vertex a
edge a
b c
a b
tetrahedron a
b c
d a
[a, b] triangle
[a, b, c] [a, b, c, d]
Fig. 2.14. k-simplices, 0 ≤ k ≤ 3. The orientation on the tetrahedron is shown on its
faces.
2.3.5 Orientability
Our earlier definition of orientability Definition 2.25 depended on differen- tiability. We now extend this definition to simplicial complexes, which are
not smooth. This extension further affirms that orientability is a topological property not dependent on smoothness.
Definition 2.40 orientation Let K be a simplicial complex. An orientation of a k-simplex
σ ∈ K, σ = {v , v
1
, . . . , v
k
}, v
i
∈ K, is an equivalence class of orderings of the vertices of
σ, where v
, v
1
, . . . , v
k
∼ v
τ0
, v
τ1
, . . . , v
τk
2.2 are equivalent orderings if the parity of the permutation
τ is even. We denote an oriented simplex, a simplex with an equivalence class of orderings, by
[σ]. Note that the concept of orientation derives from that fact that permutations
may be partitioned into two equivalence classes if you have forgotten these concepts, you should review Definitions 2.4 and 2.15. Orientations may be
shown graphically using arrows, as shown in Figure 2.14. We may use oriented simplices to define the concept of orientability to triangulated d-manifolds.
Definition 2.41 orientability Two k-simplices sharing a
k − 1-face σ are consistently oriented
if they induce different orientations on σ. A triangulable
d -manifold is orientable if all d-simplices can be oriented consistently. Other-
wise, the d-manifold is nonorientable
Example 2.4 rendering The surface of a three-dimensional object is a 2- manifold and may be modeled with a triangulation in a computer. In computer
graphics, these triangulations are rendered using light models that assign color to each triangle according to how it is situated with respect to the lights in the
scene and the viewer. To do this, the model needs the normal for each triangle.
But each triangle has two normals pointing in opposite directions. To get a correct rendering, we need the normals to be consistently oriented.
2.3.6 Filtrations and Signatures
All the spaces explored in this book will be simplicial complexes. We will explore them by building them incrementally, in such a way that all the subsets
generated are also complexes.
Definition 2.42 subcomplex A subcomplex of a simplicial complex K is a simplicial complex L
⊆ K.
Definition 2.43 filtration A filtration of a complex K is a nested sequence of subcomplexes,
∅ = K ⊆ K
1
⊆ K
2
⊆ . . . ⊆ K
m
= K. We call a complex K with a filtration a filtered complex.
Note that complex K
i +1
= K
i
˙ ∪ δ
i
, where δ
i
is a set of simplices. The sets δ
i
provide a partial order on the simplices of K. Most of the algorithms will require a full ordering. One method to derive a full ordering is to sort each
δ
i
according to increasing dimension, breaking all remaining ties arbitrarily.
Definition 2.44 filtration ordering A filtration ordering of a simplicial complex K is a full ordering of its simplices, such that each prefix of the or-
dering is a subcomplex.
We will index the simplices in K by their rank in a filtration ordering. We may also build a filtration of n
+ 1 complexes from a filtration ordering of n simplices,
σ
i
, 1 ≤ i ≤ n, by adding one simplex at a time. That is, K = ∅ and
for i 0, K
i
= {σ
j
| j ≤ i}. The primary output of algorithms in this book will be a signature function,
associating a topologically significant value to each complex.
Definition 2.45 signature Let K
i
be a filtration of m + 1 complexes, and let
[m] denote the set {0, 1, 2, . . . , m} of the complex indices. A signature function is a map
λ : [m] → R.
2.4 Alpha Shapes
We have now seen the types of spaces that will be examined in this book, as well as their representation. What remains is the derivation of meaningful fil-
trations, encoding the geometry of the space in the ordering. In this section, we
