{"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}]}