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-
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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
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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
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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
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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+
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+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 .
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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
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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).
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