Dual Complex Alpha Shapes

a 312 b 690 c 1,498 d 2,266 e 3,448 f 4,315 g 4,808 h 5,655 i 7,823 j 8,591 Fig. 2.18. Gramicidin A, a protein, modeled as a filtration of 8,591 α-complexes of data set 1grm in Section 12.1. Ten complexes are shown with their indices. u is the star, St u, that consists of u together with the edges and triangles that share u as a vertex. Since all vertices have different heights, each edge and triangle has a unique lowest and a unique highest vertex. Following Banchoff 1970, we use this to partition the simplices of the star into lower and upper stars. Formally: Definition 2.54 upper, lower star The lower star St u and upper star St u of vertex u for a height function h are St u = {σ ∈ St u | hv ≤ hu, ∀ vertices v ≤ σ}, 2.5 St u = {σ ∈ St u | hv ≥ hu, ∀ vertices v ≤ σ}. 2.6 These subsets of the star contain the simplices that have u as their highest or their lowest vertex, respectively. As we shall see in Chapter 6, we may examine the lower and upper stars of a vertex to determine if the vertex is a maximum, a minimum, or a saddle point in a triangulated manifold. These points are critical to our understanding of the topology of the iso-lines of a surface, as all topological changes happen when they occur. For example, a maximum vertex u is not the lowest vertex of any simplex, so St u = {u} and St u = St u. A a 20,714 b 41,428 c 62,142 d 82,856 e 103,570 f 124,284 Fig. 2.19. A filtration of the terrain of the Himalayas data set Himalayas in Sec- tion 12.5. Six out of the 124,284 complexes are shown with their indices. maximum also creates a new component of iso-lines if we sweep the manifold from above, as in Figure 1.8. We may partition K into a collection of either lower or upper stars, K = ˙ u St u = ˙ u St u. Each partition gives us a filtration. Suppose we sort the n vertices of K in order of increasing height to get the sequence u 1 , u 2 , . . ., u n , hu i hu j , for all 1 ≤ i j ≤ n. We then let K i be the union of the first i lower stars, K i = 1 ≤ j ≤ i St u j . Each simplex σ has an associated vertex u i , and we call the height of that vertex the birth time h σ = hu i of σ. This def- inition mimics the definition of birth times for alpha-shapes. The subcomplex K i of K consists of the i lowest vertices together with all edges and triangles connecting them. Clearly, the sequence K i defines a filtration of K. We may define another filtration by sorting in decreasing order and using upper stars. We show an example of such a filtration in Figure 2.19. Either filtration is geo- metrically ordered and will provide us with filtration orderings and meaningful topological results.