Discrete Gradient Line Fields on Surfaces

12/21/2017 ∙ by Thomas Lewiner, et al. ∙ UFRJ puc-rio 0

A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress flux in elasticity. We propose a discretization of a Morse-Smale line field on surfaces, extending Forman's construction for discrete vector fields. More general critical elements and their indices are defined from local matchings, for which Euler theorem and the characterization of homotopy type in terms of critical cells still hold.

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1 Introduction

A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points. Such fields model a number of physical properties, like velocity and temperature gradient in fluid flow, stress and momentum flux in elasticity. Recently, line fields have earned eminence in the nematic fields ambiance [16, 25]. In computer graphics, line field are common tools for quadrangulation [9]

, visualization of vector/line/symmetric tensor fields

[28, 8], and remeshing [1]. Curious line fields abound: some are illustrated in Figure 1

. Fingerprints are studied by the pattern recognition community

[31].

Figure 1: Lines fields: a zebra, sand, coral, fingerprints.

There are natural numerical issues, which are common to vector and line fields: how does one compute critical points or special orbits joining them? The literature is extensive (e.g. [27, 26, 30, 13, 7, 29]). We take instead a point of view which is frequent within the dynamical systems community: given a field, we consider its structurally stable properties. For this, one prescribes a metric in the space of fields in a fixed surface (induced by the topology, for ), and an equivalence relation between fields — topological equivalence — given by the existence of a homeomorphism in taking orbits of one field to another (a fine, detailed description of the many objects involved is [5]). A field is structurally stable if it is equivalent to sufficiently close fields , i.e., there is such that if then and are equivalent.

Andronov and Pontryagin [2] identified the now called Morse-Smale vector fields as being structurally stable, and Peixoto [20, 21, 22] proved that they are the only such fields on compact, oriented surfaces. Bronshteyn and Nikolaev [4, 18] defined Morse-Smale line fields and showed their structural stability. Such fields contain only finitely many critical points and cycles. Line fields admit some stable critical points with no counterpart in the continuous case, shown in Figure 2. Morse-Smale vector and line fields are dense with respect to the metric: fields arising from a physical or computational situation are well approximated by a structurally stable field.

Figure 2: The structurally stable critical points: only the first two are present in vector fields.

Peixoto also defined a fundamental combinatorial object, named the Peixoto orgraph in [5], which in a sense contains all the stable aspects: classes of Morse-Smale vector fields with respect to topological equivalence are in bijection with the possible Peixoto orgraphs. The analogous result for line fields was presented in [5].

Figure 3: Extracting a Peixoto orgraph of the sphere from a line field.

Forman [11, 10] introduced discrete vector fields as a combinatorial counterpart to Morse-Smale vector fields. For oriented surfaces, such objects give rise to (all possible) Peixoto orgraphs: in a sense, discrete and continuous vector fields are equivalent for stability purposes. Forman’s theory has been used in computational geometry ([15, 14, 19]), yielding a robust approach to (discrete) Morse functions and the computation of Peixoto orgraphs [23, 24].

In this text we propose a discretization of a subclass of Morse-Smale line field on compact surfaces, extending Forman’s construction. We only allow critical points given by the first four cases in Figure 2 and those presented in Figure 4, of possible physical interest, despite of the fact that they are not structurally stable.

Figure 4: Non structurally stable critical points amenable to discretization

As for the continuous cases and Forman’s theory, our construction obtains an Euler-Poincaré theorem and a Peixoto orgraph which encapsulates the underlying topology.

2 Basic vocabulary

We follow [12]. An embedding of a graph into a connected, compact surface is a - continuous map. Two embeddings and of in a surface are equivalent if there exists a homeomorphism such that . A 2-cell embedding is an embedding for which the components of the set are homeomorphic to open discs, the 2-cells. A 2-cell embedding induces a (cell) decomposition of , where in the obvious way and are the 2-cells of . Given a 2-cell , the edges and vertices in the boundary of can be arranged as a closed walk along the edges of , the boundary walk of .

The Hasse diagram of is a graph embedding whose vertices consist of a unique point in each cell of , and edges are disjoint lines connecting points of adjacent cells whose dimension differ by one (see Figure 5). We abuse language slightly and think of the vertices of as the cells of . The edges of split into those containing vertices or faces of , inducing the subgraphs and .

Figure 5: , its Hasse diagram and two representations of a Morse matching.

A matching in is a collection of disjoint edges in the Hasse diagram . A discrete vector field is a pair . Clearly is the disjoint union of the sets of edges and containing elements respectively in and in . Following [10, 6], a Morse matching is a matching in for which neither or contains a set of alternate edges of a closed cycle of . A discrete gradient vector field is a pair for a Morse matching . Such fields correspond to Forman’s discrete, acyclic Morse-Smale vector fields.

An unmatched cell in a Morse matching is a critical cell and its index is its dimension. The discrete version on the Poincaré-Euler formula is due to Forman [11].

[Euler-Poincaré] Let be a discrete gradient vector field of a compact surface , and be the number of unmatched -cells. Then

Forman also discretized the basic homotopy theorem of Morse theory [17].

[Homotopy] Any discrete gradient vector field of is homotopy equivalent to a decomposition whose -cells are the critical -cells of .

The basic ingredient in the construction of is Forman’s definition ([10]) of a (discrete) gradient path of dimension , which is a sequence of -cells in , such that for each there is a -cell which contains and satisfying and . The vertices of the Hasse diagram are the critical cells. Given a gradient path of dimension , for which belongs to a critical -cell and is also critical, we introduce an edge of joining both critical cells.

Following [12], a rotational system on a finite, connected graph consists of a choice of a cyclic ordering on each set of edges with a common vertex. A vertex which belongs to a single edge is associated to the ordering : we suppose there are two copies of . We denote a graph endowed with a rotational system by .

We say that and equivalent if and are isomorphic and the isomorphism either preserves or reverses the cyclic orderings for all vertices.

Similarly, one may suppose that a decomposition of a connected oriented surface admits cyclic orderings of the edges sharing a vertex — the orientability induces one preferred rotational system in the graph , which we denote by . The next theorem states that we reconstruct from .

Every rotational system on a finite graph induces a 2-cell embedding in some oriented surface with a decomposition . All such embeddings are equivalent.

In a nutshell, discrete and continuous gradient vector fields are related by the following operations. From a discrete gradient vector field of an oriented surface , we obtain the simpler field by Theorem 3, and its Hasse diagram is endowed of the natural rotational system induced by , giving rise to . One can construct directly from by considering the gradient paths between critical points defined above — they play the role of separatrices in the discrete context. On the other hand, following [22], the equivalence class under topological equivalence of a gradient vector field corresponds to the so called Peixoto orgraph in [5].

It is not hard to identify and . If these graphs have only two vertices, is a sphere and the identification is trivial: we handle the more general situation. Both graphs are tripartite: the Hasse diagram by cell dimension and the Peixoto orgraph, by minima, saddle points and maxima of . Edges in the — the 1-cells of — connect to two 0-cells and two 2-cells. The same happens for the Peixoto orgraph: saddles connect to two minima and two maxima. Again, both graphs induce 2-cell embeddings in with 4 edges in the boundary walk of each face. Since is oriented, both rotational systems are compatible.

3 Discrete gradient line fields

Let be a decomposition of a compact surface and be a Morse matching on the subgraph of . The pair is discrete (restricted) gradient line field. The relationship between this concept and the appropriate subclass of continuous line fields is not obvious at this point. Here we prove the expected basic results.

Figure 6: A sphere, a decomposition, discrete and continuous line fields.

A vertex is critical if it is unmatched in . Its index is if it is critical and otherwise. Let be the number of unmatched edges in the boundary walk of a face . Then is critical if . Its index is . As an example, let and be a spanning tree on the graph . Then induces a Morse matching in , hence a discrete gradient line field with a single critical cell, the root of . In Figure 6, the pentagon is a critical face.

[Euler-Poincaré] Let be a discrete gradient line field on . Then

[Homotopy] The decomposition of a discrete gradient line field is homotopy equivalent to a decomposition of a line field whose -cells, for , are the critical -cells of . The indices of the cells are preserved.

Figure 7: and , two homotopy equivalent decompositions.

Informally, our main result states that discrete restricted gradient line fields codify equivalence classes of (restricted) acyclic line fields which do not admit the last two types of critical points listed in Figure 2. We follow the line of thought presented at the end of the previous section. The first arrow in the diagram is Theorem 3. The Hasse diagram has to be replaced by a radial graph, to be defined below.

In analogy to the equivalence classes defined in the continuous contexts, we identify two discrete line fields and with isomorphic radial graphs and .

In a tradition dating back to the construction of Riemann surfaces, Bronshtein and Nikolaev [5] approached the study of a Morse-Smale line field by considering a double covering of with branches at the non-orientable critical points (the last four cases in Figure 2) on which the field suspends to a vector field . Clearly, a vector field obtained in such a way gives rise to a Peixoto orgraph with an additional symmetry, incorporated in their definition as an involution. Equivalence classes of gradient line fields are in correspondence with Peixoto orgraphs [5]. We might follow their approach, by defining double coverings of discrete line fields . Instead, taking into account the possible implementation of the constructions we suggest in this paper, we try to avoid the doubling of cells: we project to obtain a folded orgraph , which we define and show to be a radial graph in the section dedicated to proofs.

As in [3], we define the radial graph of a decomposition of . Its vertices consist of the vertices of and a point for each face of . The edges indicate the adjacency relations between vertices and faces, and are represented by disjoint arcs in . Thus is embedded in . The next result is a characterization of radial graphs [3].

Embed a bipartite graph on a surface yielding a decomposition . Suppose that either all 2-cells of have four edges along their boundary walk, or contains a single 2-cell, and the corresponding boundary walk traverses a single edge twice (in this case, is a 2-sphere). Then, up to equivalence, there are only two decompositions and of whose radial graphs are isomorphic to .

The decompositions in the statement are obtained as follows — we handle the case when is not the 2-sphere. For , split using that is bipartite. We construct . Its set of vertices is . Edges are diagonals of each 2-cell joining two vertices in . Finally, faces are components of , and are identified with the vertices in .


As in the previous section, where we constructed the Hasse diagram of in terms of , we may construct the radial graph of directly from by using the gradient paths of dimension . The vertices of are the critical cells (vertices and faces). Each edge corresponds to a pair consisting of a critical face and a critical vertex such that there is a gradient path connecting with some vertex in . Finally, is obtained from by introducing the natural rotational system induced by . From Theorem 3, the map is a bijection. The first picture in Figure 8 is a line field for which we indicated a gradient path, or equivalently, an edge of . In the second picture, all such edges are represented and the rotational system gives rise to 2-cells. Finally the third picture shows the associated radial graph.

Every equivalence class of restricted acyclic line fields admits a folded orgraph isomorphic to a radial graph, which in turn corresponds to an equivalence class of discrete line fields.

Figure 8: Discrete gradient line field . Radial graph . Folded orgraph. Line field.

4 Proofs

This final section is dedicated to sketches of proofs of the main theorems concerning discrete line fields: the homotopy Theorem 3, from which we derive the Euler-Poincaré formula, Theorem 6, and Theorem 3.

We use an auxiliary construction — a Morse matching on the Hasse diagram as defined above Theorem 5 — which enables us to convert the proof of Theorem 3 into Forman’s argument yielding Theorem 5.

Let be a discrete gradient line field on a compact, oriented surface . Define a graph as follows: the vertices are the unmatched edges in , and the edges are the non critical faces (i.e., faces with exactly two unmatched edges in its boundary walk). As an example, consider Figure 9.

Figure 9: Critical cells of . The graph . The induced Morse matching.

For each connected component of , a spanning tree induces a Morse matching in restricted to the cells in , as described before Theorem 6. The Morse matching is just the union of the ’s. Finally the induced Morse matching on is defined by . We treat two annoying cases separately.

The decomposition induced by a critical face of index is sphere.

Proof.

If the index of a critical face is (when ), all edges in the boundary walk of are matched with vertices in , and the matching has to be acyclic, because is acyclic. In other words, the complex induced by the edges in the boundary of is a tree. Applying Theorem 5 in the complex induced by and the matching is a critical vertex and a critical face, which is a sphere. Hence the next lemma follows. ∎

Let be a connected component of which contains a cycle , a collection of faces and edges. Then the closure of is a sphere .

Figure 10: A cycle , the Morse matching induced by and what’s left after simplifying as in Forman’s homotopy theorem.
Proof.

Take a cycle in the graph . The matching above has exactly one critical face and one edge in (see Figure 10), and the rest of edges on the boundary walk of the faces in are matched, since all face has just two unmatched edges and they belong to . Each edge in belongs to exactly two faces in . Applying Theorem 5 in the complex induced by , we are left with the decomposition of a sphere. In particular, all faces of the decomposition of belong to the cycle . and is a sphere or a projective plane. Since is orientable, the latter case does not happen. ∎


Proof of Theorem 3.

By Theorem 5 applied to the induced Morse matching in , we learn that is homotopy equivalent to another decomposition with exactly the same critical cells that . To complete the proof, we must prove that and have the same critical cells and indices. The cases handled in the two lemmas above imply trivial decompositions of .

The critical vertices of and are the same: by definition and is a matching in .

Let be a critical face of . The case is treated in Lemma 8 and is a noncritical face. Suppose then . By construction, is not contained on the graph . By the definition of the Morse matching , must be unmatched and hence corresponds to a face of . Also, , where we label the unmatched edges on the boundary of as . Each edge belongs to a connected component of . For each there is exactly one critical edge and exactly one gradient path from to . Thus , since the adjacency of faces in are given by the gradient paths of from edges in the boundary of to critical edges. ∎

Proof of Theorem 6.

From Theorem 3, it is enough to prove the case . By Euler’s formula,

Since all vertices are critical, . In the sum

each edge is contained in the boundary walk of exactly two faces in and the sum on the right hand side then is the number of edges in . ∎

We now prepare for the proof of Theorem 3. Following [5], a 2-branched covering induces the suspension of a Morse-Smale line field to a vector field with branches at the non-orientable critical points of . The vector field in turn gives rise to a Peixoto orgraph , an orgraph with a rotational system . More precisely, the orgraph is a tripartite graph, consisting of an upper level of maximum (critical) points, a lower level of minimum points and a middle level containing the saddles with 2, 4 or 6 separatrices. Additional properties follow from the fact that was obtained from a double covering. Thus, there is an involution in the set of vertices which induces a bijection . Also, restricts to , keeping fixed the saddles with 2 and 6 separatrices and coupling saddles with 4 separatrices. The edge set of consist of the separatrices joining saddles to vertices in the extremal levels.

The orientation of the orbits in induces naturally a orientation in the edges of . If then consists of a pair maximum-minimum connected by an edge — is a sphere. Finally the rotational system is given by the orientation of the surface and induces, as in Theorem 2, a collection of faces.

Every face obtained from the Peixoto orgraph of an acyclic Morse-Smale line field with has 4 edges in its boundary walk.

Proof.

Let be a face of . Clearly, the are no separatrices between saddles (since the fields are Morse-Smale) and between maxima and minimum. In particular, the number of edges in the boundary walk of is even. Now attach two copies of along its boundary, which becomes an equator to a sphere with copies of the field in on each hemisphere. In the sphere, the vertices of which represent saddle points of have index zero (they are destroyed by small perturbations) and maxima and minima are maintained. Since the Euler characteristic of is 2, the total number of maxima and minima must be 2. ∎

We define the folded Peixoto orgraph of a Morse-Smale line field as the graph combined with a rotational system obtained by projecting the Peixoto orgraph with the 2-branched covering .

Proof of Theorem 3.

Consider a Peixoto orgraph with . We must prove that the folded Peixoto orgraph is a radial graph, that is, a bipartite graph with a rotational system giving rise to faces with boundary walks with four edges.

The involution on the vertices of leads a partition of the vertices of on two levels, and the projection .

A face obtained from the Peixoto orgraph is a disk with four edges in its boundary walk, by the previous lemma. The map restricted to the interior of is a homeomorphism since all the branched points are concentrated in the vertices of . Thus the projection in is a face which 4 edges in its boundary walk. ∎

Notice that the presence of the non-stable critical points is irrelevant for the argument.

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