Beta Functions

Each combinatorial map of dimension N defines N beta functions linking the set of darts together (e.g. a 2-map contains β₁ and β₂). These functions model the topology of the map, giving information about connections of the different cells of the map / mesh. In our case, we mostly use:

• β₁, a (partial) permutation,
• β₂, β₃, two (partial) involutions

Additionally, we define β₀ as the inverse of β₁, i.e. β₀(β₁(d)) = d. This comes from a practical consideration for performances and efficiency of the implementation.

The β_i functions can be interpreted as navigation functions along the i-th dimensions: β₁ makes you navigate along the edges, β₂ along the faces, etc. This can be generalized to N dimensions, but we are only interested in 2D and 3D at the moment.

Properties

For a given dart d, we define two properties:

• d is i-free if β_i(d) = ∅, ∅ being the null dart
• d is free if it is i-free for all i

Construction Example