The concept of alpha shapes formalizes the intuitive notion of "shape'' for spatial point set data, which occurs frequently in the computational sciences. An alpha shape is a concrete geometric object that is uniquely defined for a particular point set. It thus stands in sharp contrast to many common concepts in computer graphics, such as isosurfaces, which are approximate by definition and the exact form depends on the algorithm used to construct them.

Alpha shapes are generalizations of the convex hull. Given a finite point set S, and a real parameter alpha, the alpha shape of S is a polytope which is neither necessarily convex nor necessarily connected. The set of all real numbers alpha leads to a family of shapes capturing the intuitive notion of "crude'' versus "fine'' shape of a point set. For sufficiently large alpha, the alpha shape is identical to the convex hull of S. As alpha decreases, the shape shrinks and gradually develops cavities. These cavities may join to form tunnels and voids. For sufficiently small alpha, the alpha shape is empty.

A related combinatorial concept is the alpha complex. Given a point set S, it consists of vertices, edges, triangles, and tetrahedra in space. For each alpha, the alpha complex is a subcomplex of the 3-dimensional Delaunay simplicial complex.

In the original (unweighted) definition, a piece of the polytope disappears when alpha becomes small enough so that a ball with radius alpha, or several such balls, can occupy its space without enclosing any of the points of S. Think of R^3 filled with Styrofoam and the points of S made of solid rock. Now imagine an eraser in the form of a ball with radius alpha. It is omnipresent in the sense that it carves out Styrofoam at all positions where the Styrofoam particle does not contain any of the sprinkled rocks, that is, points of S. The resulting object is called the alpha hull. For good reasons we straighten the surface by substituting straight edges for circular arcs and triangles for spherical caps. The resulting object is the alpha shape of S. It is a polytope in a fairly general sense: it can be concave and even disconnected, it can contain two-dimensional patches of triangles and one-dimensional strings of edges, and its components can be as small as single points.

A question that was repeatedly asked in the past is whether it is possible to construct a shape that represents different levels of detail. This is indeed possible by assigning a weight to each point. Intuitively, a large weight favors and a small weight discourages connections to neighboring points. We refer to the resulting concept as the weighted alpha shape. If all weights are zero it is the same as the original, unweighted alpha shape. This software is general enough to handle weights. Refer to Edelsbrunner and Mucke [3] for the formal definition of unweighted alpha shapes and to [7, 11] for the extension including weights.

A related notion is the space-filling diagram, which is used in biology and chemistry to model or represent molecular structures. The space-filling diagram is the union of balls centered at the points. The radius of a ball is the same as the weight of the corresponding point. It is geometrically dual to the alpha shape. Moreover, a natural decomposition into convex cells, one per atom or ball, is dual to the alpha complex. The discrete nature of the complex has computational advantages exploited in computing surface area and volume of a space-filling diagram and in localizing and measuring voids. The latter is useful in studying water molecules residing inside a protein. Alpha Shapes have been used in geometric modeling, grid generation, medical image analysis, and visualizing the structure of earthquake data.

Aids for using the software: Alpha README, Tutorial | Delaunay README.
Additional README files are available in the download.