Efficient representation and derivation of fundamental transformation of relationships using Euler angles and quaternions

E icien Rep esen a ion and De i a ion o undamen al T ans o ma ion o Rela ionships using Eule Angles and Qua e nions HANA CHUD ´ A Tomas Ba a Uni e si y in Zlin, Facul y o Applied In o ma ics Depa men o Ma hema ics Nad S ´ anˇ emi 4511,76005 Zlin CZECH REPUBLIC chuda@u b.cz Abs ac : This pape in oduces and de ines wo p incipal o a ional me hods; he Eule angles and he qua e nions heo ies wi h a b ie insigh in o hei de ini ions and algeb aic p ope ies. These me hods a e widely used in a ious scien i ic ields, only ma ginally in he ai c a indus y, he obo ics, he quan um mechanics, he elec o mechanics, he came as sys ems, he compu e g aphics, he hea y indus y and o he . The main pa o his pape is de o ed o he de i a ion o basic equa ions o he ec o o a ion a ound each o a ional x,y,zaxis using bo h o a ional me hods. Then, he gene al h ee-dimensional o a ion ma ix and he gene al ope a o o he qua e nion o a ion a e de i ed. Finally he u iliza ion o he ma ices and qua e nion equa ions a e demons a ed on a simple example. Key–Wo ds: Eule angles, qua e nion, o a ion ma ix, equa ions o o a ion, gene al ope a o o qua e nion o a- ion. 1 In oduc ion A la ge numbe o scien i ic disciplines sol e he p oblem o inding a new objec posi ion in space a - e elemen a y ans o ma ion, b ie ly in he ai c a indus y, he obo ics, he quan um mechanics, he elec o mechanics, he came as sys ems, he compu e g aphics, he hea y indus y, he opology, he di e - en ial geome y and o he . In his publica ion, we ha e ocused ou a en ion on wo widesp ead me hods; inding a new objec posi ion using o a ing ma ices used by Eule angles and he second me hod is qua e - nion heo y. The main pa is de o ed o he de i a ion o basic equa ions o he ec o o a ion a ound each o a ional x,y,zaxis using bo h o a ional me hods. The au ho o he i s me hod o he objec o a ion used is Leonha d Eule - L. Eule was a Swiss ma h- ema ician and physicis , who made key con ibu ions o he ields o in ini esimal calculus and g aph he- o y. Many de elopmen s a e a ibu ed o him includ- ing se e al designa ed as he Eule s Theo em. He e, on o he in e es s highligh ed by M.J. Ama uso s a es: Any wo independen o hono mal coo dina e ames can be ela ed by a sequence o o a ions (no mo e han h ee) abou coo dina e axes, whe e no wo suc- cessi e o a ions may be abou he same axis. The angles o hese h ee o a ions a e commonly de ined as he Eule angles and he axes o o a ion designa ed as axes x, y and z. The o de in which he axes o o a ion a e aken is e e ed o as he Eule o a ion wel e sequence. The de elopmen o he second used me hod, qua e - nions, is a ibu ed o W. R. Hamil on and yea 1843. The g ea ma hema ician Si W. R. Hamil on had been in e es ed in complex numbe s in he o m a+bi, whe e numbe s a, b a e eal and he uni iis imagi- na y. The ank o complex numbe s in he plane is 2. Some ma hema icians sough o he ma hema ical sys- ems o e he complex numbe s he ank mo e han 2. Si Hamil on o o e 10 yea s ied o ex end concep s o complex numbe s in he plane in o de o de ine a complex olume by sea ching o he second imagi- na y axis. And on 16 h Oc obe 1843 he in en ed he so-called hype -complex numbe s o he ank 4wi h 3 imagina y uni s needed. 2 Eule angles heo y We assume he exis ence o app op ia e coo dina e sys ems (x, y, z), which is combina ion o he ine - ial coo dina e sys em ixed in he Euclidean space and he body coo dina e sys em a ached and mo es oge he wi h he mo ing poin in he wo- and h ee- WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 221 Volume 18, 2019 dimensional Euclidean space. O ien a ion o a mo - ing poin in he wo- and h ee-dimensional Euclidean space can be desc ibed by u iliza ion, h ee angles measu ed om mixed axis o he o a ion sys em known as Eule angles α, β and γ. The Eule angles a e h ee angles desc ibing he o ien a ion o he igid body wi h he espec o he gi en coo dina e sys em. They can ep esen he o ien a ion o a gene al basis in he h ee-dimensional linea algeb a. Any o ien- a ion can be achie ed by composing h ee elemen al o a ions, i.e., o a ions abou he axes o a coo dina e sys em (abou z, y and xaxes). The Eule angles can be de ined by h ee o hese o a ions. They can also be de ined by he elemen al geome y, and he geome - ical de ini ion demons a es ha h ee o a ions a e always su icien o each any posi ion. A well-known is a ac ha he elemen a y o a ions may be ex insic o in insic. The posi ion o he objec , acco ding o he gi en coo dina e sys em, changes. This change is called he ans o ma ion. The ans o ma ion means changing some posi ion o he objec in o some hing else by applying ules. We can ha e a ious ypes o ans- o ma ions such as he ansla ion, he scaling and he o a ion. When he ans o ma ion akes place on he wo-dimensional plane, i is called he wo- dimensional ans o ma ion, o place on he h ee- dimensional plane, i is called he h ee-dimensional ans o ma ion. T ans o ma ions play an impo an ole in he compu e g aphics o eposi ion he g aph- ics on he sc een and change hei size o o ien a ion. 2.1 Two-dimensional o a ion This ans o ma ions a e wo king wi h 2- coo dina ions o he objec s which a e coo dina ion xand coo dina ion y. The objec s can be poin s, line and shapes ha a e p esen ed on hose axis. The basic geome ic ans o ma ion, he Ro a ion, is desc ibed as below. An objec ha is eposi ioned along a ci cula pa h in he xy-plane called he o a ion. The Figu e 2.1 shows ha o a ion by angle γ. The o a- ion poin o posi ion is desc ip ion o he o igin as A and is he cons an dis ance o he poin om he o igin, angle δis he o iginal angula posi ion o he poin om he ho izon al and γis he added o a ion angle. Using he s anda d igonome ic iden i ies can be exp ess by he ans o med coo dina es in e m o he angles γand δ. We de i e he basic ans o ma ion equa ions o he posi ion o he o a ed poin in wo-dimensional Eu- clidean space, om he basics assump ions: x0= cos(γ+δ) = cos γcos δ− sin γsin δ, y0= sin(γ+δ) = sin γcos δ+ cos γsin δ. Fig.1Two-dimensional o a ion. The o iginal coo dina es o he poin s on plane a e x= cos δ, y= sin δ. Then, he inal ans o ma ion equa ion o o a ing he poin a posi ion (x,y) h ough he angle γ o he inding (x,y) posi ion x0=xcos γ−ysin γ, y0=xsin γ+ycos γ. The e o e, he o a ed ans o ma ion can be o mu- la ed in o ma ix o m "x0 y0#="cos γ−sin γ sin γcos γ#·"x y#.(1) 2.1.1 Homogeneous coo dina es As men ioned abo e, ha he h ee basic geome ic ans o ma ions a e ep esen ed as he ansla ion, o- a ion and scaling ha a e combina ions o he mul i- plica i e and addi i e equa ions. Un o una ely, he ansla ion is ea ed di e en ly (as an addi ion) by scaling and o a ion (as mul iplica ions). As he esul , some di icul y occu s when he e is need o combine mo e han one ma ix o he ans o ma ion. The e- o e all h ee ans o ma ions need o be ea ed in consis en way by expanding hem o 3×3ma ix. Then, he column o he ans o ma ion ma ix can be used by he ansla ion e m and all ans o ma ions can be exp essed as he ma ix mul iplica ions by ho- mogenous coo dina e. The homogeneous coo dina e is he s anda d echnique o expand each o he wo- dimensional coo dina e posi ion ep esen a ion (x,y) o he h ee-elemen ep esen a ion (xh,yh,h) whe e he homogeneous pa ame e his a nonze o alue o be p esen in he same coo dina e. In o de o ge wo se s o homogenous coo dina es (x,y,h) and (x,y,h) ep esen ing he same poin hand hcoo dina e which is nonze o, we can no mally di ide h ough he coo - dina e: (x,y,h) and (x,y,h) ep esen he same poin WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 222 Volume 18, 2019 as (x/h,y/h,1) and (x/h,y/h,1). The numbe s (x/h,y/h) and (x/h,y/h) a e called he Ca esian co- o dina es o he homogeneous poin . The poin s wi h hand h= 0 a e called poin s a in ini y which will no appea e y o en in he discussion. The e o e, he homogeneous-coo dina e app oach can be exp essed in wo-dimensional o a ion as he ollowing ma ix mul iplica ion:    x0 y0 1  =   cos γ−sin γ0 sin γcos γ0 0 0 1   ·   x y 1  .(2) 2.2 Th ee-dimensional o a ion The h ee-dimensional ans o ma ion is addi ional me hod o ex ending o he wo-dimensional ans o - ma ion, whe e zis added on he coo dina es. Using homogeneous coo dina es, h ee-dimensional ans- o ma ion is p esen ed by he 4×4ma ices. Thus, ins ead o ep esen ing a poin as (x,y,z), i ep e- sen s i as (x,y,z,w), whe e wo o hese quad uples ep esen he same poin i one is a nonze o mul iple o he o he one; he quad uple (0,0,0,0) is no al- lowed as in wo-dimensional ans o ma ion. The h ee-dimensional coo dina e sys em can be used in wo sys ems which a e igh -handed and le - handed. The igh -handed will gi e he posi i e o a- ion om he posi i e axis owa ds he o igin, a 90◦ coun e clockwise o a ion will ans o m one posi i e axis in o he o he one. Whe eas, he le -handed will gi e he opposi e esul , which is clockwise nega i e o a ion om he nega i e axis owa ds he o igin o 90◦. Fig.2The igh -handed sys em. As al eady men ioned, any o ien a ion can be achie ed by composing h ee elemen al o a- ions(abou z, y and xaxes). The Eule angles can be de ined by h ee o hese o a ions. Each o hese o a ions is illus a ed wi h he unique o a ion ma ix, z-axis o a ion wi h he ma ix R(γ)xy,y-axis o a ion wi h he ma ix R(β)xz and x-axis o a ion Fig.3The le -handed sys em. wi h he ma ix R(α)yz. The ollowing o mulas a e alid o he igh -hand sys em, which is he con en ion used in almos all enginee ing and physics disciplines. Z-axis o a ion equa ions in homogeneous coo di- na es a e easily ex ended o h ee dimensions as:      x0 y0 z0 1      =     cos γ−sin γ0 0 sin γcos γ0 0 0 0 1 0 0 0 0 1      ·     x y z 1      .(3) Y-axis o a ion equa ions in homogeneous coo di- na es a e in he ollowing o m:      x0 y0 z0 1      =     cos β0 sin β0 0 1 0 0 −sin β0 cos β0 0 0 0 1      ·     x y z 1      .(4) X-axis o a ion equa ions in homogeneous coo di- na es can be exp essed in he o m bellow:      x0 y0 z0 1      =     1 0 0 0 0 cos α−sin α0 0 sin αcos α0 0 0 0 1      ·     x y z 1      .(5) 2.3 Composi ion o h ee-dimensional o a- ions The Eule angles a e a mechanism o c ea ing a o a- ion h ough a sequence o h ee simple o a ions, we called hem he oll, pi ch, and yaw. Objec s a e i s o a ed by he angle γin he xy-plane, hen by angle βin he zx-plane, and hi d by he angle αin he yz- plane. The numbe γis called he yaw,βis called he pi ch and αis called he oll. The gene al ma ix T WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 223 Volume 18, 2019 consis o om he mul iplying simpli ied o a ional ma ices R(γ)xy,R(β)xz and R(α)yz. T=R(γ)xy ·R(β)zx ·R(α)yz (6) T=     cos γ−sin γ0 0 sin γcos γ0 0 0 0 1 0 0 0 0 1     ·    cos β0 sin β0 0 1 0 0 −sin β0 cos β0 0 0 0 1     · ·    1 0 0 0 0 cos α−sin α0 0 sin αcos α0 0 0 0 1     (7) T=     cβcγ−cαsγ+sαsβcγsαsγ+cαsβcγ0 cβsγcαcγ+sαsβsγ−sαcγ+cαsβsγ0 −sβsαcβcαcβ0 0 0 0 1     , whe e c angle ep esen s cos angle and s angle ep- esen s sin angle. The angle chose om Eule angles α, β, γ se . 3 Qua e nion heo y I was men ioned, ha he de elopmen o qua e - nions was a ibu ed o W. R. Hamil on on 16 h Oc- obe 1843. He in en ed he so-called hype -complex numbe s o he ank 4wi h 3imagina y uni s needed. 3.1 Algeb a o qua e nions 3.1.1 De ini ion o qua e nions The de ini ion o he eal qua e nion is exp essed in he o m q=q1+q2i+q3j+q4k(8) whe e q1, q2, q3, q4a e eal numbe s and i,j,ko q a e he imagina y uni s o qua e nions, which sa is y he equali ies i2=j2=k2=ijk =−1; ij =−ji =k; ki =−ik =j; jk =−kj =i. (9) Se o all qua e nions a e deno ed H. The qua e - nion, q∈His de ined as a pai (S(q),V(q)), whe e S(q)=q1∈Ris he scala pa o qua e nion qand V(q)=q2i+q3j+q4k, is he ec o pa o he qua e nion. q=S(q) + V(q). 3.1.2 Addi ion o qua e nions The addi ion ule o wo qua e nions is componen - wise addi ion. This ule p ese es he associa i i y and he commu a i i y p ope ies o addi ion: p+q= (p1+p2i+p3j+p4k) + (10) + (q1+q2i+q3j+q4k)=(p1+q1) + +i(p2+q2) + j(p3+q3) + k(p4+q4). 3.1.3 Mul iplica ion o qua e nions The mul iplica ion ule o he qua e nions is he same as o he polynomials, ex ended by he mul iplica i e p ope ies o he elemen s i,j,kgi en abo e. We ha e: p·q= (p1+p2i+p3j+p4k)⊗ ⊗(q1+q2i+q3j+q4k) = = (p1q1−p2q2−p3q3−p4q4) + +i(p1q2+p2q1+p3q4−p4q3) + (11) +j(p1q3+p3q1+p4q2−p2q4) + +k(p1q4+p4q1+p2q3−p3q2). The o egoing e m e eals ha he commu a i i y canno be p ese ed. The associa i i y and he dis- ibu i e p ope y o e addi ion a e p ese ed. 3.1.4 Conjuga es o qua e nions Consis en wi h he complex numbe s, he de ini ion o he conjuga e ope a ion on a gi en qua e nion qis q= (q1+q2i+q3j+q4k) = (12) =q1−q2i−q3j−q4k. As wi h he complex numbe s, no e ha bo h (q+q) and (q·q) a e he eal numbe s. Mo eo e , de ining he absolu e alue o he no m he equa ion is o be |q|=qq12+q22+q32+q42.(13) Then e iden ly (q·q)=(q·q) = q2. The conjuga e ope a ion is dis ibu i e o e addi ion. 3.1.5 Uni qua e nion The subspace o he uni qua e nions, sa is ying he condi ion |q|= 1, ha e some impo an p ope ies. A i ially hold |q|=|q|= 1 and q·q=q·q= 1 WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 224 Volume 18, 2019 And a e y use ul o m is q=S(q)·cos θ+V(q)·sin θ= cos θ+V(q)·sin θ, whe e S(q)=(1,0,0,0)is he scala pa o he uni qua e nion, V(q)=(0,q2i,q3j,q4k)is he ec o pa o he uni qua e nion and θis he eal numbe . 3.1.6 In e se qua e nions We de ine he in e se qua e nion in he ollowing o m: q−1=q1−q2i−q3j−q4k |q|2=q |q|2,(14) whe e |q|=pq12+q22+q32+q42is absolu e alue o he qua e nion and q=q1−q2i−q3j−q4k is he conjuga e qua e nion. This exp ession was in- oduced by he equa ion q·q−1=q−1·q= 1. 3.1.7 Vec o p ope ies o qua e nions The qua e nion q=q1+q2i+q3j+q4kcan be in e - p e ed as he scala pa q1∈Rand he ec o pa q2i+q3j+q4k, whe e he elemen s i,jand ka e gi en he added geome ic in e p e a ion as he uni ec o s along he x,y,zaxes. The e o e, he sub- space o he eal qua e nions may be ega ded as be- ing equi alen o he eal numbe s and subspace o he ec o qua e nions may be ega ded as being equi a- len o he o dina y ec o s q≡qxi+qyj+qzk.(15) This a ibu e is u he used in ou calcula ions. 3.1.8 Poin as qua e nion I he poin P= (x,y,z) is ep esen ed as he posi ion ec o , i can be ep esen ed as he qua e nion q≡0 + xi+yj+zk.(16) 3.1.9 P oduc o ec o qua e nions The p oduc o wo ec o qua e nions has an in e es - ing p ope y p·q= (p2i+p3j+p4k)·(q2i+q3j+q4k) = =−(p2q2+p3q3+p4q4) + +i(p3q4−p4q3) + (17) +j(p4q2−p2q4) + +k(p2q3−p3q2) = =−p·q+p×q, whe e ”.”is an ope a o o he eal pa o he qua e - nion and ”×”is an ope a o o he ec o pa s o he qua e nions. 3.2 Qua e nion o a ion The qua e nion, which ep esen s he o a ion o he θ a ound he axis n= (n1,n2,n3) is gi en by q= cos θ+n·sin θ= = cos θ+ (n1i+n2j+n3k)·sin θ, (18) whe e qis he uni qua e nion, also nis he uni ec- o o he uni qua e nion q. Fo any uni qua e nion q= cos θ+n·sin θand o any ec o p∈R3he ac ion o he ope a o Rq(p) = q·p·q(19) may be in e p e ed geome ically as he o a ion o he ec o p h ough he angle 2θa ound he qas he axis o he o a ion. Fig.4Ro a ion ope a o geome y. 3.2.1 Qua e nion o a ion a ound he z-axis by γ The o a ion axis ep esen s he uni qua e nion n=0i+ 0 j+ 1 kwhile he o a ion ope a o is gi en by q= cos γ 2+n·sin γ 2= cos γ 2+k·sin γ 2. Using he o a ion ope a o on o any ec o p=xi+yj+zk,p∈R3: Rq(p)z=q·p·q= = (cos γ 2+k·sin γ 2)·(xi+yj+zk)⊗ ⊗(cos γ 2−k·sin γ 2) = =xicos2γ 2+yjcos2γ 2+zkcos2γ 2+ WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 225 Volume 18, 2019 +xk i sin γ 2cos γ 2+yk j sin γ 2cos γ 2+ +zk k sin γ 2cos γ 2−xi k sin γ 2cos γ 2− −yj k sin γ 2cos γ 2−zk k sin γ 2cos γ 2− −xk i k sin2γ 2−ykjk sin2γ 2− −zk k k sin2γ 2. Equa ion o he o a ion ope a o Rq(p)z: Rq(p)z=ihxcos2γ 2−sin2γ 2−2ysin γ 2cos γ 2i+ +jhycos2γ 2−sin2γ 2+ 2xsin γ 2cos γ 2i+ +khzcos2γ 2+ sin2γ 2i.(20) 3.2.2 Qua e nion o a ion a ound he y-axis by β The o a ion axis ep esen s he uni qua e nion n=0i+ 1 j+ 0 kwhile he o a ion ope a o is gi en by q= cos β 2+n·sin β 2= cos β 2+j·sin β 2. Using he o a ion ope a o on o any ec o p=xi+yj+zk,p∈R3: Rq(p)y=q·p·q= = (cos β 2+j·sin β 2)·(xi+yj+zk)⊗ ⊗(cos β 2−j·sin β 2) = =xicos2β 2+yjcos2β 2+zkcos2β 2+ +xj i sin β 2cos β 2+yj j sin β 2cos β 2+ +zj k sin β 2cos β 2−xi j sin β 2cos β 2− −yj j sin β 2cos β 2−zk j sin β 2cos β 2− −xj i j sin2β 2−yj j j sin2β 2− −zj k j sin2β 2. Equa ion o he o a ion ope a o Rq(p)y: Rq(p)y=ixcos2β 2−sin2β 2+ 2zsin β 2cos β 2+ +jycos2β 2+ sin2β 2+(21) +kzcos2β 2−sin2β 2−2xsin β 2cos β 2. 3.2.3 Qua e nion o a ion a ound he x-axis by α The o a ion axis ep esen s he uni qua e nion n=1i+ 0 j+ 0 kwhile he o a ion ope a o is gi en by q= cos α 2+n·sin α 2= cos α 2+i·sin α 2. Using he o a ion ope a o on o any ec o p=xi+yj+zk,p∈R3: Rq(p)x=q·p·q= = (cos α 2+i·sin α 2)·(xi+yj+zk)⊗ ⊗(cos α 2−i·sin α 2) = =xicos2α 2+yjcos2α 2+zkcos2α 2+ +xi i sin α 2cos α 2+yi j sin α 2cos α 2− −zi k sin α 2cos α 2−xi i sin α 2cos α 2− −yj i sin α 2cos α 2−zk i sin α 2cos α 2− −xi i i sin2α 2−yiji sin2α 2 −zi k i sin2α 2. Equa ion o he o a ion ope a o Rq(p)x: Rq(p)x=ihxcos2α 2+ sin2α 2i+(22) +jhycos2α 2−sin2α 2−2zsin α 2cos α 2i+ +khzcos2α 2−sin2α 2+ 2ysin α 2cos α 2i. 3.2.4 Ope a o o composi ion Le qIand qII be wo uni qua e nions (14). The op- e a o Rq(p)Iis i s applied o he ec o p. Then we apply he ope a o Rq(p)II and ob ain he ope a o Rq(p)I, II . Equi alen ly, he composi ion RqI◦RqII o he wo ope a o s can be applied: Rq(Rq(p)I) = qII ·(qIpqI)·qII = = (qII qI)·p·(qIqII ) = (23) = (qII qI)·p·(qII qI) = =Rq(p)I, II . WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 226 Volume 18, 2019 Because qIand qII a e he uni qua e nions, same as he p oduc qII ·qI. Hence he abo e equa ion (23) desc ibes he o a ion ope a o de ining qua e nion is he p oduc o he wo qua e nions qIand qII . The ollowing equa ion desc ibes he ope a o Rq(p)z y x o h ee uni qua e nions qz,qyand qx. These qua e - nions ep esen he uni qua e nions o a ions a ound he belonging axes x,yand z, espec i ely, o he gene al p=xi+yj+zk,p∈R3 Rq(p)z y x = (qzqyqx)·p·(qzqyqx) = = [(cos γ 2+k·sin γ 2)(cos β 2+j·sin β 2)⊗ ⊗(cos α 2+i·sin α 2)] ⊗(24) ⊗(xi+yj+zk)⊗[(cos γ 2−k·sin γ 2)⊗ ⊗(cos β 2−j·sin β 2)(cos α 2−i·sin α 2)]. Compound qua e nion: (qzqyqx) = = (cos α 2cos β 2cos γ 2+ sin α 2sin β 2sin γ 2) + +i(sin α 2cos β 2cos γ 2−cos α 2sin β 2sin γ 2)+ +j(cos α 2sin β 2cos γ 2+ sin α 2cos β 2sin γ 2)+ +k(cos α 2cos β 2sin γ 2−sin α 2sin β 2cos γ 2). Conjuga ed compound qua e nion: (qzqyqx) = = (cos α 2cos β 2cos γ 2+ sin α 2sin β 2sin γ 2)− −i(sin α 2cos β 2cos γ 2−cos α 2sin β 2sin γ 2)− −j(cos α 2sin β 2cos γ 2+ sin α 2cos β 2sin γ 2)− −k(cos α 2cos β 2sin γ 2−sin α 2sin β 2cos γ 2). The compound and he conjuga ed compound qua e - nions is pu in o he ela ionship o he Rq(p)z y x ; and a e he subs i u ion (25) o a,b,cand d, ol- lowing is ob ained: a= (sin α 2cos β 2cos γ 2−cos α 2sin β 2sin γ 2), b= (cos α 2sin β 2cos γ 2+ sin α 2cos β 2sin γ 2),(25) c= (cos α 2cos β 2sin γ 2−sin α 2sin β 2cos γ 2), d= (cos α 2cos β 2cos γ 2+ sin α 2sin β 2sin γ 2). Then he gene al ope a o o he qua e nion o a ion is in he o m: Rq(p)z y x =i  x(a2−b2−c2+d2)+ +2y(a·b−c·d)+ +2z(b·d+a·c) +(26) +j  2x(a·b+c·d)+ +y(−a2+b2−c2+d2)+ +2z(b·c−a·d) + +k  2x(a·c−b·d)+ +2y(b·c+a·d)+ +z(−a2−b2+c2+d2) . 4 P ac ical using and conclusions o submi ed me hods In p e ious sec ions, bo h om wo p incipal o a- ional me hods we e in oduced: one o hem is he o a ion de ined by he Eule angles ep esen ed by he o a ion ma ices, me hod, ha is well known and he o he one is de ined by he qua e nions. In his sec- ion, we will desc ibe ad an ages and disad an ages o hese me hods. Fi s , he Eule angles a e easy o unde s and and use, compa ed o he qua e nions and o aional ma ices, so can be a good choice o a use in e ace. E icien , easy o use wi h only h ee com- ponen s, any o a ion can be ep esen ed. On he o he hand, he mos discussed disad an age is he Gim- bal lock and uniqueness o he Eule angles calcu- la ions, which miss he in e se o a ion in he h ee- dimensional space. O e lea , he ime qua e nions a e no so easy o be ep esen ed ma hema ically seem o be complica ed. The ep esen a ion o he o a ions by he qua e nions has se e al ad an ages o e he o he possible ep esen a ion by he Eule angles. The pa ame iza ion o he o a ions using he qua e nions in ol e only he angle and he axis o he o a ion. In he heo y o he qua e nions, q and q co espond o he same o a ion. O he ad an age o his app oach is ha he qua e nion o a ion is no in luenced by he choice o he coo dina e sys em. Fu he , he Gimbal lock p oblem does no appea in he qua e nion ep- esen a ion. In conclusion, he qua e nions o e he bes choice o ep esen a ion o o a ions. Fo a be e unde s anding o his opis an exam- ple is b ing o wa d. Fo he pu pose o simplici y, he heo y o Eule angles and qua e nions is demon- s a ed. The calcula ions a e pe o med wi h a espec o he p esen ed heo y and he ma hema ical no a ion. Le ha e wo poin s, o example, B(200;0;0) and C(100;100;0) o Euclidean space. We wan o o- a e hem by γ= 10,02895 deg ees a ound only he WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 227 Volume 18, 2019 z-axis. New coo dina es, using he heo y o Eule an- gles a e p esen ed in he Fig.5and he esul s ob ained wi h qua e nions heo y, a e depic ed in he Fig.6. Fig.5Gene al o a ion using Eule angles a ound z- axis by γangle. Fig.6Gene al ope a o o qua e nion o a ion a ound z-axis o a ion by γangle. Acknowledgemen s: The esea ch was suppo ed by he G an o TBU in Zlin (g an No. 30196041025). Re e ences: [1] A. Wa , M. Wa , Ad anced Anima ion and Ren- de ing Techniques, ACM P ess, San F ancisco 1992 [2] M. J. Amo uso, Eule angles and qua e nions in six deg ee o eedomsimula ions o p ojec- iles, A my A mamen Resea ch De elopmen- and Enginee ing Cen e Pica inny A senal NJ A mamen Enginee ing Di ec o a e, Tech. Rep., 1996 [3] L. Pe umal, Qua e nion and I s Applica ion in Ro a ion Using Se s o Regions, IJETI 1,2011, pp. 35 −52. [4] L. Vicci, Qua e nions and Ro a ions in 3-Space: The Algeb a and i s Geome ic In e p e a ion, Depa men o Compu e Science UNC Chapel Hill,2001, pp. 1− −11. [5] B.K.P. Ho n, Closed- o m solu ion o absolu e o ien a ion using uni qua e nions, JOSA 4(4), 1987, pp. 629 −642. [6] E.B. Dam, M. Koch, M. Lillholm, Qua e - nions, In e pola ion and Anima ion, Uni e si y o Copenhagen P ess, Copenhagen 1998 [7] W.R. Hamil on, On qua e nions; o on a new sys em o imagnia ies in algeb a. London, Edin- bu gh, and Dublin Philosophical Magazine and Jou nal o Science 25(3), 1844, pp. 489 −495. [8] J.B. Kuipe s, Qua e nions and Ro a ion Se- quences, P ince on Uni e si y P ess, P ince on 1999 [9] M. Ben-A i, A Tu o ial on Eule Angles and Qua e nions. A ailable om: < h p : //www.weizmann.ac.il/sci ea/bena i/si es/sci− ea.bena i/ iles/uploads/so wa eAndLea ning Ma e ials.pd > [10] Y.B. Jia, Qua e nion and Ro a ion, Com S No es 477/577 15,2017 [11] J. Vince,Qua e nions o Compu e G aph- ics, Sp inge –Ve lag, Be lin–Heidelbe g–New Yo k–Tokyo 2011 [12] B. Wi en, J. Sh agge,Qua e nion based Signal P ocessing, S and o d Uni e si y, New O leans, 2006 [13] J. Diebel, Rep esen ing a i ude: Eule angles, uni qua e nions, and o a ion ec o s. Ma ix 58, 2006, pp. 1–35 [14] S. Zomo odi, Qua e nions App oach in S udy- ing Ro a ion.A ailable om: < h ps : //www.academia.edu/32250200/Qua e nions App oachinS udyingRo a ion?au odownload > [15] J.G. Campbell,No es on Ma hema ics o 2D and 3D G aphics. A ailable om: < h p ://www.jgcampbell.com/msc2d3d/ g ma hs.pd > [16] B. Saleh, Compu e G aphicsFunda- men al: 2D and 3D A ine T ans- o ma ions.A ailable om: < h ps : //s3.amazonaws.com/academia.edu.documen s /53228060/CG2Dand3DA ineT ans o ma ion .pd ?AWSAccessKeyIdAKIAIWOWY Y GZ2Y 53UL3AExpi es1559638120qgQS4aOXi38SRT z5pRKSKUzP 2B43.pd > WSEAS TRANSACTIONS on SYSTEMS Hana Chuda E-ISSN: 2224-2678 228 Volume 18, 2019