Hypergraph creation from a B-Rep model

In chapter 3, the data structure of STEP format has been introduced. Within this hierarchical data structure, ADVANCED FACE at level 8, ORIENTED EDGE at level 5 and VERTEX POINT at level 2 are key topological entities describing a B-Rep model topology. Level 3 entities contain the geometric description of surfaces and curves. The CARTESIAN POINT contains the coordinates of a vertex. Other levels contain either geometric information or complementary topological entities, e.g.

edges without orientation, edge loops, which contribute to the definition of the key entities listed previously.

The STEP format is a typical B-Rep data structure. Depending on the model translation process from a geometric modeler to the STEP file format, depending on the internal model of this geometric modeler, the detailed model data contained in a STEP file may differ. Anyhow, a B-Rep model stored under a STEP format contains topological as well as geometric entities. A hypergraph points at the topological entities which are represented as F_HG, E_HG and V_HG. The geometric entities are attached to the topological ones. The connections between the topological entities are coming from the parent-child relationships contained in the STEP format. In this case, from the hypergraphs, it is possible to get all geometric parameters of the faces, edges and vertices (see Figure4.4).

Because different users of different CAD software often produce distinct shape generation processes of the same object model, i.e. each shape is visually identical

Figure 4.4: Hypergraph data structure.

Hypergraph creation from a B-Rep model 69

(a) (b)

Figure 4.5: Spherical surface represented by different entities: (a) 3D representation of the sphere; (b) parametric space of a patch with a singular edge.

to the others, these models may contain different entities. More precisely, configura- tions with coinciding vertices or otherwise stated, non-manifold configurations around vertices, produce visually equivalent objects though they contain a different set of entities. Other configurations originate from the parametric space of surfaces, which are bi-parametric and described by square or rectangular domains in their parametric space. Some of their vertices must coincide when describing a surface of revolution:

see Figure 4.5 the pole of the sphere behaving like the apex of a cone in 3D because vertices coincide together with a zero length edge. Consequently, visually similar ob- jects may be represented by different entities of B-Rep models, some with coinciding vertices and others with zero length edges, etc. Indeed, the edge lengthlE of an edge E may not be zero, as long as it is a curve shorter than the modeler tolerance,ε. Con- sequently, its two vertices, V1, V2 may not coincide exactly, rather their distance is smaller than the modeler tolerance, }ÝÝÑV1V2}  ε. These entities are usually created as a result of algorithm computations. It can be also the result of internal computations for the model translation between STEP standard and the internal model of a mod- eler or the opposite. Because of zero length edges and coinciding vertices, they impact the symmetry analysis of the input shape. To avoid such an influence and obtain a generic boundary description process, the zero length edges and coinciding vertices should be ignored prior to the generation of the hypergraphs forming the description of the input modelM_I. The criteria supporting these transformations are:

Criterion 1 If the edge length of E is smaller than the user-prescribed tolerance

l_E  ε_u, this edge will not be collected as an HG EDGE.

Criterion 2 If the distance between two vertices V1, V2, is smaller than the user- prescribed tolerance }ÝÝÑV1V2}   ε_u, the second vertex V2 will not be collected as a HG VERTEX and its neighbor connections are moved to the first one V1.

To initialize the hypergraphs, the algorithm could be created based on criteria 1 and 2 in a first place. Then, it scans all topological faces in the B-Rep model input.

For each face, it traces the boundary edges and an arc is added inG21 for each such edge. Using each edge of the B-Rep model, its extremities can be identified and the corresponding vertices are created in G10. Using the faces and vertices obtained in G21 and G10, they are inserted inG20 to populate it. After this process, the B-Rep model is transferred to the hypergraph datastructure where all its entities are valid within the scope of ε_u. It is a consistent topological representation since extraneous entities (edges, vertices) linked to the modeler behavior have been removed and these hypergraphs form the basic datastructure of the object boundaryBM_I to be processed for symmetry analysis. This gives a precise definition of the content ofM_I where its hypergraph description contains no edge smaller than εu and no vertices within a similar distance.

As a consequence of this process, loop edges can appear when vertices are merged that belong to tangent curves. The corresponding graph structure gets modified and the dual graph around a vertex V_i no longer reduces to a simple loop since a face can appear several times around V_i. It can become a set of simple loops connected through the faces appearing several times (see Figure 4.6) or exhibit dangling arcs when edges reduce to loops. The content of hypergraphs is no longer able to define the orientation ofM_I but the hypergraphs are not oriented and the orientation ofM_I is still available through the references to B-Rep entities accessible from the hypergraph entities. Further details about the relationship between a dual graph at a vertex and hypergraphs are formally given at section4.6.

Hypergraphs transformation operators 71

Figure 4.6: Illustration of the effect of vertex merge operations on the dual graphGD

around a vertex V1 and V2.