{"title": "The Geometry of Eye Rotations and Listing's Law", "book": "Advances in Neural Information Processing Systems", "page_first": 117, "page_last": 123, "abstract": null, "full_text": "The Geometry of Eye Rotations \n\nand Listing's Law \n\nAmir A. Handzel* \n\nTamar Flasht \n\nDepartment of Applied Mathematics and Computer Science \n\nWeizmann Institute of Science \n\nRehovot, 76100 Israel \n\nAbstract \n\nWe analyse the geometry of eye rotations, and in particular \nsaccades, using basic Lie group theory and differential geome(cid:173)\ntry. Various parameterizations of rotations are related through \na unifying mathematical treatment, and transformations between \nco-ordinate systems are computed using the Campbell-Baker(cid:173)\nHausdorff formula. Next, we describe Listing's law by means of \nthe Lie algebra so(3). This enables us to demonstrate a direct \nconnection to Donders' law, by showing that eye orientations are \nrestricted to the quotient space 80(3)/80(2). The latter is equiv(cid:173)\nalent to the sphere S2, which is exactly the space of gaze directions. \nOur analysis provides a mathematical framework for studying the \noculomotor system and could also be extended to investigate the \ngeometry of mUlti-joint arm movements. \n\n1 \n\nINTRODUCTION \n\n1.1 SACCADES AND LISTING'S LAW \n\nSaccades are fast eye movements, bringing objects of interest into the center of \nthe visual field. It is known that eye positions are restricted to a subset of those \nwhich are anatomically possible, both during saccades and fixation (Tweed & Vilis, \n1990). According to Donders' law, the eye's gaze direction determines its orientation \nuniquely, and moreover, the orientation does not depend on the history of eye motion \nwhich has led to the given gaze direction . A precise specification of the \"allowed\" \nsubspace of position is given by Listing's law: the observed orientations of the eye \nare those which can be reached from the distinguished orientation called primary \n\n*hand@wisdom.weizmann.ac.il \nt tamar@wisdom.weizmann.ac.il \n\n\f118 \n\nA. A. HANDZEL, T. FLASH \n\nposition through a single rotation about an axis which lies in the plane perpendicular \nto the gaze direction at the primary position (Listing's plane). We say then that \nthe orientation of the eye has zero torsion. Recently, the domain of validity of \nListing's law has been extended to include eye vergence by employing a suitable \nmathematical treatment (Van Rijn & Van Den Berg, 1993). \n\nTweed and Vilis used quaternion calculus to demonstrate, in addition, that in order \nto move from one allowed position to another in a single rotation, the rotation axis \nitself lies outside Listing's plane (Tweed & Vilis, 1987). Indeed, normal saccades are \nperformed approximately about a single axis. However, the validity of Listing's law \ndoes not depend on the rotation having a single axis, as was shown in double-step \ntarget displacement experiments (Minken, Van Opstal & Van Gisbergen, 1993): \neven when the axis of rotation itself changes during the saccade, Listing's law is \nobeyed at each and every point along the trajectory which is traced by the eye. \n\nPrevious analyses of eye rotations (and in particular of Listing's law) have been \nbased on various representations of rotations: quaternions (Westheimer, 1957), ro(cid:173)\ntation vectors (Hepp, 1990), spinors (Hestenes, 1994) and 3 x 3 rotation matrices; \nhowever, they are all related through the same underlying mathematical object -\nthe three dimensional (3D) rotation group. In this work we analyse the geometry of \nsaccades using the Lie algebra of the rotation group and the group structure. Next, \nwe briefly describe the basic mathematical notions which will be needed later. This \nis followed by Section 2 in which we analyse various parameterizations of rotations \nfrom the point of view of group theory; Section 3 contains a detailed mathematical \nanalysis of Listing's law and its connection to Donders' law based on the group \nstructure; in Section 4 we briefly discuss the issue of angular velocity vectors or \naxes of rotation ending with a short conclusion. \n\n1.2 THE ROTATION GROUP AND ITS LIE ALGEBRA \nThe group of rotations in three dimensions, G = 80(3), (where '80' stands for \nspecial orthogonal transformations) is used both to describe actual rotations and \nto denote eye positions by means of a unique virtual rotation from the primary \nposition. The identity operation leaves the eye at the primary position, therefore, \nwe identify this position with the unit element of the group e E 80(3). A rotation \ncan be parameterized by a 3D axis and the angle of rotation about it. Each axis \n\"generates\" a continuous set of rotations through increasing angles. Formally, if n \nis a unit axis of rotation, then \n\nEXP(O\u00b7 n) \n\n(1) \nis a continuous one-parameter subgroup (in G) of rotations through angles () in the \nplane that is perpendicular to n. Such a subgroup is denoted as 80(2) C 80(3). \nWe can take an explicit representation of n as a matrix and the exponent can \nbe calculated as a Taylor series expansion. Let us look, for example, at the one \nparameter subgroup of rotations in the y- z plane, i.e. rotations about the x axis \nwhich is represented in this case by the matrix \no \no \n-1 \n\n(2) \n\nA direct computation of this rotation by an angle () gives \n\no \ncos () \n- sin () \n\no \nsin () \ncos () \n\n) \n\n(3) \n\n\fThe Geometry of Eye Rotations and Listing's Law \n\n119 \n\nwhere I is the identity matrix. Thus, the rotation matrix R( 0) can be constructed \nfrom the axis and angle of rotation. The same rotation, however, could also be \nachieved using A Lx instead of Lx, where A is any scalar, while rescaling the angle \nto 0/ A. The collection of matrices ALx is a one dimensional linear space whose \nelements are the generators of rotations in the y-z plane. \n\nThe set of all the generators constitutes the Lie algebra of a group. For the full \nspace of 3D rotations, the Lie algebra is the three dimensional vector space that is \nspanned by the standard orthonormal basis comprising the three direction vectors \nof the principal axes: \n\n(4) \nEvery axis n can be expressed as a linear combination of this basis. Elements of \nthe Lie algebra can also be represented in matrix form and the corresponding basis \nfor the matrix space is \n\n0 \n0 \n0 \n-1 0 \n\n0 n L, = ( ~1 1 D; \n8, ) U: ) \n\nOx \n0 \n\n0 \n0 \n\n+-------t \n\n(5) \n\n(6) \n\nL.= 0 0 D L, = ( \n\n0 \n-1 \n\nhence we have the isomorphism \n\n( -~, \n\n-Oy \n\nOz \n0 \n-Ox \n\nThanks to its linear structure, the Lie algebra is often more convenient for analysis \nthan the group itself. In addition to the linear structure, the Lie algebra has a \nbilinear antisymmetric operation defined between its elements which is called the \nbracket or commutator. The bracket operation between vectors in g is the usual \nvector cross product . When the elements of the Lie algebra are written as matrices , \nthe bracket operation becomes a commutation relation, i.e. \n\n[A,B] == AB - BA. \n\n(7) \n\nAs expected, the commutation relations of the basis matrices of the Lie algebra (of \nthe 3D rotation group) are equivalent to the vector product: \n\n(8) \n\nFinally, in accordance with (1), every rotation matrix is obtained by exponentiation: \n\nR(8) = EXP(OxLx +OyLy +OzLz). \n\n(9) \n\nwhere 8 stands for the three component angles . \n\n2 CO-ORDINATE SYSTEMS FOR ROTATIONS \n\nIn linear spaces the \"position\" of a point is simply parameterized by the co-ordinates \nw.r.t. the principal axes (a chosen orthonormal basis). For a non-linear space (such \nas the rotation group) we define local co-ordinate charts that look like pieces of \na vector space ~ n. Several co-ordinate systems for rotations are based on the \nfact that group elements can be written as exponents of elements of the Lie al(cid:173)\ngebra (1). The angles 8 appearing in the exponent serve as the co-ordinates. \nThe underlying property which is essential for comparing these systems is the non(cid:173)\ncommutativity of rotations. For usual real numbers, e.g. Cl and C2, commutativ(cid:173)\nity implies expCI expC2 = expCI +C2. A corresponding equation for non-commuting \nelements is the Campbell-Baker-Hausdorff formula (CBH) which is a Taylor series \n\n\f120 \n\nA. A. HANDZEL. T. FLASH \n\nexpansion using repeated commutators between the elements of the Lie algebra. \nThe expansion to third order is (Choquet-Bruhat et al., 1982): \n\nEXP(Xl)EXP(X2) = EXP (Xl + X2 + ~[Xl' X2] + 112 [Xl - X2, [Xl, X2]]) \n\n(10) \n\nwhere Xl, X2 are variables that stand for elements of the Lie algebra. \n\nOne natural parameterization uses the representation of a rotation by the axis and \nthe angle of rotation. The angles which appear in (9) are then called canonical \nco-ordinates of the first kind (Varadarajan, 1974). Gimbal systems constitute a \nsecond type of parameterization where the overall rotation is obtained by a series \nof consecutive rotations about the principal axes. The component angles are then \ncalled canonical co-ordinates of the second kind. In the present context, the first \ntype of co-ordinates are advantageous because they correspond to single axis rota(cid:173)\ntions which in turn represent natural eye movements. For convenience, we will use \nthe name canonical co-ordinates for those of the first kind, whereas those of the \nsecond type will simply be called gimbals. The gimbals of Fick and Helmholtz are \ncommonly used in the study of oculomotor control (Van Opstal, 1993). A rotation \nmatrix in Fick gimbals is \n\nRF(Bx,Oy,Oz) = EXP(OzLz ) . EXP(ByLy) . EXP(OxLx), \n\nand in Helmholtz gimbals the order of rotations is different: \n\nRH(Ox, By,Oz) = EXP(ByLy) . EXP(OzLz) . EXP(OxLx). \n\n(11) \n\n(12) \n\nThe CBH formula (10) can be used as a general tool for obtaining transformations \nbetween various co-ordinate systems (Gilmore, 1974) such as (9,11,12). In particu(cid:173)\nlar, we apply (10) to the product of the two right-most terms in (11) and then again \nto the product of the result with the third term. We thus arrive at an expression \nwhose form is the same as the right hand side of (10). By equating it with the \nexpression for canonical angles (9) and then taking the log of the exponents on \nboth sides of the equation, we obtain the transformation formula from Fick angles \nto canonical angles. Repeating this calculation for (12) gives the equivalent formula \nfor Helmholtz angles l . Both transformations are given by the following three equa(cid:173)\ntions where OF,H stands for an angle either in Fick or in Helmholtz co-ordinates; for \nHelmholtz angles there is a plus sign in front of the last term of the first equation \nand a minus sign in the case of Fick angles: \n\nx \n\n-\n\nx \n\nY \n\n12 \n\nBe - OF,H (1 _ ...L ((BF,H)2 + (OF,H)2)) \u00b1 lOF,H OF,H \n/2 (( O;,H)2 + (O:,H)2) ) + ~O;,H O:,H \nOf = O:,H ( 1 -\n/2 (( B;,H? + (B:,H)2)) - !O;,H O:,H \nOf = O;,H ( 1 -\n\n2 Y \n\nz \n\nz \n\n(13) \n\nThe error caused by the above approximation is smaller than 0.1 degree within most \nof the oculomotor range. \n\nWe mention in closing two additional parameterizations, namely quaternions and \nrotation vectors. Unit quaternions lie on the 3D sphere S3 (embedded in lR 4) which \nconstitutes the same manifold as the group of unitary rotations SU(2). The latter \nis the double covering group of SO(3) having the same local structure. This enables \nto use quaternions to parameterize rotations. The popular rotation vectors (written \nas tan(Oj2)n, n being the axis of rotation and B its angle) are closely related to \n\n1 In contrast to this third order expansion, second order approximations usually appear \n\nin the literature; see for example equation B2 in (Van Rijn & Van Den Berg, 1993). \n\n\fThe Geometry of Eye Rotations and Listing's Law \n\n121 \n\nquaternions because they are central (gnomonic) projections of a hemisphere of S3 \nonto the 3D affine space tangent to the quaternion qe = (1,0,0,0) E ]R4. 2 \n\n3 \n\nLISTING'S LAW AND DONDERS' LAW \n\nA customary choice of a head fixed coordinate system is the following: ex IS III \nthe straight ahead direction in the horizontal plane, ey is in the lateral direction \nand ez points upwards in the vertical direction. ex and e z thus define the mid(cid:173)\nsagittal plane; e y and ez define the coronal plane. The principal axes of rotations \n(Lx, Ly, Lz) are set parallel to the head fixed co-ordinate system. A reference eye ori(cid:173)\nentation called the primary position is chosen with the gaze direction being (1,0,0) \nin the above co-ordinates. How is Listing's law expressed in terms of the Lie algebra \nof SO(3)? The allowed positions are generated by linear combinations of Lz and \nLy only. This 2D subspace of the Lie algebra, \n\n(14) \nis Listing's plane. Denoting Span{ Lx} by h, we have a decomposition of the Lie \nalgebra so(3) into a direct sum of two linear subspaces: \n\n1 = Span{Ly, Lz }, \n\n9 = 1 EB h. \n\n(15) \n\nEvery vector v E 9 can be projected onto its component which is in I: \n\n(16) \nUntil now, only the linear structure has been considered. In addition, h is closed \nunder the bracket operation: \n\nV = VI + Vh ----t VI. \n\nproj. \n\n(17) \n\nand because h is closed both under vector addition and the Lie bracket, it is a \nsub algebra of g. In contrast, I is not a sub algebra because it is not closed under \ncommutation (8) . The fact that h stands as an algebra on its own implies that it \nhas a corresponding group H, just as 9 = so(3) corresponds to G = SO(3). The \nsubalgebra h generates rotations about the x axis, and therefore H is SO(2), the \ngroup of rotations in a plane. \nThe group G = SO(3) does not have a linear structure. We may still ask whether \nsome kind of decomposition and projection can be achieved in G in analogy to \n(15,16). The answer is positive and the projection is performed as follows: take any \nelement of the group , a E G, and multiply it by all the elements of the subgroup H. \nThis gives a subset in G which is considered as a single object a called a coset: \n\na = {ab I bEH} . \n\nThe set of all cosets constitutes the quotient space. It is written as \n\nS == G / H = SO(3)/ SO(2) \n\n(18) \n\n(19) \n\nbecause mapping the group to the quotient space can be understood as dividing G \nby H. The quotient space is not a group , and this corresponds to the fact that the \nsubspace I above (14) is not a sub algebra. The quotient space has been constructed \nalgebraically but is difficult to visualize; however, it is mathematically equivalent \n\n2 Geometrically, each point q E S3 can be connected to the center of the sphere by a \nline. Another line runs from qe in the direction parallel to the vector part of q within the \ntangent space. The intersection of the two lines is the projected point. Numerically, one \nsimply takes the vector part of q divided by its scalar part. \n\n\f122 \n\nA. A.HANDZEL,T. FLASH \n\nTable 1: Summary table of biological notions and the corresponding mathematical \nrepresentation, both in terms of the rotation group and its Lie algebra. \nRotation Group \n\nBiological notion \n\nLie Algebra \n\ngeneral eye position \n\nprimary position \n\neye torsion \n\"allowed\" eye \n\npositions \n\nO.q E 9 \n\n9 = so(3) = h El71 \nh = Span{Lx} \n1= Span{ Ly, LzJ S = ~/H = SO(3)/SO(2) \n~ S2 (Donders' sphere \n(Listing's plane) \nof gaze directions) \n\nG = SO(3) \nH = SO(2) \n\neE G \n\nthe unit sphere S2 (embedded in ~3). This equivalence can be \nto another space -\nseen in the following way: a unit vector in ~3, e.g. e = (1,0,0), can be rotated so \nthat its head reaches every point on the unit sphere S2; however, for any such point \nthere are infinitely many rotations by which the point can be reached. Moreover, \nall the rotations around the x axis leave the vector e above invariant. We therefore \nhave to \"factor out\" these rotations (of H =SO(2\u00bb in order to eliminate the above \ndegeneracy and to obtain a one-to-one correspondence between the required subset \nof rotations and the sphere. This is achieved by going to the quotient space. \n\nThe matrix of a torsion less rotation (generated by elements in Listing's plane) is \nobtained by setting Ox = 0 in (9): \n\nR = \n\n(\n\ncosO \n\n- sin 0 sin ljJ \n- sin 0 cos ljJ \n\nsin 0 sin ljJ \n\ncos 0 + (1 - cos 0) cos 2 ljJ \n\ncos ljJ sin ljJ(l - cos 0) \n\nsin 0 cos ljJ \n\ncos ljJ sin ljJ(l - cos 0) \ncos 0 + (1 - cos 0) sin 2 ljJ \n\n) \n\n,(20) \n\nwhere 0 = .)0;+0; is the total angle of rotation and ljJ is the angle between 0 and \nthe y axis in the Oy -Oz plane, i.e. (0, ljJ) are polar co-ordinates in Listing's plane. \nNotice that the first column on the left constitutes the Cartesian co-ordinates of a \npoint on a sphere of unit radius (Gilmore, 1974). \n\nAs we have just seen, there is an exact correspondence between the group level and \nthe Lie algebra level. In fact, the two describe the same reality, the former in a \nglobal manner and the latter in an infinitesimal one. Table 1 summarizes the impor(cid:173)\ntant biological notions concerning Listing's law together with their corresponding \nmathematical representations. The connection between Donders' law and Listing's \nlaw can now be seen in a clear and intuitive way. The sphere, which was obtained \nby eliminating torsion, is the space of gaze directions. Recall that Donders' law \nstates that the orientation of the eye is determined uniquely by its gaze direction. \nListing's law implies that we need only take into consideration the gaze direction \nand disregard torsion. In order to emphasize this point, we use the fact that locally, \nSO(3) looks like a product of topological spaces: 3 \n\np = u x SO(2) \n\nwhere \n\n(21) \n\nU parameterizes gaze direction and SO(2) -\ntorsion. Donders' law restricts eye \norientation to an unknown 2D submanifold of the product space P. Listing's law \nshows that the submanifold is U, a piece of the sphere. This representation is \nadvantageous for biological modelling, because it mathematically sets apart the \ndegrees of freedom of gaze orientation from torsion, which also differ functionally. \n\n350(3) is a principal bundle over S2 with fiber 50(2). \n\n\fThe Geometry of Eye Rotations and Listing's Law \n\n123 \n\n4 AXES OF ROTATION FOR LISTING'S LAW \n\nAs mentioned in the introduction, moving between two (non-primary) positions \nrequires a rotation whose axis (i.e. angular velocity vector) lies outside Listing's \nplane. This is a result of the group structure of SO(3). Had the axis of rotation \nbeen contained within Listing's plane, the matrices of the quotient space (20) should \nhave been closed under multiplication so as to form a subgroup of SO(3). In other \nwords, if ri and rJ are matrices representing the current and target orientations of \nthe eye corresponding to axes in Listing's plane, then rJ . r;l should have been a \nmatrix of the same form (20); however, as explained in Section 3, this condition is \nnot fulfilled. \n\nFinally, since normal saccades involve rotations about a single axis, they are one(cid:173)\nparameter subgroups generated by a single element of the Lie algebra (1). In addi(cid:173)\ntion, they have the property of being geodesic curves in the group manifold under \nthe natural metric which is given by the bilinear Cartan-Killing form of the group \n(Choquet-Bruhat et al., 1982). \n\n5 CONCLUSION \n\nWe have analysed the geometry of eye rotations using basic Lie group theory and \ndifferential geometry. The unifying view presented here can serve to improve the \nunderstanding of the oculomotor system. It may also be extended to study the \nthree dimensional rotations of the joints of the upper limb. \n\nAcknowledgements \n\nWe would like to thank Stephen Gelbart, Dragana Todoric and Yosef Yomdin for \ninstructive conversations on the mathematical background and Dario Liebermann \nfor fruitful discussions. Special thanks go to Stan Gielen for conversations which \ninitiated this work. \n\nReferences \n\nChoquet-Bruhat Y., De Witt-Morette C. & Dillard-Bleick M., Analysis, Manifolds \nand Physics, North-Holland (1982). \nGilmore R.,LieGroups, Lie Algebras, and Some of Their Applications, Wiley (1974). \n\nHepp K., Commun. Math. Phys. 132 (1990) 285-292. \n\nHestenes D., Neural Networks 7, No.1 (1994) 65-77. \nMinken A.W.H. Van Opstal A.J. & Van Gisbergen J.A .M., Exp. Brain Research \n93 (1993) 521-533. \nTweed, D. & Vilis T., J. Neurophysiology 58 (1987) 832-849. \nTweed D. & Vilis T., Vision Research 30 (1990) 111-127. \nVan Opstal J., \"Representations of Eye Positions in Three Dimensions\", in Multi(cid:173)\nsensory Control of Movement, ed. Berthoz A., (1993) 27-4l. \nVan Rijn L.J. & Van Den Berg A.V., Vision Research 33, No. 5/6 (1993) 691-708. \n\nVaradarajan V.S., Lie Groups, Lie Algebras, and Their Reps., Prentice-Hall (1974). \n\nWestheimer G., Journal of the Optical Society of America 47 (1957) 967-974. \n\n\f", "award": [], "sourceid": 1041, "authors": [{"given_name": "Amir", "family_name": "Handzel", "institution": null}, {"given_name": "Tamar", "family_name": "Flash", "institution": null}]}