The hyperbolic space can be represented in three different models.
- [Hyperboloid](@ref hyperboloid_model) which is the default model, i.e. is used when using arbitrary array types for points and tangent vectors
- [Poincaré ball](@ref poincare_ball) with separate types for points and tangent vectors and a [visualization](@ref poincare_ball_plot) for the two-dimensional case
- [Poincaré half space](@ref poincare_halfspace) with separate types for points and tangent vectors and a [visualization](@ref poincare_half_plane_plot) for the two-dimensional cae.
In the following the common functions are collected.
A function in this general section uses vectors interpreted as if in the [hyperboloid model](@ref hyperboloid_model), and other representations usually just convert to this representation to use these general functions.
Modules = [Manifolds]
Pages = ["manifolds/Hyperbolic.jl"]
Order = [:type, :function]
Modules = [Manifolds]
Private = false
Pages = ["manifolds/HyperbolicHyperboloid.jl"]
Order = [:type, :function]
For the case of Hyperbolic(2) there is plotting available based on a PlottingRecipe. You can easily plot points, connecting geodesics as well as tangent vectors.
!!! note The recipes are only loaded if Plots.jl or RecipesBase.jl is loaded.
If we consider a set of points, we can first plot these and their connecting
geodesics using the geodesic_interpolation for the points. This variable specifies with how many points a geodesic between two successive points is sampled (per default it's -1, which deactivates geodesics) and the line style is set to be a path.
In general you can plot the surface of the hyperboloid either as wireframe (wireframe=true) additionally specifying wires (or wires_x and wires_y) to change the density of wires and a wireframe_color. The same holds for the plot as a surface (which is false by default) and its surface_resolution (or surface_resolution_x or surface_resolution_y) and a surface_color.
using Manifolds, Plots
M = Hyperbolic(2)
pts = [ [0.85*cos(φ), 0.85*sin(φ), sqrt(0.85^2+1)] for φ ∈ range(0,2π,length=11) ]
scene = plot(M, pts; geodesic_interpolation=100)
To just plot the points atop, we can just omit the geodesic_interpolation parameter to obtain a scatter plot. Note that we avoid redrawing the wireframe in the following plot! calls.
plot!(scene, M, pts; wireframe=false)
We can further generate tangent vectors in these spaces and use a plot for there. Keep in mind that a tangent vector in plotting always requires its base point.
pts2 = [ [0.45 .*cos(φ + 6π/11), 0.45 .*sin(φ + 6π/11), sqrt(0.45^2+1) ] for φ ∈ range(0,2π,length=11)]
vecs = log.(Ref(M),pts,pts2)
plot!(scene, M, pts, vecs; wireframe=false)
Just to illustrate, for the first point the tangent vector is pointing along the following geodesic
plot!(scene, M, [pts[1], pts2[1]]; geodesic_interpolation=100, wireframe=false)
The following functions are available for internal use to construct points in the hyperboloid model
Modules = [Manifolds]
Public = false
Pages = ["manifolds/HyperbolicHyperboloid.jl"]
Order = [:type, :function]
Modules = [Manifolds]
Pages = ["manifolds/HyperbolicPoincareBall.jl"]
Order = [:type, :function]
For the case of Hyperbolic(2) there is a plotting available based on a PlottingRecipe you can easily plot points, connecting geodesics as well as tangent vectors.
!!! note The recipes are only loaded if Plots.jl or RecipesBase.jl is loaded.
If we consider a set of points, we can first plot these and their connecting
geodesics using the geodesic_interpolation For the points. This variable specifies with how many points a geodesic between two successive points is sampled (per default it's -1, which deactivates geodesics) and the line style is set to be a path.
Another keyword argument added is the border of the Poincaré disc, namely
circle_points = 720 resolution of the drawn boundary (every hlaf angle) as well as its color, hyperbolic_border_color = RGBA(0.0, 0.0, 0.0, 1.0).
using Manifolds, Plots
M = Hyperbolic(2)
pts = PoincareBallPoint.( [0.85 .* [cos(φ), sin(φ)] for φ ∈ range(0,2π,length=11)])
scene = plot(M, pts, geodesic_interpolation = 100)
To just plot the points atop, we can just omit the geodesic_interpolation parameter to obtain a scatter plot
plot!(scene, M, pts)
We can further generate tangent vectors in these spaces and use a plot for there. Keep in mind, that a tangent vector in plotting always requires its base point
pts2 = PoincareBallPoint.( [0.45 .* [cos(φ + 6π/11), sin(φ + 6π/11)] for φ ∈ range(0,2π,length=11)])
vecs = log.(Ref(M),pts,pts2)
plot!(scene, M, pts,vecs)
Just to illustrate, for the first point the tangent vector is pointing along the following geodesic
plot!(scene, M, [pts[1], pts2[1]], geodesic_interpolation=100)
Modules = [Manifolds]
Pages = ["manifolds/HyperbolicPoincareHalfspace.jl"]
Order = [:type, :function]
For the case of Hyperbolic(2) there is a plotting available based on a PlottingRecipe you can easily plot points, connecting geodesics as well as tangent vectors.
!!! note The recipes are only loaded if Plots.jl or RecipesBase.jl is loaded.
We again have two different recipes, one for points, one for tangent vectors, where the first one again can be equipped with geodesics between the points. In the following example we generate 7 points on an ellipse in the Hyperboloid model.
using Manifolds, Plots
M = Hyperbolic(2)
pre_pts = [2.0 .* [5.0*cos(φ), sin(φ)] for φ ∈ range(0,2π,length=7)]
pts = convert.(
Ref(PoincareHalfSpacePoint),
Manifolds._hyperbolize.(Ref(M), pre_pts)
)
scene = plot(M, pts, geodesic_interpolation = 100)
To just plot the points atop, we can just omit the geodesic_interpolation parameter to obtain a scatter plot
plot!(scene, M, pts)
We can further generate tangent vectors in these spaces and use a plot for there. Keep in mind, that a tangent vector in plotting always requires its base point.
Here we would like to look at the tangent vectors pointing to the origin
origin = PoincareHalfSpacePoint([0.0,1.0])
vecs = [log(M,p,origin) for p ∈ pts]
scene = plot!(scene, M, pts, vecs)
And we can again look at the corresponding geodesics, for example
plot!(scene, M, [pts[1], origin], geodesic_interpolation=100)
plot!(scene, M, [pts[2], origin], geodesic_interpolation=100)
Pages = ["hyperbolic.md"]
Canonical=false