HYPERBOLIC(1)          Mathematica package          HYPERBOLIC(1)



NAME
     Projective, PoincareBall, Minkowski, UpperHalfSpace, Hemi-
     sphere, ModelQ, UnitCircle, Vector, Convert, Distance, Vec-
     torLength, Angle, Drag, Frame, CompleteFrame, Isometry,
     Peel, ResolveHyperbolic


SYNOPSIS
     <<Hyperbolic`

     x = Projective[{.1,.2}],
     y = PoincareBall[{.1,.2}],
     v = UpperHalfSpace[Vector[{0,1},{1,1.5}]],
     w = UpperHalfSpace[Vector[{0,1},{1,2.0}]], ...

     ModelQ[Projective], ModelQ[PoincareBall], ...

     Show[Graphics[{UnitCircle, ... }]],

     Convert[x, PoincareBall],
     Convert[v, Hemisphere], ...

     Angle[v,w],

     Drag[v],
     Drag[v, 5],

     Minkowski[Frame[{{1,0,0},{0,1,0},{0,0,1}}]],

     CompleteFrame[v],
     CompleteFrame[v,-1],

     Isometry[frame1, frame2],

     Peel[x],

     Show[Graphics[{Line[{x,y}], ... }]],
     {Point[x], Line[{x,y}], ... } /. ResolveHyperbolic.


DESCRIPTION
     The following Mathematica functions are defined by loading
     the package "Hyperbolic.m".

     Projective[x],
         where x is a list of length n, is a point in hyperbolic
         n-space, in the projective model. Projective is a coor-
         dinate chart for hyperbolic space with domain the unit
         ball in euclidean n-space. Projective[Vector[x,y]] is
         the image under the coordinate chart of the tangent vec-
         tor y at the point x.

     PoincareBall[x] and PoincareBall[Vector[x,y]]
         are like their counterparts with head Projective.  As
         with Projective, PoincareBall is a coordinate chart with
         domain the unit ball in euclidean n-space.

     UpperHalfSpace[x] and UpperHalfSpace[Vector[x,y]]
         are like their counterparts with head Projective.
         UpperHalfSpace is a coordinate chart with domain the
         subset of euclidean n-space such that the last coordi-
         nate is non-negative.

     Minkowski[x]
         where x is a list of length n+1, describes a point in
         hyperbolic n-space, in the hyperboloid model. Minkowski
         is a coordinate chart for hyperbolic space with domain
         the hyperboloid x_1^2+...+x_n^2-x_(n+1)^2=-1 in
         euclidean n+1 space. Minkowski[Vector[x,y]] is the image
         under the coordinate chart of the vector y, tangent to
         the hyperboloid at x, that is, x_1y_1+...+x_n y_n-
         x_(n+1)y_(n+1)=0.

     ModelQ[keyword]
         tests whether the keyword is one of the five models
         known to the package: Minkowski, Projective, Hemisphere,
         PoincareBall, and UpperHalfSpace. This is typically used
         in defining functions whose arguments are to be points
         in hyperbolic space. For example:
         Distance[mx_?ModelQ[x_], my_?ModelQ[y_]]:= ...

     UnitCircle
         is a convenient shorthand for the 2D graphics Cir-
         cle[{0,0},1].

     Convert[x,(options)],
         where x has head Projective, PoincareBall, etc., con-
         verts x to another model, specified by the Model option.
         The default Model is Projective. If the model is speci-
         fied as Automatic no conversion is performed.
         Convert[x,model] is the same as Convert[x,Model->model].
         For example:

         In[2]:= Convert[UpperHalfSpace[{.5,.5}], Minkowski]

         Out[2]= Minkowski[{1., -0.5, 1.5}]

     Vector[x,y]
         denotes the Euclidean vector y based at x.

     Distance[model1[x],model2[y]],
         gives the hyperbolic distance between model1[x] and
         model2[y].

     VectorLength[model[Vector[x,y]]]
         gives the hyperbolic length of the given tangent vector.

     Angle[model1[Vector[x1,y1]],model2[Vector[x2,y2]]]
         gives the angle between two tangent vectors at the same
         point.  Model1[x1] and model2[x2] must represent the
         same point and y1 and y2 must be nonzero.

     Drag[model[Vector[x,y]]]
         gives the vector obtained by parallel transport of
         Vector[x,y] along the geodesic it determines, by a dis-
         tance equal to the length of y.  Drag[model[Vector[x,
         y]],d] drags Vector[x,y] a distance d.

     WARNING: the syntax for the next three commands is likely to
     change

     Minkowski[Frame[x]],
         where x is an (n+1) by (n+1) matrix, representing an
         orthonormal basis for (n+1)-dimensional Minkowski space.
         The last element should have length -1, the others
         length 1, and all inner products should be 0. In hyper-
         bolic space this corresponds to giving a point and an
         orthonormal basis for the tangent space at that point.

     CompleteFrame[model[Vector[x,y]],sign_:1]
         yields a frame having x and y (more precisely their
         counterparts in Minkowski space) in the last and last-
         but-one positions. The optional argument sign gives the
         sign of the frame's determinant.

     Isometry[frame1,frame2],
         where frame1 and frame2 are of the form
         Minkowski[Frame[x]], gives the unique isometry that
         takes frame1 to frame2. Isometry[,] . model[x], where x
         is a point or a vector, gives the image of x under the
         isometry. Isometries can be composed using the . opera-
         tor, and inverted using Inverse[].

     Peel[x]
         removes the head of the expression x, which should have
         only one element. This should be used with caution as it
         removes essential information about the meaning of x.

     ResolveHyperbolic
         contains a set of replacement rules for turning Point,
         Line, Polygon and Text expressions containing hyperbolic
         coordinates into standard Mathematica graphics primi-
         tives. Coordinates may be in the Projective, Poincare-
         Ball or the UpperHalfSpace model and in either two or
         three dimensions with the following exceptions: Lines in
         the UpperHalfSpace model and Polygons (in any model) are
         only supported in dimension two. In practice, as the
         following paragraph describes, it should never be neces-
         sary to make direct use of ResolveHyperbolic.

     Mathematica's built-in Display function is overloaded to
     apply the current default Convert function followed by
     ResolveHyperbolic whenever it encounters points in hyper-
     bolic space. Display is called automatically whenever Graph-
     ics or Graphics3D objects are shown. For example, to draw a
     line in the poincare ball model:

     In[4]:= SetOptions[Convert, Model -> PoincareBall]

     Out[4]= {Model -> PoincareBall}

     In[5]:= Show[Graphics[Line[{Projective[{1,0}], Projective[{0,.5}]}]]]

     Out[5]= -Graphics-

SEE ALSO
     Manual entry "hypintro" for an introduction to the facili-
     ties of the package.