HYPINTRO(1)            Mathematica package            HYPINTRO(1)



NAME
     hypintro - "An introduction to the hyperbolic geometry pack-
     age Hyperbolic.m"

DESCRIPTION
     If you have a NeXT workstation have a look at the notebook
     "HyperbolicDemo.ma" first. Other users should look at the
     session in "HyperbolicDemo.txt".

     MODELS:

     The following models of hyperbolic space are supported in
     arbitrary dimension: Projective (or Klein), PoincareBall,
     UpperHalfSpace, Minkowski and Hemisphere.

     POINTS AND VECTORS:

     The user can specify a point in any of these models by wrap-
     ping the appropriate head around a coordinate vector. (Eg.
     Projective[{0,0}] is the coordinate of the point at the
     center of the Projective model for hyperbolic 2-space.) The
     user can also specify a tangent vector "v" at point "x" with
     the notation "Model[Vector[x, v]]" where "Model" is any of
     the supported models.

     Points in Minkowski space are given in homogeneous coordi-
     nates. Thus Minkowski[{0,0,1}] and Minkowski[{0,0,2}]
     represent the same point in hyperbolic 2-space. However most
     functions assume that points given in the Minkowski model
     have been normalized, ie. satisfy Minkowski[x].Minkowski[x]
     == -1. (The "." here evaluates as the Minkowski dot pro-
     duct.) Normalize[Minkowski[x]] returns a normalized
     representative for its argument.

     Minkowski[Vector[x, v]] should always satisfy
     Minkowski[x].Minkowski[x] == -1 and
     Minkowski[x].Minkowski[v] == 0. No function is provided to
     put vectors into this form.

     STANDARD MAPPINGS (OR CONVERSIONS):

     The package provides standard mappings between the different
     models or, equally, conversions between different represen-
     tations of points. Thus Convert[PoincareBall[{.5,.5}], Model
     -> Projective] will apply a standard mapping to map the
     given point from the Poincare ball model to the projective
     model giving the answer Projective[{0.666667, 0.666667}].
     Vectors can similarly be mapped from model to model by use
     of the "Convert" function.

     DISTANCES, LENGTHS AND ANGLES:

     Functions Distance, VectorLength and Angle are provided for
     calculating the hyperbolic distance between two points, the
     length of a vector and the angle between two vectors based
     at the same point respectively.

     TRANSFORMATION GEOMETRY:

     A tangent vector at a point determines a directed geodesic
     through that point parametrized with a certain velocity.
     Drag[Model[Vector[x, v]]] gives the vector obtained by
     parallel transport of Vector[x, v] along the geodesic it
     determines, by a distance equal to the length of the vector.

     A frame consists of a point and orthonormal basis for the
     tangent space at the point. Two frames determine a hyper-
     bolic isometry taking one to the other. Given a vector it is
     possible to extend it to a frame at the basepoint.

     At the moment frames are supported only in the Minkowski
     model. A frame in hyperbolic n-1 space is represented by an
     expression of the form Minkowski[Frame[{x1, ...  ,xn}]]
     where x1, ... ,x(n-1) have Minkowski norm 1 and xn has Min-
     kowski norm squared of -1. So xn represents the point at
     which the frame is based.

     CompleteFrame[...] constructs a frame from a vector and
     Isometry[...] constructs an isometry of hyperbolic space
     from two frames. The isometry is given by an orthogonal n by
     n transformation matrix for Minkowski space.

     GRAPHICS:

     Graphics are available in 2 and 3D. Point, Text, Line and
     (2D)Polygon primitives are supported. Pictures may be drawn
     in the Projective, PoincareBall and UpperHalfSpace models.

     To select which model is used to display graphics the user
     must set the appropriate default option for Convert. Eg.
     SetOptions[Convert, Model -> PoincareBall]. To display
     graphics Show[...] is used in the normal way. (The Display
     function, through which all graphics passes, is overloaded
     appropriately when the package is first read.)

     Where a graphics primitive would normally take a coordinate
     vector it should instead be supplied with a hyperbolic coor-
     dinate.  Thus Line[{Minkowski[{0,0,1}],
     Minkowski[{1,0,Sqrt[2]}]}] would be a legal primitive. There
     is no need for the model(s) in which points are specified to
     match the model in which graphics are rendered (via Convert
     options).


SEE ALSO
     Manual entry "hyperbolic" for a complete listing of func-
     tions defined in the package "Hyperbolic.m".