### Boundary recognition

Based on the idea of alpha shape, one can devise a boundary recognition scheme by eliminating the elements whose $\alpha$  circumsphere contains no nodes which are within the boundary. This means, first one must have those nodes identified, then those nodes and the coefficient $\alpha$ (defines the extended version of circumsphere) are used for boundary recognition. Here is a example of identifying the boundary of those nodes lying inside a cube but outside its inscribed sphere.

The first four plots show original meshed cube with inside view, the next six plots are when elements inside the sphere are eliminated. The rest is what we have when we change $\alpha$, i.e. the coefficient determining whether one tetrahedron should be eliminated because its extended circumsphere, whose radius depends on $\alpha$, contains nodes within the boundary. Here $\alpha$ is reduces, resulting in more elements erased from the picture.