Suppose we grow all balls
simultaneously without changing Voronoi cells.
In this case, all cells of the decomposition **C** of
can only
grow, and the dual complex can only get larger.
The result is a one-parametric family of simplicial complexes.

More formally, let ,
,
and
.
For the radius of
is its weight,
and for the radius is
.
By construction, the Voronoi cell of in
is the same as the Voronoi cell of in **B**.
The *-complex*,
,
is the dual complex of .
The *-shape* is
.
By monotonicity of the cells
,
we have

whenever .
For sufficiently large ,
geometrically
realizes the nerve of the collection of Voronoi cells.
We refer to this complex as the
*(weighted) Delaunay simplicial complex* or
*(weighted) Delaunay triangulation* of **B**,
and denote this by .
All are subcomplexes of .

Since is a finite complex,
there are only finitely many subcomplexes,
and the ones we are interested in are naturally ordered by inclusion.
We index the complexes from 0 through , and
define as the **r**-th complex after the trivial complex,
.
The last complex is .

The linear structure is essential in obtaining algorithms that efficiently deal with the entire family of alpha complexes and shapes. As a consequence, the most important step in the computation is the construction of . Many efficient algorithms have been described in the literature. The implementation provided as part of our software distribution builds incrementally, adding each point by a sequence of flips [9]. The points are added in random order.

The linear sequence of alpha complexes can be understood as assembling one simplex at a time. This is not quite correct because and may differ by more than only one simplex. In such a case, we can add the simplices in one by one, lower dimensions first. The resulting sequence of simplices,

is called a *filter*, and the sequence of complexes,

,
is a *filtration*.
For each , t
here is a
and an
with .

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