Euler characteristic on 2D and 3D mesh
Author fatimp
0 Stars
Updated Last
1 Year Ago
Started In
August 2021



EulerCharacteristic.jl computes Euler characteristic of objects defined on two- or three dimensional regular grid (FIXME: elaborate on this).


According to this and this articles, the Euler characteristic for a closed surface is defined as χ = 2 - 2g where g is genus of that surface. For a surface with boundary this formula becomes χ = 2 - 2g - b where b is a number of connected components of the boundary. For a surface embedded in three-dimensional space plus its interior the Euler characteristic is equal to half of the Euler characteristic of that surface.

Here are some examples of Euler characteristic:

Object Description g (surface) b (surface) χ
disk A disk 0 1 1
disk A disk with a hole 0 2 0
disk A disk with two holes 0 3 -1
disk A ball (crossection shown) 0 0 2/2 = 1
disk A double torus with interior(crossection shown) 2 0 -2/2 = -1


There is euler_characteristic function in the package which takes a binary array and returns Euler characteristic. The value true counts as a vertex.

There is also EulerTracker type which is a subtype of AbstractArray and allows for fast recalculation of Euler characteristic when an element of EulerTracker is changed.

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