possible area of a model boundary to structure the global symmetry properties of the object: it is the conquer phase.
Apart from symmetry analysis, asymmetry is also very interesting and is more useful for design and PDPs since many components are not globally symmetric, rather symmetry exists only at the level of a boundary subset and the loss of symmetry can be used as a means to evaluate shape transformations that could be useful for simplification purposes. The symmetry analysis proposed in the current approach intend to address this issue since the propagation process helps identify and structure areas where symmetry properties are valid.
Finally, the objectives of the approach can be summarized as the algorithms to answer the following questions:
1. Is a B-Rep NURBS model symmetric with respect to some symmetry planes or symmetry axes? If so, where are located these symmetry planes and symmetry axes?
2. If a B-Rep NURBS model has no global symmetry property, doest it benefit local symmetry planes or axes? Where are located these symmetry planes or axes and what are their extents of validity over the model boundary?
3. How does these symmetry properties can be obtained at various steps of the design and PDPs? Into which extent they can be obtained with a process in- trinsic to the object shape and how robust is this process under a wide diversity of shapes?
Hypotheses and shape range 41
(a) (b) (c)
Figure 2.1: Illustrations of example objects part of the volume category addressed by the current approach.
point located strictly inside the volume, i.e. not on its boundary, has a neighborhood defined as a ball. Then, any point located on the boundary of the volume has a neighborhood that is topologically equivalent to a disk. The corresponding class of objects is designated as 2-manifolds. The Euler-Poincar´e theorem is applicable to them and extends the notion of volume under the form:
#V #E #F 2p#s#hq, (2.1)
where the quantities are respectively, the numbers of vertices, edges and faces forming the boundary of the object and the numbers of partitions and holes of this boundary.
In particular, #sextends the concept to objects dividing the 3D space into more than two partitions. Industrial CAD modelers behave differently regarding this extension, some allowing a subset with ‘cavities’ only, others conform exactly to this general concept.
Here, as a first step of the proposed approach, volumes have been restricted to
‘manufactured’ objects only, i.e. when they are bounded by one partition only.
If there exist point neighborhoods of the object boundary that are topologically equivalent to several disks or to a disk and segments (see Figure2.2), the corresponding class of objects is of type non-manifold. This class of objects is often appearing in geometric models devoted to simulations like finite element ones. Here, they are out of the scope of the proposed approach. However, non-manifold configurations have to be distinguished from that of Figure2.1c where the two small cylinders are geometrically tangent to each other but not connected through an edge of the object boundary.
This configuration is indeed manifold and within the scope of the proposed approach.
In Figure2.1c, the two top cylinders are tangent. Between them, if at their common tangent linear edge and vertex exist, they must be two instances of the same group of edge and vertices superimposed and forming an open chain, otherwise the model is non-manifold.
(a) (b)
(c) (d)
Figure 2.2: Illustrations of object examples falling outside the scope.
2.2.2 Range of objects addressed
As stated in the previous section, the shape category addressed is of type vol- ume and follows the restrictions stated there. Industrial CAD modelers however, can produce different object categories, sometime under the same computer type they designate as volume, which is not accurate enough for the purpose of the proposed approach. Here, the goal is to define the object range which is not part of the previ- ously strictly defined volume category but that can be transformed into this category without perturbing the symmetry analysis.
As a refinement of the non-manifold category illustrated before, let us consider configurations where non-manifold singularities occur at vertices only and are further restricted to configurations where the neighborhood of these points are formed by sets of disks. Indeed, such a range of configurations is named pseudo-manifolds and is able to describe volumes as stated previously even if they incorporate these non- manifold singularities. Pseudo-manifold can appear as ‘extreme’ configurations in CAD software and STEP files. For the sake of completeness, these configurations can be converted into manifold ones through vertex duplication and fall in the scope of the proposed approach without changing the object shape, hence without influence on the symmetry properties of the object.
The analysis can be extended to non-manifold configurations where the singular entity is an edge. Figure 2.1c can be interpreted in that way if a unique edge exists as common generatrix of the two cylinders. Consequently, if the duplication of these
Hypotheses and shape range 43 edges and some of their extreme vertices modify the boundary decomposition of the object without modifying the number of partitions, i.e. #s 1, these transformed objects have preserved their initial geometry and now fall into the object category addressed here.
2.2.3 Shape geometry and reference surfaces
The previous sections have concentrated on the topological aspects of the shape range addressed. Here, the purpose is to focus on the geometry of the models covered in the proposed approach. The main context of the present work concentrates on PDPs and mechanical engineering applications. There, objects are generally created through constructive approaches using sketches and simple primitives. Line segments and arc of circles are combined to form most of the sketches content and they are translated or rotated to be extruded or revolved to form primitive volumes that can added to or removed from an existing one. Processing geometry that way forms already a wide range of manufactured objects whose boundary surfaces are derived from the previous observations. Such faces are now designated as Reference faces.
In the framework of CAD software, 2D sketches in arbitrary planes as basis for extrusion or revolution operators form the main method to create 3D volumes. Blends and chamfers come afterwards as local modification of an object boundary. As a result, the boundary surfaces of 3D volumes fall into the following configurations when combining segments and arcs with extrusion (translation) and revolution (rotation):
• Planes: they can originate from the extrusion of a segment belonging to a sketch, from the revolution of a segment orthogonal to the rotation axis or from a closed planar contour of a sketch defining a face of a volume primitive;
• Cylinders: they can be obtained by revolution of a segment parallel to the rotation axis or as extrusion of a circular arc orthogonally to the plane containing the arc;
• Cones: they are generated by rotating a segment that is not parallel to the rotation axis;
• Spheres: they are obtained by revolving a circular arc whose center is located on the revolution axis;
• Tori: they originate from revolving a circular arc whose center does not lie on the revolution axis.
As a result, assuming that sketches only contain segments and circular arcs covers a large amount of mechanical objects. Hence, the five surface types listed above
form the basic configurations of 3D volume boundary surfaces. These five surfaces are called reference surfaces and notedS. Because it covers a large diversity of CAD volumes, it is assumed that 3D volume boundaries considered here combine only these reference surfaces. There is no restriction placed on intersection curves between these surfaces. Consequently, configurations on blending radii with constant or variable radius located on arbitrary intersection curve between reference surfaces can generate free-form surfaces and are not part of the present approach.
This restriction is a trade-off between a wide enough range of objects covered and the complexity of the approach. Perspectives still exist to widen further the current object range.