Defining a Euclidean Motion

from Initial Conditions

Chris E. Bryant

Department of MathematicsOregon State Universitycbryant@math.orst.edu

May 12, 2003

Given the following Second-Order Linear Ordinary Differential Equation (SLODE) over C (with meromorphiccoefficients),

y′′ + p(z) y′ + q(z) y = 0 , (1)

we define the associated minimal immersion (with respect to any nonsingular point, zo , of (1)), Xode : D −→ R3 ,using the qualitative-EW representation:

Xode(z) = <e

Co

∫ z

zo

1− µ2

i(1 + µ2

)2µ

e− ∫p(η) dη dη

, (2)

Here, µ = q − 1

2p′ − 1

4p2 is the invariant of (1), Co ∈ C−{0} is an arbitrary constant, and D ⊆ C is chosen

to make the integral in (2) well-defined. We seek to accomplish the following:

Given the corresponding Initial Value Problem:

y′′ + p(z) y′ + q(z) y = 0 ,y(zo) = yo = ω1 + iω2

y′(zo) = y′o = ω3 + iω4

, (3)

we wish to construct a translation, Xo = X(zo) , and a rotation, Ro , uniquely from the initial conditions, yo ∈ Cand y′o ∈ C , which are, at the same time, continuous functions of yo and y′o . Then, the 4 real-parameter fam-ily of minimal immersions corresponding to the initial value problem (3) may be defined, by extension of (1), to be

X(z) = Xo + Ro Xode(z) . (4)

Determining a choice for Xo is straightforward. Let σ : C∞ −→ S2 be stereographic projection (with respect tothe South pole), and define Xo : C −→ R3 by

Xo = Xo (yo) = | yo | σ ( yo) , (5)

from which it follows that for any Xo ∈ Xo (C) ,

yo =

0 ⇐⇒ Xo = 0

σ−1(

Xo

‖Xo‖

), otherwise .

Crucially, note that (5) is a continuous injection of y(zo) = yo , by construction.

1

To construct a rotation, Ro , from the initial conditions, we first define a more general rotation-map, R : C×S1 −→SO(3) , as follows.

For any w ∈ C , its stereographic projection (with respect to the South pole), σ(w) , is a point on S2 −{( 0, 0, −1 )T} . If w = 0 , it is elementary to complete σ(0) to an ordered, right-handed orthonormal basis:Choose any α on the equator, S1 (the great circle orthogonal to σ(0) ), and select its orthogonal “partner”, iα ,on S1 . It then follows that {σ(α), σ(iα), σ(0) } forms a right-handed orthonormal basis for R3 . (This becomestransparent when we insert α = eiϑ .) For the general case, we employ a Mobius map, mw , which rotates0 to w on the Riemann sphere. Then—by conformality and preservation of circles—the equator is rotated toσ(mw(S1)

), the great circle orthogonal to σ(w) . Moreover, since Mobius maps also preserve orientation, the

ordered set, {σ(mw(α)), σ(mw(iα)), σ(w) } , forms a right-handed, orthonormal basis for R3 , fully determinedby w ∈ C and α ∈ S1 . (Observe that w = mw(0) , by construction.)

As Gauss discovered in 1819 (see [ N ], p. 162), the most general rotation of the Riemann sphere is given by

r(z) =az + b

−bz + a, where |a|2 + |b|2 = 1 . (6)

Since w 6=∞ , our rotation is non-inversive (i.e., a 6= 0 ). Therefore, (6) takes the form

r(z) =

a

az +

b

a

1− b

az

=

( aaz)

+b

a

1− b

a

( aaz) , (7)

or, upon setting ζ =a

az ,

m(ζ) = r

(a

aζ

)=

ζ +b

a

1− b

aζ

. (8)

Now, if m takes 0 to w , then it must also take ∞ to −1/w . Hence, the desired map is:

mw = mw(ζ) =ζ + w

1− w ζ. (9)

The aforementioned rotation-map, R : C× S1 −→ SO(3) , may then be defined by

R(w, α ) =(σ ◦mw(α), σ ◦mw(iα), σ ◦mw(0)

)(10)

=

(σ

(α+ w

1− wα

), σ

(iα+ w,

1− iw α

), σ(w)

),

where, explicitly, σ : C∞ −→ S2 is given by

σ(w) =

2<e {w}1 + |w|2

2=m {w}1 + |w|2

1− |w|2

1 + |w|2

. (11)

To facilitate the computation of σ (mw(α)) (note: after computing this first column, replace α with iα toquickly obtain the second), we offer the following three “key identities” (regarding which, recall |α| = 1 ) :

| 1− wα |2 + |α+ w |2 = 2(1 + |w|2

)(12)

| 1− wα |2 − |α+ w |2 = −4<e {wα} (13)

| 1− wα |2(

α+ w

1− wα

)= α− w2 α . (14)

In full detail, the rotation-map (10) is given by

R(w, α ) =

<e{ (

1− w2)α}

1 + |w|2=m

{ (1− w2

)α}

1 + |w|22<e {w}1 + |w|2

−=m

{ (1 + w2

)α}

1 + |w|2<e{ (

1 + w2)α}

1 + |w|22=m {w}1 + |w|2

− 2<e {wα}1 + |w|2

− 2=m {wα}1 + |w|2

1− |w|2

1 + |w|2

. (15)

By construction, the third column of (15) is precisely σ(w) . Thus, given two such rotations, R(w1, α1) andR(w2, α2) , it follows that

R(w1, α1) = R(w2, α2) =⇒ σ(w1) = σ(w2) ⇐⇒ w1 = w2 . (16)

Given (16), with w := w1 = w2 , we find (since their first columns must also coincide) that

σ (mw(α1)) = σ (mw(α2)) ⇐⇒ mw(α1) = mw(α2) ⇐⇒ α1 = α2 . (17)

This shows that R : C× S1 −→ SO(3) is injective. Furthermore, it is clearly continuous—since each column is acomposition of continuous functions (see definition (10)).

A rather remarkable fact about the rotation-map—which may be verified using (15)—is that it naturally fac-tors into a product of decoupled rotations:

R(w, α ) = R(w, 1 ) R( 0, α ) (18)

= R(w, idS1 ) R( idC , α ) .

The right-hand side of factorization (18) is essentially the starting point for the authors in [ L P ] (whose intriguingapproach rekindled an interest in earlier work done by this author on rotations in R3 , and ultimately inspiredthis paper). An immediate consequence of (18) is the corresponding (decoupled) inversion formula:

RT(w, α ) = RT( 0, α ) RT(w, 1 ) (19)

= R( 0, α ) R(−w, 1 ) .

To form Ro , we first set w = y′o in (15)—so that y′o uniquely determines Ro , by (16). Next, we construct ageometrically meaningful αo ∈ S1 as follows.

A natural angle function, ϑ : C∞ × C∞ −→ [ 0, π ] , exists for the Reimann sphere, namely, the angle betweenσ(z) and σ(ξ) (where σ is again stereographic projection, and z , ξ ∈ C∞ ). From the fact that ϑ satisfies

cosϑ = σ(z)T σ(ξ) =| 1 + z ξ |2 − | z − ξ |2

(1 + |z|2) (1 + |ξ|2), (20)

it follows that

sinϑ

2=

| z − ξ |√(1 + |z|2) (1 + |ξ|2)

:= 12 χ ( z, ξ ) (χ = the chordal metric ) , (21)

cosϑ

2=

| 1 + z ξ |√(1 + |z|2) (1 + |ξ|2)

= 12 χ (−1/z, ξ ) . (22)

Hence,

ϑ = ϑ(z, ξ) = 2 arctan

∣∣∣∣ z − ξ1 + z ξ

∣∣∣∣ ∈ [ 0, π ] , (23)

which corresponds to the spherical arclength between σ(z) and σ(ξ) on S2 . Upon doubling this angle, andinserting the initial conditions, we obtain

ϕo := 2ϑ( yo, y′o ) = 4 arctan

∣∣∣∣ yo − y′o1 + yo y′o

∣∣∣∣ ∈ [ 0, 2π ] , (24)

using which we defineαo := eiϕo = e2iϑ( yo, y

′o ) . (25)

Finally, by (21) and (22) (and De Moivre’s theorem), (25) may be expressed entirely in terms of the chordalmetric:

αo =1

24

(χ (−1/yo, y

′o ) + i χ ( yo, y

′o )

)4

∈ S1 . (26)

Therefore, with αo defined by either (25) and (24), or by (26), our rotation matrix—constructed from the initialvalues—becomes

Ro = Ro( yo, y′o ) := R( y′o , αo ) = R( y′o, 1 ) R( 0, αo )

=

<e{ (

1− ( y′o )2)αo}

1 + | y′o |2=m

{ (1− ( y′o )2

)αo}

1 + | y′o |22<e { y′o}1 + | y′o |2

−=m

{ (1 + ( y′o )2

)αo}

1 + | y′o |2<e{ (

1 + ( y′o )2)αo}

1 + | y′o |22=m { y′o}1 + | y′o |2

− 2<e { y′o αo}1 + | y′o |2

− 2=m { y′o αo}1 + | y′o |2

1− | y′o |2

1 + | y′o |2

. (27)

Clearly, Ro = I if and only if y′o = 0 and ϕo ∈ { 0, 2π } . However, putting y′o = 0 in (24) givesϕo = 4 arctan | yo | ∈ [ 0 , 2π ) , which then forces yo = 0 ( 2π is excluded since yo 6= ∞ ). Thus we havethat Ro = I if and only if yo = y′o = 0 .

Since Euclidean motions are ordered pairs, say (Xo , Ro ) , two such motions are distinct if and only if eithertheir translations are distinct or their rotations are distinct. Since Xo is uniquely determined by yo in (5),and Ro is injective in y′o (see (16)), the initial conditions uniquely determine the Euclidean motion, (Xo , Ro ) .Furthermore, since both constructions (5) and (27) are continuous, the motion in (4) depends continuously on theinitial conditions.

Regarding the case w = ∞ (which is singular in the differential geometric sense): one may return to (6),set a = 0 (the inversive rotation) and reparametrize to find:

m∞ = m∞(ζ) =1

ζ. (28)

Then, for any α ∈ S1 , it follows directly from (10) that

R(∞, α ) =(σ ◦m∞(α), σ ◦m∞(iα), σ ◦m∞(0)

)=

(σ (α ) , σ

(iα), σ(∞)

)

=

<e {α} −=m {α} 0

−=m {α} −<e {α} 0

0 0 −1

(29)

=

1 0 0

0 −1 0

0 0 −1

R( 0, α ) = R(∞, 1 ) R( 0, α ) . (30)

(Observe that R(∞, 1 ) represents a rotation by π about the e1 -axis, which clearly rotates the North to theSouth pole.)

Note The actual factors used by the authors in [ L T ] to define their product of (decoupled) rotations cor-responds to our version (18) as follows

R1(w) = R( 0, i ) R(w, 1 ) R( 0, −i ) , (31)

R2(z) = R( 0, e−iz ) (where, in [ L T ] , z ∈ R ) . (32)

References

[ N ] Tristan Needham, (1997) Visual Complex Analysis,Oxford University Press Inc., New York.

[ L T ] J. M. Longuski, P. Tsiotras, (1995) A New Parameterization of the Attitude Kinematics,The Journal of the Astronautical Sciences, Vol. 43, No. 3, pp. 243-262.