Usage

Getting Started

To use the FreudenthalTriangulations module, begin your code with

using FreudenthalTriangulations

FreudenthalTriangulation

The FreudenthalTriangulation struct can be used to specify the dimension of the space which is represented by n and the m represents the granularity of the triangulation.

Finding Freudenthal Vertices

To find a Freudenthal triangulation of a space of dimension n with granularity m use the vertices function which takes in a FreudenthalTriangulation struct and returns a list of vertices. Each of these vertices are represented by n dimensional vectors. Thus if we set T = FreudenthalTriangulation(n, m) and

V = vertices(T)

then V is a list of vertices comprising a freudenthal triangulation of an n dimensional space.

Finding Simplex and Barycentric Coordinates

To find the vertices of the simplex around a point x in Freudenthal space and the barycentric coordinates, run

V, coords = simplex(x)

Then we have that V is the simplex vertices in the Freudenthal space and coords will be the barycentric coordinates with respect to the simplex.

For these functions, the requirements are

• x::Vector{Float64} The point in Freudenthal space
• V::Vector{Vector{Float64}} The vertices of the simplex around x in Freudenthal space
• coords::Vector{Float64} The barycentric coordinates of x with respect to the simplex

Operating in Belief Space

This package provides two functions to operate in belief space. First to convert a freudenthal triangulation into belief space use the function

bv = belief_vertices(T)

We see that bv is a set of vertices in belief space corresponding to a freudenthal triangulation in Freudenthal space. Given any point in belief space we can find a simplex of points in these belief vertices bv and from there approximate the value function at this belief.

This leads us into the second function over the belief space, belief_simplex. This function allows us to calculate the belief simplex of a belief.

B, coords = belief_simplex(T, b)

where T is a FreudenthalTriangulation struct and b is a belief. Then we have that B is a vector containing a set of beliefs corresponding to the belief simplex and coords is the barycentric coordinates of the belief in the belief simplex.

For these functions, the requirements are

• T::FreudenthalTriangulation FreudenthalTriangulation struct
• b::Vector{Float64} A belief in the belief space
• bv::Vector{Float64} The belief vertices of a triangulation T
• B::Vector{Vector{Float64}} The vector of vertices of the belief simplex of a belief b
• coords::Vector{Float64} The barycentric coordinates of the belief in the belief simplex

Example: Interpolating a Function in Belief Space

For an example of function approximation using this package, consider the function U(x,y) = ((x-1)^2 + (y-1)^2)^0.5. However, assume that we do not have access to U(x,y) but instead, we only know U for values [x,y] in belief_vertices(FreudenthalTriangulation(2, m)) for some granularity m. Assume for this example that the maximum granularity is m = 3. Thus we have access to a function U_vertices(x, y) which returns U(x,y) if [x,y] in belief_vertices(FreudenthalTriangulation(2, 3)) and None otherwise.

Using U_vertices, we can create a function U_interp which will approximate U(x,y) by interpolating on the Freudenthal space. To do this, we will use the freudenthal_simplex_and_coords function.

function U_interp(x, y)
	X = [x, y]
	T = FreudenthalTriangulation(2, 3)
	B, coords = belief_simplex(T, X)

	interp_val = 0
	for (b, coord) in zip(B, coords)
		interp_val += U_vertices(v[1], v[2]) * coord
	end
	return interp_val
end		

Thus U_interp allows us to approximate U at real coordinates based on the U_vertices function.