Alpha shape

by Yi Zhang

Alpha shape is a generalized way determining the geometric objects by a set of points, of which convex hull is a special case. It can be applied to FEM for boundary recognition. Essentially this method identifies the boundary of a set of points by eliminating those triangles (generated by triangulation of the points) which can be put into empty circles whose diagram are defined by threshold alpha. It is intuitive for a more simple criteria: eliminating the triangles whose circumcircles contains no points in the defined set. Below is a simple demo by Mathematica which tries to identify the boundary of a subset of points within circle radius 5 from a set points randomly distributed. The red arc isolates the point set P  inside, the first step is to eliminate the triangles whose circumcirlces contains no points in P, the next step is to eliminate the triangles contain nodes other than those in P. More delicate results can be reached by tuning alpha circle instead of using simple circumcirlces.