EulerCharacteristic.jl
EulerCharacteristic.jl computes Euler characteristic of objects defined on
two- or three dimensional regular grid (FIXME: elaborate on this).
Examples
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 |
|---|---|---|---|---|
 |
A disk | 0 | 1 | 1 |
 |
A disk with a hole | 0 | 2 | 0 |
 |
A disk with two holes | 0 | 3 | -1 |
|
A ball (crossection shown) | 0 | 0 | 2/2 = 1 |
 |
A double torus with interior(crossection shown) | 2 | 0 | -2/2 = -1 |
Usage
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.