The Geometry of Eye Rotations 
and Listings Law 
Amir A. Handzel* Tamar Flash t 
Department of Applied Mathematics and Computer Science 
Weizmann Institute of Science 
Rehovot, 76100 Israel 
Abstract 
We analyse the geometry of eye rotations, and in particular 
saccarles, using basic Lie group theory and differential geome- 
try. Various parameterizations of rotations are related through 
a unifying mathematical treatment, and transformations between 
co-ordinate systems are computed using the Campbell-Baker- 
Hausdorff formula. Next, we describe Listing's law by means of 
the Lie algebra so(3). This enables us to demonstrate a direct 
connection to Dontiers' law, by showing that eye orientations are 
restricted to the quotient space S0(3)/S0(2). The latter is equiv- 
alent to the sphere $2, which is exactly the space of gaze directions. 
Our analysis provides a mathematical framework for studying the 
oculomotor system and could also be extended to investigate the 
geometry of multi-joint arm movements. 
I INTRODUCTION 
1.1 SACCADES AND LISTING'S LAW 
Saccades are fast eye movements, bringing objects of interest into the center of 
the visual field. It is known that eye positions are restricted to a subset of those 
which are anatomically possible, both during saccarles and fixation (Tweed & Vilis, 
1990). According to Dontiers' law, the eye's gaze direction determines its orientation 
uniquely, and moreover, the brientation does not depend on the history of eye motion 
which has led to the given gaze direction. A precise specification of the "allowed" 
subspace of position is given by Listing's law: the observed orientations of the eye 
are those which can be reached from the distinguished orientation called primary 
* hand@wisdm'weizmann'ac'il 
ttamar@wisdom.weizmann.ac.il 
118 A.A. HANDZEL, T. FLASH 
position through a single rotation about an axis which lies in the plane perpendicular 
to the gaze direction at the primary position (Listing's plane). We say then that 
the orientation of the eye has zero torsion. Recently, the domain of validity of 
Listing's law has been extended to include eye vergence by employing a suitable 
mathematical treatment (Van Rijn & Van Den Berg, 1993). 
Tweed and Vilis used quaternion calculus to demonstrate, in addition, that in order 
to move from one allowed position to another in a single rotation, the rotation axis 
itself lies outside Listing's plane (Tweed & Vilis, 1987). Indeed, normal saccarles are 
performed approximately about a single axis. However, the validity of Listing's law 
does not depend on the rotation having a single axis, as was shown in double-step 
target displacement experiments (Minken, Van Opstal & Van Gisbergen, 1993): 
even when the axis of rotation itself changes during the saccade, Listing's law is 
obeyed at each and every point along the trajectory which is traced by the eye. 
Previous analyses of eye rotations (and in particular of Listing's law) have been 
based on various representations of rotations: quaternions (Westheimer, 1957), ro- 
tation vectors (Hepp, 1990), spinors (Hestenes, 1994) and 3 x 3 rotation matrices; 
however, they are all related through the same underlying mathematical object -- 
the three dimensional (3D) rotation group. In this work we analyse the geometry of 
saccarles using the Lie algebra of the rotation group and the group structure. Next, 
we briefly describe the basic mathematical notions which will be needed later. This 
is followed by Section 2 in which we analyse various parameterizations of rotations 
from the point of view of group theory; Section 3 contains a detailed mathematical 
analysis of Listing's law and its connection to Dontiers' law based on the group 
structure; in Section 4 we briefly discuss the issue of angular velocity vectors or 
axes of rotation ending with a short conclusion. 
1.2 THE ROTATION GROUP AND ITS LIE ALGEBRA 
The group of rotations in three dimensions, G = $0(3), (where 'SO' stands for 
special orthogonal transformations) is used both to describe actual rotations and 
to denote eye positions by means of a unique virtual rotation from the primary 
position. The identity operation leaves the eye at the primary position, therefore, 
we identify this position with the unit element of the group e  SO(3). A rotation 
can be parameterized by a 3D axis and the angle of rotation about it. Each axis 
"generates" a continuous set of rotations through increasing angles. Formally, if n 
is a unit axis of rotation, then 
EXP(0  n) (1) 
is a continuous one-parameter subgroup (in G) of rotations through angles 0 in the 
plane that is perpendicular to n. Such a subgroup is denoted as SO(2) C SO(3). 
We can take an explicit representation of n as a matrix and the exponent can 
be calculated as a Taylor series expansion. Let us look, for example, at the one 
parameter subgroup of rotations in the y-z plane, i.e. rotations about the x axis 
which is represented in this case by the matrix 
0 0 O) 
L-- 0 0 i . (2) 
0 -1 0 
A direct computation of this rotation by an angle 0 gives 
EXP(OL,) = I+OL,+2(OL,)2+...+ 
1 0 
nfi--7(OL,)'+... = 0 cosO 
 0 -sinO 
o) 
sinO (3) 
COS 0 
The Geometry of Eye Rotations and Listing's Law 119 
where I is the identity matrix. Thus, the rotation matrix R(O) can be constructed 
from the axis and angle of rotation. The same rotation, however, could also be 
achieved using L instead of L, where  is any scalar, while rescaling the angle 
to 0/. The collection of matrices L is a one dimensional linear space whose 
elements are the generators of rotations in the y-z plane. 
The set of all the generators constitutes the Lie algebra of a group. For the full 
space of 3D rotations, the Lie algebra is the three dimensional vector space that is 
spanned by the standard orthonormal basis comprising the three direction vectors 
of the principal axes: 
g = so(3) = Span{e, %, e ). (4) 
Every axis n can be expressed as a linear combination of this basis. Elements of 
the Lie algebra can also be represented in matrix form and the corresponding basis 
for the matrix space is 
L = 0 1 L u = 0 0 L, = - 0 0 ; 
0 - 0 - 0 0 0 0 0 
(5) 
hence we have the isomorphism 
-0, 0 0 , , 0 u . (6) 
-0y -0 0 
Thanks to its linear structure, the Lie algebra is often more convenient for analysis 
than the group itself. In addition to the linear structure, the Lie algebra has a 
bilinear antisymmetric operation defined between its elements which is called the 
bracket or commutator. The bracket operation between vectors in g is the usual 
vector cross product. When the elements of the Lie algebra are written as matrices, 
the bracket operation becomes a commutation relation, i.e. 
[A, B] = AB- BA. (7) 
As expected, the commutation relations of the basis matrices of the Lie algebra (of 
the 3D rotation group) are equivalent to the vector product: 
[Li, Lj] = ,jkLk (8) 
Finally, in accordance with (1), every rotation matrix is obtained by exponentiation: 
R(0) = EXP(0L +OyLy +OzL,). (9) 
where 0 stands for the three component angles. 
2 CO-ORDINATE SYSTEMS FOR ROTATIONS 
In linear spaces the "position" of a point is simply parameterized by the co-ordinates 
w.r.t. the principal axes (a chosen orthonormal basis). For a non-linear space (such 
as the rotation group) we define local co-ordinate charts that look like pieces of 
a vector space ll . Several co-ordinate systems for rotations are based on the 
fact that group elements can be written as exponents of elements of the Lie al- 
gebra (1). The angles 0 appearing in the exponent serve as the co-ordinates. 
The underlying property which is essential for comparing these systems is the non- 
commutativity of rotations. For usual real numbers, e.g. Cl and c2, commutativ- 
ity implies exp c exp c = exp c+c. A corresponding equation for non-commuting 
elements is the Campbell-Baker-Hausdorff formula (CBH) which is a Taylor series 
12 0 A.A. HANDZEL, T. FLASH 
expansion using repeated commutators between the elements of the Lie algebra. 
The expansion to third order is (Choquet-Bruhat et al., 1982): 
EXP(x)EXP(x2) = EXP(x +x2+ [x, x2] + 2[x- x2, [x, x2]]) (10) 
where zl, z2 are variables that stand for elements of the Lie algebra. 
One natural parameterization uses the representation of a rotation by the axis and 
the angle of rotation. The angles which appear in (9) are then called canonical 
co-ordinates of the first kind (Varadarajan, 1974). Gimbal systems constitute a 
second type of parameterization where the overall rotation is obtained by a series 
of consecutive rotations about the principal axes. The component angles are then 
called canonical co-ordinates of the second kind. In the present context, the first 
type of co-ordinates are advantageous because they correspond to single axis rota- 
tions which in turn represent natural eye movements. For convenience, we will use 
the name canonical co-ordinates for those of the first kind, whereas those of the 
second type will simply be called gimbals. The gimbals of Fick and Helmholtz are 
commonly used in the study of oculomotor control (Van Opstal, 1993). A rotation 
matrix in Fick gimbals is 
RF(O,Oy,O,)=EXP(O,L,) EXP(OyLy) EXP(0L), 
(11) 
and in Helmholtz gimbals the order of rotations is different: 
RH(O,Oy,Oz) = EXP(OyLy)  EXP(0, L,) EXP(0L). (12) 
The CBH formula (10) can be used as a general tool for obtaining transformations 
between various co-ordinate systems (Gilmore, 1974) such as (9,11,12). In particu- 
lar, we apply (10) to the product of the two right-most terms in (11) and then again 
to the product of the result with the third term. We thus arrive at an expression 
whose form is the same as the right hand side of (10). By equating it with the 
expression for canonical angles (9) and then taking the log of the exponents on 
both sides of the equation, we obtain the transformation formula from Fick angles 
to canonical angles. Repeating this calculation for (12) gives the equivalent formula 
for Helmholtz angles . Both transformations are given by the following three equa- 
tions where 0 F,H stands for an angle either in Fick or in Helmholtz co-ordinates; for 
Helmholtz angles there is a plus sign in front of the last term of the first equation 
and a minus sign in the case of Fick angles: 
07 = 07 (1 - 
C oF, H ( 
O = 1- 
0 7 '-- 0 F'H (1 - 
(13) 
The error caused by the above approximation is smaller than 0.1 degree within most 
of the oculomotor range. 
We mention in closing two additional parameterizations, namely quaternions and 
rotation vectors. Unit quaternions lie on the 3D sphere $a (embedded in ll 4) which 
constitutes the same manifold as the group of unitary rotations SU(2). The latter 
is the double covering group of SO(3) having the same local structure. This enables 
to use quaternions to parameterize rotations. The popular rotation vectors (written 
as tan(O/2)n, n being the axis of rotation and 0 its angle) are closely related to 
1in contrast to this third order expansion, second order approximations usually appear 
in the literature; see for example equation B2 in (Van Rijn 8z Van Den Berg, 1993). 
The Geometry of Eye Rotations and Listing's Law 121 
quaternions because they are central (gnomonic) projections of a hemisphere of $3 
onto the 3D affine space tangent to the quaternion qe = (1, 0, 0, 0)  11 4. 2 
3 LISTING'S LAW AND DONDERS LAW 
A customary choice of a head fixed coordinate system is the following: e is in 
the straight ahead direction in the horizontal plane, e u is in the lateral direction 
and ez points upwards in the vertical direction. e and e, thus define the mid- 
sagittal plane; e u and e, define the coronal plane. The principal axes of rotations 
(L, Ly, Lz) are set parallel to the head fixed co-ordinate system. A reference eye ori- 
entation called the primary position is chosen with the gaze direction being (1, 0, 0) 
in the above co-ordinates. How is Listing's law expressed in terms of the Lie algebra 
of SO(3)? The allowed positions are generated by linear combinations of L, and 
L u only. This 2D subspace of the Lie algebra, 
l-- Span{Lu, Lz} , (14) 
is Listing's plane. Denoting Span{L} by h, we have a decomposition of the Lie 
algebra so(3) into a direct sum of two linear subspaces: 
g =lh. (15) 
Every vector v E g can be projected onto its component which is in l: 
v = vt + vh > vt. (16) 
Until now, only the linear structure has been considered. In addition, h is closed 
under the bracket operation: 
[L, L] = 0 6 h, 
(17) 
and because h is closed both under vector addition and the Lie bracket, it is a 
subalgebra of #. In contrast, I is not a subalgebra because it is not closed under 
commutation (8). The fact that h stands as an algebra on its own implies that it 
has a corresponding group H, just as g = so(3) corresponds to G = SO(3). The 
subalgebra h generates rotations about the x axis, and therefore H is SO(2), the 
group of rotations in a plane. 
The group G = SO(3) does not have a linear structure. We may still ask whether 
some kind of decomposition and projection can be achieved in G in analogy to 
(15,16). The answer is positive and the projection is performed as follows: take any 
element of the group, a  G, and multiply it by all the elements of the subgroup H. 
This gives a subset in G which is considered as a single object a called a coset: 
a = {ab I be H). (18) 
The set of all cosets constitutes the quotient space. It is written as 
x = G/u: xo(a)/xo(2) 
(19) 
because mapping the group to the quotient space can be understood as dividing G 
by H. The quotient space is not a group, and this corresponds to the fact that the 
subspace I above (14) is not a subalgebra. The quotient space has been constructed 
algebraically but is difficult to visualize; however, it is mathematically equivalent 
2Geometrically, each point q  $3 can be connected to the center of the sphere by a 
line. Another line runs from q in the direction parallel to the vector part of q within the 
tangent space. The intersection of the two lines is the projected point. Numerically, one 
simply takes the vector part of q divided by its scalar part. 
12 2 A.A. HANDZEL, T. FLASH 
Table 1' Summary table of biological notions and the corresponding mathematical 
representation, both in terms of the rotation group and its Lie algebra. 
Biological notion Lie Algebra Rotation Group 
general eye position g = so(3) = h l G = SO(3) 
primary position 0g 
eye torsion h = Span{L,} H = SO(2) 
"allowed" eye l- Span{Ly,Lz} 
positions (Listing's plane) _ $2 (Dontiers' sphere 
of gaze directions) 
to another space -- the unit sphere ,52 (embedded in I3). This equivalence can be 
seen in the following way: a unit vector in l a, e.g. e = (1, 0, 0), can be rotated so 
that its head reaches every point on the unit sphere ,52; however, for any such point 
there are infinitely many rotations by which the point can be reached. Moreover, 
all the rotations around the x axis leave the vector e above invariant. We therefore 
have to "factor out" these rotations (of H = SO(2)) in order to eliminate the above 
degeneracy and to obtain a one-to-one correspondence between the required subset 
of rotations and the sphere. This is achieved by going to the quotient space. 
The matrix of a torsionless rotation (generated by elements in Listing's plane) is 
obtained by setting 0 =0 in (9): 
cos 0  
R = - sin 0 sin 
sin 0 cos 
sin 0 sin 4 
cos 0 + (1 - cos 0) cos 2 4 
cos 4 sin 4(1 - cos 0) 
sin 0 cos 4 ) 
cos 4 sin 4(1 - cos0) 
cos 0 + (1 - cos 0) sin 2 4 
, (20) 
where 0= V/0+0 2 is the total angle of rotation and 4 is the angle between 0 and 
the y axis in the 0y-0, plane, i.e. (0, 4) are polar co-ordinates in Listing's plane. 
Notice that the first column on the left constitutes the Cartesian co-ordinates of a 
point on a sphere of unit radius (Gilmore, 1974). 
As we have just seen, there is an exact correspondence between the group level and 
the Lie algebra level. In fact, the two describe the same reality, the former in a 
global manner and the latter in an infinitesimal one. Table 1 summarizes the impor- 
tant biological notions concerning Listing's law together with their corresponding 
mathematical representations. The connection between Donders' law and Listing's 
law can now be seen in a clear and intuitive way. The sphere, which was obtained 
by eliminating torsion, is the space of gaze directions. Recall that Donders' law 
states that the orientation of the eye is determined uniquely by its gaze direction. 
Listing's law implies that we need only take into consideration the gaze direction 
and disregard torsion. In order to emphasize this point, we use the fact that locally, 
SO(3) looks like a product of topological spaces: 3 
P = U x SO(2) where U C ,52. (21) 
U parameterizes gaze direction and SO(2) -- torsion. Donders' law restricts eye 
orientation to an unknown 2D submanifold of the product space P. Listing's law 
shows that the submanifold is U, a piece of the sphere. This representation is 
advantageous for biological modelling, because it mathematically sets apart the 
degrees of freedom of gaze orientation from torsion, which also differ functionally. 
3SO(3) is a principal bundle over $2 with fiber $0(2). 
The Geometry of Eye Rotations and Listing's Law 123 
4 AXES OF ROTATION FOR LISTING'S LAW 
As mentioned in the introduction, moving between two (non-primary) positions 
requires a rotation whose axis (i.e. angular velocity vector) lies outside Listing's 
plane. This is a result of the group structure of SO(3). Had the axis of rotation 
been contained within Listing's plane, the matrices of the quotient space (20) should 
have been closed under multiplication so as to form a subgroup of SO(3). In other 
words, if ri and r! are matrices representing the current and target orientations of 
the eye corresponding to axes in Listing's plane, then r!  r - should have been a 
matrix of the same form (20); however, as explained in Section 3, this condition is 
not fulfilled. 
Finally, since normal saccades involve rotations about a single axis, they are one- 
parameter subgroups generated by a single element of the Lie algebra (1). In addi- 
tion, they have the property of being geodesic curves in the group manifold under 
the natural metric which is given by the bilinear Cartan-Killing form of the group 
(Choquet-Bruhat et al., 1982). 
5 CONCLUSION 
We have analysed the geometry of eye rotations using basic Lie group theory and 
differential geometry. The unifying view presented here can serve to improve the 
understanding of the oculomotor system. It may also be extended to study the 
three dimensional rotations of the joints of the upper limb. 
Acknowledgement s 
We would like to thank Stephen Gelbart, Dragana Todori6 and Yosef Yomdin for 
instructive conversations on the mathematical background and Dario Liehermann 
for fruitful discussions. Special thanks go to Stan Gielen for conversations which 
initiated this work. 
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