CayleyMengerDistanceMatrix{T,Sz}(simplex_dimensions::Int, square_distances::Vector{T}) where {T<:Real,Sz::Union{Int,Dynamic}}
CayleyMengerDistanceMatrix(points::Vararg{NTuple{N,T},N+1})::CayleyMengerDistanceMatrix{T,N} where {T<:Real,N::Int}
CayleyMengerDistanceMatrix(P::AbstractMatrix{T})::CayleyMengerDistanceMatrix{T,Dynamic} where {T<:Real}

The zero-diagonal symmetric matrix of square distances among the points of an N-simplex, backed by efficient linear storage.

When providing points, it must be N+1 tuples of N values each, which are the coordinates of the points of the N-simplex. When providing P, it must be a tall near-square matrix with N+1 rows and N columns, where the rows are the coordinates of the points of the N-simplex. When providing simplex_dimensions and square_distances directly, simplex_dimensions must be the integer number of dimension N of the N-simplex, andsquaredistancesmust be the precomputed backing linear storage of the square distances among the points of theN-simplex and so must have lengthbinomial2(simplexdimensions + 1)`.

binomial2(x::Union{Int,StaticInt,Dynamic}) -> Union{Int,StaticInt,Dynamic}

Compute the second binomial coefficient of x.

This function is a convenience shorthand for binomial(x, 2).

julia> binomial2(4)
6

CayleyMengerDeterminant.index_triangular_nodiag(ixA::Int, ixB::Int) -> Int

For row ixA and column ixB, compute the linear index into the flat vector storage for a symmetric or lower-triangular matrix with a zero diagonal.

ixA and ixB are both 1-based and contiguous, and ixA is assumed to be greater than or equal to ixB.

julia> CayleyMengerDeterminant.index_triangular_nodiag(4,2)
5

inverse_binomial2(x::Union{Int,StaticInt,Dynamic) -> Union{Int,StaticInt,Dynamic}

Compute the integer value that x is the second binomial coefficient of.

This function is the inverse of binomial2.

julia> inverse_binomial2(6)
4

Computes only on all non-Dynamic() arguments unless all arguments are Dynamic(), in which case it returns Dynamic().

simplex_volume(distances::CayleyMengerDistanceMatrix{T,Sz}; distance_type::Union{Nothing,Type{<:Real}} = nothing) where {N,T<:Real}
simplex_volume(points::Vararg{NTuple{N,T},N+1}; distance_type::Union{Nothing,Type{<:Real}} = nothing) where {N,T<:Real}
simplex_volume(P::AbstractMatrix{T}; distance_type::Union{Nothing,Type{<:Real}} = nothing) where {T<:Real}

Calculates the (interior) measure of an N-simplex given the coordinates of its points, using the Cayley-Menger determinant involving the squared distances between the points.

For example, if N == 2, then this function calculates the area of a triangle given the 2-dimensional coordinates of its 3 points; likewise, if N == 3, then then this function calculates the volume of a tetrahedron given the 3-dimensional coordinates of its 4 points.

When providing points, it must be N+1 tuples of N values each, which are the coordinates of the points of the N-simplex. When providing P, it must be a tall near-square matrix with N+1 rows and N columns, where the rows are the coordinates of the points of the N-simplex.

If distance_type is provided and is not nothing, then the internal calculations on the squared distances will be done using distance_type as the type; otherwise, the type used for the internal calculations will be automatically derived from T.