Vertex splitting at regular and non-regular points 89 Indeed, both of them can be used and will be discussed in chapter 5. In the present chapter, Crit.8 is left as it is, which produces the largest possible edges.
(a) (b)
Figure 4.20: A continuous edge with a sharp point after the maximal edge generation:
(a),P1connects to two curves, (b) from a topological point of view, Σ1,2 forms a single edgeE1,2.
by tangent loops and homologous faces connected to each other through a vertex characterize the criteria of the manifold vertex splitting operation.
(a) (b)
Figure 4.21: An example configuration where curves bounding surfaces are tangent to each other. The continuity of the corresponding edges is broken atP1: (a) the initial B-Rep modelMM I, (b) the maximal boundary modelMM ax.
The interesting thing is that not all the similar cases produce a manifold vertex split operation. For example, in Figure 4.22, the local structure looks similar to Figure4.21a. Indeed, when analyzing the faces surrounding point P1, the cylindrical facesFM3 and FM5 are homologous. P1 connects with S2 twice, i.e. on both sides of P1. Obviously, FM2 is homologous to itself. Also, the planar surfaces FM1 and FM4
are homologous. Now, the difference between Figure4.21a and4.22a is that the pairs of homologous surfaces in Figure 4.22a are crossing each other, i.e. pS3, S5q ‘crosses’
pS1, S4q,pS3, S5q‘crosses’pS2, S2qwhen considering their adjacencies aroundP1. IfV1
is split and some pairs of homologous surfaces are merged, the continuity properties of other surface pair(s) cannot be achieved. Currently, there is no criterion available to select a pair of homologous faces to be merged or to be broken (see Figure4.22b and c).
Because the vertex splitting operation is not unique in the present configuration and there is no criterion appearing to justify the choice of one solution, no vertex split should take place. Consequently, the point P1 is called a non-regular vertex and if the point could be split it is called a regular vertex. To distinguish the regular and non-regular configurations at vertices, the definition is as follows:
Definition 3 Considering a vertexVα ofMM I, Vα is surrounded bynordered sectors Φi and n ordered edges ηi, i P r1, ns, n ¥ 3, based on the adjacency relationships between the edges attached at Vα. A sector of a face FM j is formed by two adjacent edges ofFM j whose common vertex isVα. Based on the input modelMM I, there exists an order of sectors available from the real B-Rep datastructure of MM I that defines the disk around Vα as part of the B-Rep model, i.e. its orientation is consistent with that of the B-Rep. Because the sector and edge numberings differ for each one, a dual graph based on Φi and ηi, restricted to Vα, can be generated that forms a simple loop around Vα.
Vertex splitting at regular and non-regular points 91 Now, let us consider that there exists at least a couple of homologous sectors:
DpΦi,Φjq {rΦi rΦj, pi, j¤nq, i j.
If there exists at least another couple of homologous sectors: DpΦk,Φqq {rΦk rΦq, pk, q¤nq,pΦi,ΦjqandpΦk,Φqqare said to cross each other ifi k jandj q¤n or 1¤ q i. This configuration is qualified as non-regular configuration at vertex Vα and Vα cannot be split forpΦi,Φjq andpΦk,Φqq.
If there exists one or more homologous sectors only: DpΦk, . . . ,Φqq {rΦk . . . Φrq, k, q, . . . ¡ i and k, q, . . . j, pΦk, . . . ,Φqq are not crossing pΦi,Φjq and form a regular configuration at Vα. Vα can be split to assign its copy to the sectors Φt, tP tpi 1q, . . . ,pj1qu
Otherwise, the configuration at Vα is said neutral because the vertex split is not meaningful for it.
It has to be noted that each sectorΦi maps to one face FM l of MM I butFM l may generate several sectors Φi, Φj, . . . .
(a) (b) (c)
Figure 4.22: An illustration of non-regular vertex P1: (a) the original B-Rep model MM I, (b) and (c) are two possible maximal configurations.
Figure4.22is an illustration of non-regular configuration. In this figure, surround- ing pointP1,Φr3 rΦ6,Φr1 rΦ5 and Φr2 rΦ4. pΦ3,Φ6q is crossingpΦ1,Φ5qas well as pΦ2,Φ4q. Hence, this is a non-regular configuration atP1.
Back to the example in the Figure4.21. Φr1 rΦ5 and Φr2 rΦ4, but both of them are not crossing each other. Hence, this example is a regular configuration at vertex P1.
Given this regular configuration at pointP1, the face merging process is applied as follows. Its surrounding faces can be detected easily. In Figure4.24b, the correspond- ing sectors arepΦ1,Φ2,Φ3,Φ4,Φ5,Φ6,Φ7q. Because sectors Φ2 and Φ4are homologous as well as Φ1 and Φ5 and these couples are not crossing each other, vertexV1 can be split and face FM1 can be merged with FM4. Also, the splitting operation removes the tangent loop in FM2.
Based on the previous qualitative analysis, the purpose is now to formalize the
Figure 4.23: Dual graph based on sectors ΦiatVαand corresponding transformations.
characterization of the configurations and set the corresponding operators. Starting from the distinct sectors mentioned in definition3that form a simple loop, the purpose is to connect the transformations of this loop through the homologous sectors to the maximal faces ofMM I. To this end, let us consider any two homologous sectors ˜Φi, Φ˜j,ij. Because they are homologous to each other, this property can be expressed in the simple loop by merging Φi and Φj to produce Φi,j. Consequently, this simple loop becomes two tangent simple loops at the node Φi,j if|ij| ¡1 (see Figure4.23):
Φi,j, Φi 1, . . . , Φj1 and Φi,j, Φj 1, . . . , Φn, Φ1, . . . , Φi1. Indeed, |ij| ¡ 1 holds because Φi and Φj cannot be adjacent otherwise they would belong to the same maximal face, which contradicts the fact that every sector belongs to a maximal face.
Now, let us consider the following consequences:
a If Φi and Φj belongs to the same maximal faceFM k, this merging operation sets a one to one mapping between Φi,j and FM k;
b If two of the edgesηtattached to Φi and Φj are mapped to the same edgeEp of MM I, then these edges ηr and ηs may be merged to set a one to one mapping withEp. Ifηr andηsconnect to the same sector of index such thati 1j1, i.e. there is only one sector between Φi and Φj, then ηr and ηs can be merged and form a dangling arc to connect Φi,j to Φi 1. Otherwise, if the sectors Φk, Φl connecting to ηr and ηs are mapped to non homologous faces FM p, FM q, these edges cannot be merged since merging would produce a hyperarc, which contradicts the fact that the neighborhood of Vα is manifold.
Assuming that all homologous sectors ˜Φi, ˜Φj producing a one to one mapping between Φi,j and FM k are effectively merged and that all edgesηi that can be merged
Vertex splitting at regular and non-regular points 93 are effectively merged, the resulting graph structure has a one to one mapping between each of its nodes and a face FM k and a one to one mapping between each of its arcs and an edgeEp. Consequently, the resulting graph pΦ, ηqis isomorphic to a subset of hypergraph G21 restricted to the faces and edges connected at Vα: G21|Vα. Because of this isomorphism, the subsequent analysis is performed on G21|Vα. Based on the B-Rep datastructure ofMM I, it can be assumed that the simple loops tangent to each other and the possible dangling arcs are available to structureG21|Vα.
Now, repeating the face merging process for any set ofm homologous facesFM p, FM q, . . . , in G21|Vα produces new loops and two configurations may occur:
• All the homologous faces belong to the same simple loop, then the merge oper- ation creates mnew simple loops only from this initial loop;
• Homologous faces belong to different simple loops, then the merge operation creates m new loops among which there can be simple loops and at least one connection between existing simple loops.
Applying a vertex split operator to this graph structure transforms it as follows:
the split has to be associated with a face FM k of G21|Vα and is submitted to the configurations regarding its surroundingb edges Ei:
• If there existhedgesEj,El, . . . whose ranks inG10,RjG10RlG10. . .1, i.e. these edges are loop edges defined by a single vertex, they may generate up toh split operations of FM k;
• Either h0 and b is even orh¥0 and bh is even, then Ei generates either db{21 ord pb hq{21 split operations, respectively.
Then, the effect of the split is the d-times duplication of FM k whose rank is RN k: FM k Ñ pFM k1 , . . . , FM kd q and the connection of each FM kj with a subset of Ei to terminate each elementary split. Then, each termination of elementary split with an increment of the number of components in G21|Vα characterizes a regular con- figuration and each elementary split without this increment is not terminating and characterizes a non-regular configuration. A completed elementary split creates a new component in G21|Vα because it is assigned to the new copy of FM k, showing that the neighborhood of this new vertex can be independent of that ofVα, which is the purpose of the splitting mechanism. Similarly, a non terminating elementary split reflects a configuration where no independent neighborhood can be found in Vα that can be assigned to a new vertex. Hence, the vertex split criterion can be stated as follows:
Criterion 9 Having merged all the sets of homologous faces connected at vertex Vα, the hypergraph G21|Vα restricted to the neighborhood of Vα is transformed with new
connections. The type and number of edges Ei connected at Vα generate dcandidate elementary vertex split at nodesFM k. The ith elementary vertex split characterizes a regular configuration ofFM k at Vα if it increments by one the number of connected components in G21|Vα. As a result, the homologous faces merged to generate this configuration can be effectively used to generate a independent neighborhood for the new vertex generated by the elementary split.
If an edge Ei is a loop edge: RiG10 1 and if it belongs to a dangling edge or, more generally, a branch ofG21|Vα, any copy ofFM k: FM kj connects toEi, is assigned a rankRNj
k 1 and creates a new componentinG21|Vα, which really terminates this elementary split. If Ei is a loop edge and belongs to a loop of G21|Vα, a copy FM kj connects to it and the new content ofG21|Vα is scanned to look for an increment in the number of components. If its number of components has not been modified, this elementary split is not performed and characterizes a non-regular configuration.
Repeating this processh1 times if b 0 or h times if b ¡0 leaves bh edges to connect to copies ofFM k.
Based on the current content ofG21|Vα, each copyFM kph 1q orFM kh , . . . ofFM k left is connected to two edges ofEi in accordance with the loop structure associated with G21|Vα. If this copy and its new connections in G21|Vα create a new component, this elementary split terminates and characterizes aregularconfiguration. Otherwise, the configuration isnon-regular. If the configuration is non-regular, some of the homol- ogous face merge are no longer necessary because they are not meaningful. Therefore, it is necessary to restore these loops to obtain a consistent face decomposition. To this end, let us first consider that all the nodes candidate to the split operation have been processed, extracting components from the initialG21|Vα. Then, the component left in the final version of G21|Vα can contain nodes that were connected to simple loops in the initial G21|Vα. Consequently, the purpose of the transformation of the current version ofG21|Vα consists in splitting all the nodes connecting multiple loops to produce simple loops as large as possible but preserving the connections between loops that were existing in the initial version ofG21|Vα.
Once all the regular and non-regular configurations have been identified through the splitting process and the non-regular configurations have led to split face nodes resulting from unnecessary face merge operations, it forms a subsetG210of the trans- formedG21|Vα. Then, the graph components generated for each regular configuration needs to be connected again to G210 to obtain the correct version of G21|Vα to be inserted inG21 (see Figure 4.24as an example of this process).
Now, coming back to examples, focusing on vertexV1 in the configuration depicted at Figure4.21, Figure4.24a, b gives the hypergraphG21|V1 as well asG20|V1 providing the set of faces surrounding V1 and G10|V1 describing the set of edges surrounding V1, as starting point of the face merging and vertex split processes. Figure 4.26a
Vertex splitting at regular and non-regular points 95 illustrates the underlying connection betweenG21|V1 and the initial dual graph based on sectors. Figure 4.24d illustrates the result of the face merge step inG21 and G20 whereFM1andFM4 are homologous to each other and becomeFM1,4. G10 represents the configuration when FM1,4 is being processed for splitting. The set of edges Ei connected toFM1,4 does not contain loop edgesph0q resulting ind1 elementary splits. Figure4.24c, illustrates the result of this elementary vertex split producingV11. Figure 4.26b contains the new structure ofG21 showing the component created with V11. Figure4.26b contains also the transformation attached to the next candidate face inG21, i.e. FM2. FM2 is a pre-existing node inG20|V1 producing tangent loops. There h0 andd1,G20 andG10 show the results of the corresponding elementary split:
V12. Figure 4.26b is the graphical representation of the split generating V12. Finally, Figure 4.26b summarizes the final configuration after performing all the vertex splits and subsequently the edge merge operation resulting in EM1,4 and EM2,3. Then, Figure 4.25d represents the new local content of G21, G20, G10 to be inserted in these hypergraphs. It can be noticed that FM2 and FM3 are now disconnected from FM1,4 in G20 but will be connected to other surrounding faces. EM2,3 becomes an edge without vertex and EM1,4 will be connected to extreme vertices ofE1 and E4.
Comparing Figure 4.26a and b shows that the orientation of MM I contained in the initial dual graph atV1 cannot be conveyed in the hypergraphG21 resulting from the face merging process. However, this orientation is not mandatory for performing the face merge and vertex splits, which is a property of this process even though it is not formally demonstrated here.
Because F2 is homologous with itself. The next work is to “merge” them and in this example the vertex V1 of Figure 4.21 can be split again. The last step is to merge the edges using criterion8, because the previous steps may create some vertices connected only to two edges (see Figure4.25d). All the operations in the hypergraphs follow the hypergraph basic editing operations described at section 4.5.1.
Another example configuration is illustrated in Figure 4.28a with a set of eight faces around V1. In this case, the homologous faces form a larger set Fr1 rF3 Fr5 rF7 (see Figure 4.28b). Within the set of faces, there is no other homologous faces. Regarding the criterion9, after merging these four faces (see Figure4.28d), the face FM1,3,5,7 is the only candidate face for vertex split with h 0 and d 3. In the present configuration, each elementary split inG21|V1 produces a new component, thus characterizing a regular configuration. Figure4.28c and d respectively, show the resulting vertex configuration and the effect of the splits inG20 and G10.
Finally, the configuration of Figure4.22 is used to illustrate how G21|V1 is trans- formed into a configuration whereFM3,5 andFM1,4 are candidate nodes for the vertex split operator but this operation cannot terminate, highlighting the non-regular con- figuration (see Figure4.27).
Through the previous sections, all the processes and criteria have been clarified to
(a) (b)
(c) (d)
Figure 4.24: The successive steps to process homologous faces at a regular vertex through merging operations and vertex split. The final edge merging operation is also illustrated (first subset).
generate a boundary decomposition from the initial B-Rep model MI that is suited for its symmetry analysis. After some processing depending on the conformity of MI with respect to a 2-manifold model, MI can be transformed into MM I with the non-manifold vertex split. Then, the boundary decomposition ofMM I is modified to meet the symmetry analysis requirements. To this end, the maximal faces are gener- ated with the face merge operator to remove edges between homologous and adjacent surfaces. Maximal faces are further extended with the merging process of faces adja- cent to each other through a vertex using a manifold vertex split. The resulting face neighborhoods are extended around their common vertex. Finally, maximal edges are produced through a vertex removal operation between adjacent edges. The resulting model is MM AX that has a boundary decomposition intrinsic to the object shape and operational for symmetry analysis. The overall process enumerated previously is summarized in Figure4.29.
It has to be reminded that the boundary description MM AX contained in the hypergraphs has no effect on the geometric description of the object. All the trans- formations performed are strictly topological and have no influence on the surfaces, curves and points describing the object. Further, the datastructure set up estab-
Vertex splitting at regular and non-regular points 97
(a) (b)
(c) (d)
Figure 4.25: The successive steps to process homologous faces at a regular vertex through merging operations and vertex split. The final edge merging operation is also illustrated (second subset).
lishes a connection between the B-Rep CAD topological description of the object and the hypergraphs structuring this object boundary in accordance with the symmetry requirements. Consequently, the hypergraphs content combined with the geometric data describing the object can be seen as a view of the object devoted to its symmetry analysis. In the framework of a PDP, it is effectively a view of the object similarly to the approach proposed by Foucault et al. [18] devoted to mesh generation constraints for finite elements simulations.
Finally, for the sake of simplicity, maximal faces and edges respectively notedFM i
and EM j in this chapter will be noted as Fi and Ej in the following chapters since all the subsequent treatments for symmetry analysis will be performed on an object decomposition using the maximal faces and edges.
(a) (b) (c)
Figure 4.26: The dual graph from Figure4.24b inG21|V1 with its surrounding sectors:
(a) initial dual graph showing that the merge faces produce regular configurations, (b) resulting graph after the face merging processes.
(a) (b) (c)
Figure 4.27: Graph transformations ofG21|V1 based on Figure4.22: (a) the status of homologous faces; (b) the homologous configurations of faces showing that the split vertex operator; (c) one homologous faces won’t split the graph.
Vertex splitting at regular and non-regular points 99
(a) (b)
(c) (d)
Figure 4.28: A special case of non-regular vertex: (a) The B-Rep model; (b) local hypergraph around the regular vertex V1; (c) the result after splitting; (d) the result of local hypergraph.
Figure 4.29: The process flow of creating hypergraphs describing a maximal boundary decomposition from an initial standard B-Rep model MI. The resulting model is MM ax, which is used for subsequent symmetry analysis.
Conclusion 101