The symmetric area surrounding a CSP or a CSA

The symmetric area surrounding a CSP or a CSA 177

property of the LS-CSP is valid for every point of Γ1, Γ2. If Γi are loop edges, it means that the symmetry property is addressed globally through the relative position and intrinsic parameters ofF2 and F3, the faces defining Γ1, Γ2 and adjacent to F1 (see Figure6.1f). Otherwise, if Γiare composite loops edges is coming from closest vertices, the symmetry property can be decomposed through two sets of entities: vertices of Γi

and then, edges. Checking the symmetry of these sets of vertices can be conducted first (see Figure6.1e) and then, using the surface parameters and locations of the faces adjacent toF1 and defining the edges of Γi, the symmetry of the sets of edges can be analyzed.

Finally, a CSA appears as a particular configuration of O-CSP where I_c is repre- sented by a loop edge with no vertex. Hence, the symmetry property is valid for the whole loop edgeE1.

All the CSP configurations involving point sets corresponding to edges have now been analyzed. Now the purpose is to focus on faces: another type of infinite point sets. Indeed, the generation of CSPs incorporate the spatial location and parameters of the reference surfaces adjacent to each edge involved in the definition of a CSP. The symmetric area around a CSP is not only along its associated edges, the surrounding surfaces of these edges is symmetric, too. Consequently, in a small area around each point on the symmetric curve(s), the symmetry property apply to the adjacent faces:

F1,F2for an O-CSP, a BS-CSP, a CSA;F1,F2,F3for a LB-CSP;F_ithe faces adjacent toF1 in case of LS-CSP. Because all the points of these adjacent faces benefit of the same geometric properties as those in the neighborhood of their common edges, the symmetry property extends to these entire surfaces, whether they are bounded or not.

Then, each face Fi of MM AX being bounded by one loop at least, it is mandatory to insert repetitively all the CSPs attached to the vertices, edges and toF_i that form the boundary of F_i. Adding new vertices and edges to a first edge, means that new faces adjacent toFi are also added until all the loops of the surface are covered. Then, the area ofF_i is closed and all the constraints are set to analyze the symmetry over F_i. In this case, the validity of the current CSP for the whole face can be evaluated through the relative positions of the faces adjacent toFi that model each edge of the loops bounding F_i. If these are symmetrically set with respect to the CSP, then F_i is effectively symmetric with respect to this CSP. However, if any of these surfaces is not symmetrically set with respect to the CSP, the propagation stops. The symmetry property is lost.

It has also to be pointed out that if the CSP is valid forFi, the symmetry property extends outsideFi to its adjacent faces, forming an open domain outsideFi where the symmetry holds.

Figure6.1a is the illustration of the symmetry area around an O-CSP.Vstartis the intersection point between E1 and the O-CSP. The gray area around E1 illustrates the extension of the symmetry property alongF1 and F2. Then, this symmetry needs

The symmetric area surrounding a CSP or a CSA 179

(a) (b)

(c) (d)

(e) (f)

Figure 6.1: Symmetry area of each CSP category: (a) symmetry area of an O-CSP;

(b) symmetry area of an LB-CSP; (c) are symmetry area of a BS-CSP; (e) and (f) are symmetry areas of an LS-CSP.

to be analyzed with respect to the complementary boundary entities ofF1 andF2. In F2, all boundary curves are symmetric with respect to the O-CSP. This plane is valid for F2. But in F1, its boundary loop is asymmetric, hence the symmetry property cannot be expanded to the finite area defined byF2.

Figure 6.1b is the illustration about the symmetry area of an LB-CSP. LB-CSP is generated from facesF1,F2,F3. The gray area shows the extent of the symmetry around V_start and the edges E1 and E2, which are symmetric with respect to the LB-CSP.

Figure 6.1c shows the symmetric area around a BS-CSP. In this chapter’s intro- duction, it has been recalled that a BS-CSP is related to an edge E1 , intersection of F1 and F2. The gray area indicates the propagation of the symmetry property, starting fromE1 and progressing over F1 andF2.

In Figure6.1d, the symmetry area represented originates from a CSA. This sym- metry axis can be seen as attached to the intersection curve E1 between F1 and F2. E1 is necessarily a loop edge without vertex corresponding to a circle. The symmetry area, in gray, expands on both sides ofE1 to propagate the symmetry axis from E1.

Figure6.1e and f depicts the symmetry areas attached to an LS-CSP. Depending on the type of the internal loops, they can be either composite or loop edges. Figure6.1e illustrates the symmetry area restricted to vertices when they are considered as an independent set of entities, distinct from the edges. The symmetry area propagates from two symmetric vertices of the two loops. In Figure 6.1f, the loops are loop edges, the LS-CSP originates from F2 and F3 So, Γ1pE1q and Γ2pE2q are symmetric with respect to the LS-CSP and the symmetry area expands from these two edges.

Considering the bounded area ofF1, the symmetry of the LS-CSP is also valid globally forF1.

When processing an LS-CSP as well as during the second level of symmetry prop- agation (see Section6.7), it is necessary to check the symmetry of vertex pairs, edges pairs and face pairs with respect to this plane that is not attached a vertex or an edge.

Evaluating the symmetry of vertices, edges and faces with respect to a symmetry plane can be addressed through a bottom-up approach. In this case, the Figure6.2is an illustration with all the successive steps to analyze the symmetry of edgesE1 and E2 and their adjacent faces, when E1 and E2 are bounded by two vertices. The first step processes the extreme points two the edges E1 and E2. Choosing arbitrary V11

as starting vertex and the other vertex asV22, if they are asymmetric, then changing it toV21, this time, the vertices must be symmetric otherwise the configuration is not symmetric (see Figure6.2a and b). Thus, assuming that the vertex pairs are symmet- ric, Figure6.2c illustrates the beginning of second step. With the help of hypergraph G21, the surfaces adjacent toE1 andE2are picked up. Then, the intrinsic parameters and locations ofF11andF21, both cylinders (Cy1and Cy2), are evaluated for symme-

The symmetric area surrounding a CSP or a CSA 181

(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 6.2: The symmetry analysis process for two edgesE1 andE2: (a) current edges and CSP; (b) symmetry analysis of extreme points; (c) adjacent surfaces at each edge;

(d) selection of one surface pairs; (e) the symmetrical cylinders; (f) the second pair of surfaces; (g) the symmetrical cones; (h) all the entities leading the symmetry of E1

and E2.

Figure 6.3: An illustration of the need to distinguish concave/convex areas during the symmetry analysis of an object.

try. Assuming the symmetry, the homologous cylinders are shown in Figure6.2e and they are symmetric. Then, selecting the other pair of surfacesF12 and F22, they are cones (Co1 and Co2), and their are symmetric with respect to the symmetry plane, as shown in Figure6.2g. Finally, this plane orCSP is valid forE1 and E2 as well as for its neighboring surfaces.

IfE1 and E2 are loop edges, the above symmetry analysis can be reduced to sur- face comparison when the edges have no vertex. Indeed, the symmetry property, if valid, applies to all the points of Cy1 and Cy2, Co1 and Co2. The B-Rep model of the object contains information to describe the used area of a surface. The paramet- ric representation of these surfaces and the external loop defined in their associated parametric space characterizes their used area. However, this information does not produce a characterization of the concavity/convexity of the corresponding face. This is mandatory to separate configurations when analyzing the symmetry of an object (see Figure 6.3). To this end, the concept of orientation index (see Section 5.4.6) is used to separate the ambiguous configurations. In addition to the geometric con- ditions stated previously, couples of faces must have the same orientation index to effectively meet the symmetry property.

InMM AX, each edge has two neighbor surfaces. So, the propagation mechanisms described in this section, combined with the adjacency relations available in the hy- pergraphs can be used to conquer, i.e. to cover, the entire boundary, BM_{M AX}. In this case, the propagation process overlaps the face boundary and when all faces of BM_{M AX} are covered without any asymmetry, the CSP is upgraded to the status of Global Symmetry Plane(GSP). Similarly, a CSA extending to the whole boundary of MM AX becomes a GSA. A first propagation process is now described in the following section.

First level propagation and CSP chains 183