nd Geometric Modeling, IEEE Computer Grphics nd Applictions, vol. 0, no., Mr./Apr. 000, pp Stephen Mnn is n ssocite professor in the School of Computer Science t the University of Wterloo. He hs BA in computer science nd pure mthemtics from the University of Cliforni, Berkeley, nd PhD in computer science nd engineering from the University of Wshington. His reserch interests re in splines nd the mthemticl foundtions of computer grphics. Further Reding There is growing body of literture on geometric lgebr. Unfortuntely, much of the more redble writing is not very ccessible nd in specilized books rther thn generl journls. Reserchers hve written little with computer science in mind becuse the initil pplictions hve been to physics. We recommend the following s nturl follow-ups to this rticle: Gble, Mtlb pckge for geometric lgebr, ccompnied by tutoril. An ppliction softwre genertor Gigen. The introductory chpters of New Foundtions of Clssicl Mechnics. 3 An introductory course intended for physicists. 4 An ppliction to bsic but involved geometry problem in computer vision, with brief introduction into geometric lgebr. 5 Ppers showing how liner lgebr becomes enriched by viewing it s prt of geometric lgebr. 6,7 Red them in pproimtely this order. We re working on tets more specificlly suited for computer grphics udience, which my first pper s Siggrph courses. References. L. Dorst, S. Mnn, nd T.A. Boum, GABLE: A Geometric Algebr Lerning Environment, uv.nl/~leo/gable/.. D. Fontijne, GAIGEN: A Geometric Algebr Implementtion Genertor, gigen/. 3. D. Hestenes, New Foundtions for Clssicl Mechnics, nd ed., D. Reidel, Dordrecht, C. Dorn nd A. Lsenby, Physicl Applictions of Geometric Algebr, 00, uk/~clifford/ptiiicourse/. 5. J. Lsenby et l., New Geometric Methods for Computer Vision, Int l J. Computer Vision, vol. 36, no. 3, 998, pp C. Dorn, A. Lsenby, nd S. Gull, Liner Algebr, Clifford (Geometric) Algebrs with Applictions in Physics, Mthemtics, nd Engineering, W.E. Bylis, ed., Birkhäuser, Boston, D. Hestenes, The Design of Liner Algebr nd Geometry, Act Applicnde Mthemtice, vol. 3, 99, pp Leo Dorstis n ssistnt professor t the Informtics Institute t the University of Amsterdm. His reserch interests include geometric lgebr nd its pplictions to computer science. He hs n MSc nd PhD in the pplied physics of computer vision from Delft University of Technology, The Netherlnds. Reders my contct Stephen Mnn t the School of Computer Science, Univ. of Wterloo, 00 University Ave. W, Wterloo, Ontrio, Cnd, emil smnn@uwterloo.c. For further informtion on this or ny other computing topic, plese visit our Digitl Librry t org/publictions/dlib. IEEE Computer Grphics nd Applictions 67