Alpha hulls and the material available on them are rapidly defeating me, Looking back on today I have gone round and round in circles trying to understand a standard (rather than my homebrew adhoc) method for creating them. Two general methods I have come across are Alpha Prisms by Jihad El-Sana and Amitabh Varshney, which have stumped me on the computation of the union of their alpha prism representation, and Alpha Shapes by Herbert Edelsbrunner and Ernst P Mucke which makes use of Delaunay Triangulation.
I have looked into 2 methods for computing the Delaunay triangulation of a surface.
Firstly using edge flipping which I understand in 2D but goes to hell in 3D although is posible according to Barry Joe "Constuction of Three-Dimensional Improved-Quality Triangulations using local transformations".
Secondly lifting maps which takes advantage of the knowledge that the convex hull of the 4D representation is when brought back down to 3D equal to the delaunay triangulation. This causes problems with the need to compute the determinant of a 5x5 matrix as well as with the mapping back into 3D space.
I suspect my best chance is the lifting maps route as there should be plenty of information of the matrix computations required and then i just need to find a way of determining the bottom faces of a 4D object for the mapping back to 3D.
I have another day to look at it tomorrow but I cant help but feel it would take a miracle for this stuff to click. Fun Times...
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