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ANTI-COMMUTATIVE DUAL COMPLEX NUMBERS AND 2D RIGID TRANSFORMATION GENKIMATSUDA,SHIZUOKAJI†,ANDHIROYUKIOCHIAI Abstract. Weintroduceanewpresentationofthetwodimensionalrigid transformationwhichismoreconciseandefficientthanthestandardma- 6 trix presentation. By modifying the ordinary dual number construction 1 for the complex numbers, we define the ring of the anti-commutative 0 dualcomplexnumbers,whichparametrizestwodimensionalrotationand 2 translation all together. With this presentation, one can easily interpo- n late or blend two or more rigid transformations at a low computational a cost.WedevelopedalibraryforC++withtheMIT-licensedsourcecode J ([13])anddemonstrateitsfacilitybyaninteractivedeformationtoolde- 8 velopedforiPad. ] R 1. RigidTransformation G . The n-dimensional rigid transformation (or Euclidean) group E(n) con- s c sists of transformations of Rn which preserves the standard metric. This [ group serves as an essential mathematical backend for many applications 1 (see[2,9]). Itiswell-known(see[6],forexample)thatanyelementofE(n) v 4 canbewrittenasacompositionofarotation,areflection,andatranslation, 5 andhence,itisrepresentedby(n+1)×(n+1)-homogeneousmatrix; 7 (cid:40) (cid:32) (cid:33) (cid:41) 1 Aˆ d 0 E(n) = A = A | AˆtAˆ = I ,d ∈ Rn . 0 1 n A . 1 0 Here, we adopt the convention that a matrix acts on a (column) vector by 6 the multiplication from the left. E(n) has two connected components. The 1 identity component SE(n) consists of those without reflection. More pre- : v cisely, i (cid:40) (cid:32) (cid:33) (cid:41) X Aˆ d SE(n) = A = A | Aˆ ∈ SO(n),d ∈ Rn , r 0 1 A a where SO(n) = {R | RtR = I ,det(R) = 1} is the special orthogonal group n composedofn-dimensionalrotations. The group SE(n) is widely used in computer graphics such as for ex- pressing motion and attitude, displacement ([10]), deformation ([1, 5, 11]), skinning ([8]), and camera control ([3]). In some cases, the matrix representation of the group SE(n) is not convenient. In particular, a linear com- bination of two matrices in SE(n) does not necessarily belong to SE(n) and Key words and phrases. 2D rigid transformation, 2D Euclidean transformation, dual numbers,dualquaternionnumbers,dualcomplexnumbers,deformation. †Correspondingauthor. 1 2 GENKIMATSUDA,SHIZUOKAJI†,ANDHIROYUKIOCHIAI it causes the notorious candy-wrapper defect in skinning. When n = 3, an- otherrepresentationofSE(3)usingthedualquaternionnumbers(DQN,for short) is considered in [8] to solve the candy-wrapper defect. In this paper, we consider the 2-dimensional case. Of course, 2D case can be handled by regarding the plane embedded in R3 and using DQN, but instead, we intro- duce the anti-commutative dual complex numbers (DCN, for short), which isspecificto2Dwithmuchmoreconciseandfasterimplementation([13]). Tosummarise,ourDCNhasthefollowingadvantages: • any number of rigid transformations can be blended/interpolated easilyusingitsalgebraicstructurewithnodegenerationdefectssuch asthecandy-wrapperdefect(see§5) • itisefficientintermsofbothmemoryandCPUusage(see§6). WebelievethatourDCNoffersachoiceforrepresentingthe2Drigidtrans- formationincertainapplicationswhichrequirestheaboveproperties. 2. Anti-commutativedualcomplexnumbers Let K denote one of the fields R,C, or H, where H is the quaternion numbers. First, we recall the standard construction of the dual numbers overK. Definition2.1. TheringofdualnumbersKˆ isthequotientringdefinedby Kˆ := K[ε]/(ε2) = {p + p ε | p ,p ∈ K}. 0 1 0 1 WeoftendenoteanelementinKˆ byasymbolwithhatsuchas pˆ. Theadditionandthemultiplicationoftwodualnumbersaregivenas (p + p ε)+(q +q ε) = (p +q )+(p +q )ε, 0 1 0 1 0 0 1 1 (p + p ε)(q +q ε) = (p q )+(p q + p q )ε. 0 1 0 1 0 0 1 0 0 1 Thefollowinginvolutionisconsideredtobethedualversionofconjuga- tion p(cid:93)+ p ε := p∗ − p∗ε, 0 1 0 1 where p∗ is the usual conjugation of p in K. (Note that in some literatures i i p˜ˆ isdenotedby pˆ∗.) Theunitdualnumbersareofspecialimportance. (cid:113) Definition 2.2. Let |pˆ| = pˆp˜ˆ for pˆ ∈ Kˆ. The unit dual numbers is defined as Kˆ := {pˆ ∈ Kˆ | |pˆ| = 1} ⊂ Kˆ. 1 Kˆ actsonKˆ byconjugationaction 1 pˆ (cid:5)qˆ := pˆqˆp˜ˆ where pˆ ∈ Kˆ ,qˆ ∈ Kˆ. 1 ANTI-COMMUTATIVEDUALCOMPLEXNUMBERSAND2DRIGIDTRANSFORMATION3 The unit dual quaternion Hˆ is successfully used for skinning in [8]; a 1 vector v = (x,y,z) ∈ R3 is identified with 1+(xi+yj+zk)ε ∈ Hˆ and the 1 conjugationactionofHˆ preservestheembeddedR3 anditsEuclideanmet- 1 ric. Infact,theconjugationactioninducesthedoublecoverHˆ → SE(3). 1 On the other hand, when K = R or C, the conjugation action is triv- ial since the corresponding dual numbers are commutative. Therefore, we define the anti-commutative dual complex numbers (DCN, for short) Cˇ by modifyingthemultiplicationofCˆ. Thatis,Cˇ = Cˆ asaset,andthealgebraic operationsarereplacedby (p + p ε)(q +q ε) = (p q )+(p q˜ + p q )ε, 0 1 0 1 0 0 1 0 0 1 p(cid:93)+ p ε = p˜ + p ε, 0 1 0 1 |p + p ε| = |p |. 0 1 0 Theadditionandtheconjugationactionarekeptunchanged. Then, Theorem 2.3. Cˇ satisfies distributive and associative laws, and hence, has a(non-commutative)ringstructure. Similarlytotheunitdualquaternionnumbers,theunitanti-commutative complexnumbersareofparticularimportance: Cˇ := {pˆ ∈ Cˇ | |pˆ| = 1} = {eiθ + p ε ∈ Cˇ | θ ∈ R,p ∈ C}. 1 1 1 Itformsagroupwithinverse (eiθ + p ε)−1 = e−iθ − p ε. 1 1 WedefineanactionofCˇ onCˇ bytheconjugation. Now,weregardC = R2 1 as usual. Identifying v ∈ C with 1 + vε ∈ Cˇ, we see that Cˇ acts on C as 1 rigidtransformation. 3. RelationtoSE(2) In the previous section, we constructed the unit anti-commutative dual complex numbers Cˇ and its action on C = R2 as rigid transformation. 1 Recallthat pˆ = p + p ε ∈ Cˇ actsonv ∈ Cby 0 1 1 (3.1) pˆ (cid:5)(1+vε) = (p + p ε)(1+vε)(p˜ + p ε) = 1+(p2v+2p p )ε, 0 1 0 1 0 0 1 thatis,vmapsto p2v+2p p . Forexample,when p = 0,v ∈ Cismapped 0 0 1 1 to p2v, which is the rotation around the origin of degree 2arg(p ) since 0 0 |p | = 1. On the other hand, when p = 1, the action corresponds to the 0 0 translationby2p . Ingeneral,wehave 1 ϕ : Cˇ → SE(2) 1   Re(p2) −Im(p2) Re(2p p ) p0 + p1ε (cid:55)→ Im(p020) Re(p200) Im(2p00p11), 0 0 1 4 GENKIMATSUDA,SHIZUOKAJI†,ANDHIROYUKIOCHIAI where Re(2p p ) (respectively, Im(2p p )) is the real (respectively, imagi- 0 1 0 1 nary) part of 2p p ∈ C. Note that this gives a surjective group homomor- 0 1 phism ϕ : Cˇ → SE(2) whose kernel is {±1}. That is, the preimage of any 1 2D rigid transformation consists of exactly two unit DCN’s with opposite signs. Example 3.1. We compute the DCN’s ±pˆ ∈ Cˇ which represent θ-rotation 1 around v ∈ C. It is the composition of the following three rigid transformations: (−v)-translation, θ-rotation around the origin, and v-translation. Thus,wehave (cid:18) v (cid:19) (cid:16) (cid:17) (cid:18) v (cid:19) (cid:18) (cid:16) (cid:17) v (cid:19) pˆ = 1+ ε · ±e2θi · 1− ε = ± e2θi + e−2θi −e2θi ε . 2 2 2 4. Relationtothedualquaternionnumbers Thefollowingringhomomorphism Cˇ (cid:51) p + p ε (cid:55)→ p + p jε ∈ Hˆ 0 1 0 1 is compatible with the involution and the conjugation, and preserves the norm. Furthermore, if we identify v = (x,y) ∈ R2 with 1 + (xj + yk)ε (= 1 + (x + yi)jε), the above map commutes with the action. From this point ofview,DCNisnothingbutasub-ringofDQN. Note also that Cˇ can be embedded in the ring of the 2×2-complex ma- tricesby (cid:32) (cid:33) p p p + p ε (cid:55)→ 0 1 . 0 1 0 p˜ 0 Then (cid:32) (cid:93) (cid:33) (cid:32) (cid:33) (cid:12)(cid:12)(cid:32) (cid:33)(cid:12)(cid:12)2 (cid:32) (cid:33) p p p˜ p (cid:12) p p (cid:12) p p 0 1 = 0 1 , (cid:12) 0 1 (cid:12) = det 0 1 . 0 p˜0 0 p0 (cid:12)(cid:12) 0 p˜0 (cid:12)(cid:12) 0 p˜0 WethushavevariousequivalentpresentationsofDCN.However,ourpre- sentationofCˇ astheanti-commutativedualnumbersiseasiertoimplement andmoreefficient. 5. Interpolationof2Drigidtransformations First, recall that for the positive real numbers x,y ∈ R , there are two >0 typicalinterpolationmethods: (1−t)x+ty, t ∈ R and (yx−1)tx, t ∈ R. ThefirstmethodcanbegeneralizedtoDQNastheDualquaternionLin- ear Blending in [8]. Similarly, a DCN version of Dual number Linear Blending(DLB,forshort)isgivenasfollows: ANTI-COMMUTATIVEDUALCOMPLEXNUMBERSAND2DRIGIDTRANSFORMATION5 Definition5.1. For pˆ ,pˆ ,...,pˆ ∈ Cˇ ,wedefine 1 2 n 1 w pˆ +w pˆ +···+w pˆ (5.1) DLB(pˆ ,pˆ ,...,pˆ ;w ,w ,...,w ) = 1 1 1 2 n n , 1 2 n 1 2 n |w pˆ +w pˆ +···+w pˆ | 1 1 1 2 n n where w ,w ,...,w ∈ R. Note that the denominator can become 0 and for 1 2 n thosesetofw’sand pˆ ’sDLBcannotbedefined. i i A significant feature of DLB is that it is distributive (it is called bi- invariantinsomeliteratures). Thatis,thefollowingholds: pˆ DLB(pˆ ,pˆ ,...,pˆ ;w ,w ,...,w ) = DLB(pˆ pˆ ,pˆ pˆ ,...,pˆ pˆ ;w ,w ,...,w ). 0 1 2 n 1 2 n 0 1 0 2 0 n 1 2 n This property is particularly important when transformations are given in a certain hierarchy such as in the case of skinning; if the transformation assigned to the root joint is modified, the skin associated to lower nodes is deformedconsistently. Next,weconsidertheinterpolationofthesecondtype. Forthis,weneed theexponentialandthelogarithmmapsforDCN. Definition5.2. For pˆ = p + p ε ∈ Cˇ,wedefine 0 1 (cid:88)∞ (p + p ε)n (ep0 −ep˜0) exppˆ = 0 1 = ep0 + p ε. n! p − p˜ 1 n=0 0 0 Whenexppˆ ∈ Cˇ ,wecanwrite p = θiforsome−π ≤ θ < π,and 1 0 sinθ exppˆ = eθi + p ε. 1 θ Forqˆ = eθi +q ε ∈ Cˇ ,wedefine 1 1 θ log(qˆ) = θi+ q ε. 1 sinθ Asusual,wehaveexp(log(qˆ)) = qˆ andlog(exp(pˆ)) = pˆ. Notethatthisgives thefollowingLiecorrespondence(see[4,2,9]) exp : dcn → Cˇ , 1 log : Cˇ → dcn, 1 where dcn = {θi + p ε ∈ Cˇ | θ ∈ R,p ∈ C} (cid:39) R3. We have the following 1 1 commutativediagram: dcn dϕ (cid:47)(cid:47) se(2) exp exp (cid:15)(cid:15) (cid:15)(cid:15) Cˇ ϕ (cid:47)(cid:47) SE(2), 1 where dϕ : dcn → se(2)   0 −2θ 2x θi+(x+yi)ε (cid:55)→ 2θ 0 2y, 0 0 0 6 GENKIMATSUDA,SHIZUOKAJI†,ANDHIROYUKIOCHIAI isanisomorphismofR-vectorspaces. ThefollowingisaDCNversionofSLERP[12]. Definition5.3. SLERPinterpolationfrom pˆ ∈ Cˇ toqˆ ∈ Cˇ isgivenby 1 1 SLERP(pˆ,qˆ;t) = (qˆpˆ−1)tpˆ = exp(tlog(qˆpˆ−1))pˆ, wheret ∈ R. This gives a uniform angular velocity interpolation of two DCN’s, while DLB can blend three or more DCN’s without the uniform angular velocity property. 6. Computationalcost Wecomparethefollowingfourmethodsfor2Drigidtransformation: our DCN, the unit dual quaternion numbers (see §4), the 3 × 3-real (homo- geneous) matrix representation of SE(2), and the 2 × 2-complex matrices describedbelow. We list the computational cost in terms of the number of floating point operationsfor • transformingapoint • composingtwotransformations • converting a transformation to the standard 3×3-real matrix representation (except for the 3 × 3-real matrix case where it shows the computationalcosttoconverttoDCNrepresentation). Wealsogivethememoryusageforeachpresentationintermsofthenumber offloatingpointunitsnecessarytostoreatransformation. transformation composition conversion memoryusage DCN 22FLOPs 20FLOPs 15FLOPs 4scalars DQN 92FLOPs 88FLOPs NA 8scalars 2×2-complexmatrix 112FLOPs 56FLOPs 15FLOPs 8scalars 3×3-realmatrix 15FLOPs 45FLOPs 18FLOPs(toDCN) 9scalars Table1. Comparisonofcomputationalcost Note that when a particular application requires to apply a single trans- formationtoalotofpoints,itisfastertofirstconverttheDCNtoa3×3-real matrix. 7. AC++Library WeimplementedourDCNinaformofC++library. Thoughitiswritten in C++, it should be easy to translate to any language. You can down- load the MIT-licensed source code at [13]. We also developed a small demo application called the 2D probe-based deformer ([7]) for tablet de- vices running OpenGL ES (OpenGL for Embedded Systems). Thanks to the efficiency of DCN and touch interface, it offers interactive and intuitive ANTI-COMMUTATIVEDUALCOMPLEXNUMBERSAND2DRIGIDTRANSFORMATION7 operation. Although we did not try, we believe DCN works well with 2D skinningjustasDQNdoeswith3Dskinning. Figure 1. Left: set up initial positions of (blue) probes. Right: the target picture is deformed according to user’s ac- tionontheprobes. Herewebrieflydiscussthealgorithmoftheapplication. Onecanplacean arbitrarynumberofprobesP ,...,P onatargetimage. Thetargetimageis 1 n represented by a textured square mesh with vertices v ,...,v . The weight 1 m w of P’s effect on v is painted by user or automatically calculated from ij i j the distance between v and P. When the probes are rotated or translated j i bytheuser’stouchgesture,theDCN pˆ iscomputedwhichmapstheinitial i stateof P toitscurrentstate. Thevertexv istransformedby i j v (cid:55)→ DLB(pˆ ,pˆ ,...,pˆ ;w ,w ,...,w )(cid:5)v . j 1 2 n 1j 2j nj j (SeeEquations(3.1)and(5.1)respectivelyforthedefinitionoftheaction(cid:5) andDLB.) Acknowledgements This work was supported by Core Research for Evolutional Science and Technology (CREST) Program “Mathematics for Computer Graphics” of JapanScienceandTechnologyAgency(JST).TheauthorsaregratefulforS. 8 GENKIMATSUDA,SHIZUOKAJI†,ANDHIROYUKIOCHIAI Hirose at OLM Digital Inc., and Y. Mizoguchi, S. Yokoyama, H. Hamada, andK.MatsushitaatKyushuUniversityfortheirvaluablecomments. References [1] M. Alexa, D. Cohen-Or, and D. Levin, As-rigid-as-possible shape interpolation, In Proceedingsofthe27thannualconferenceonComputergraphicsandinteractivetech- niques (SIGGRAPH ’00). ACM Press/Addison-Wesley Publishing Co., New York, NY,USA,157–164.DOI=10.1145/344779.344859. 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