First level propagation and CSP chains

First level propagation and CSP chains 183

Figure 6.4: LS-CSP cutting a cylindrical face without intersecting its external loops.

curve (see Figure6.4). Because these LS-CSPs produce directly a closed curve, there is no need of propagation to produce a closed path, the second propagation process can start directly using these CSPs.

This first propagation process aims at reducing the number of CSPs as soon as possible to improve the efficiency of the algorithm. If two CSPs of the above lists coincide or are co-planar, using the symmetry property of the area around a CSP, other edges and neighboring faces can be identified with the help of the adjacency relations expressed by the hypergraphs. So, this propagation process relies on a criterion of coinciding CSPs.

Now, let us study the behavior of the propagation process with respect to the various categories of CSPs. Starting with an O-CSP, the corresponding edge E1 is self symmetric. Using a neighboring face ofE1, it is possible to select a loop Γ which contains the current O-CSP and to look for another CSP coinciding with the O-CSP.

Regarding CSPs attached to entities of a loop, they are among the categories O-CSP, BS-CSP and LB-CSP. A loop having no self intersection, a property can be stated regarding the CSPs:

Property 3 CSP propagation conditions over a loop Γ: Given a loop Γ without self intersection, Γ contains n edges and lies on face F, and let us consider a first con- figuration where there exists an O-CSP attached to the edge E1. If n is even, the O-CSP must coincide with one and only one other O-CSP to able to propagate the symmetry property acrossF. Ifnis odd, the O-CSP must coincide with one and only one LB-CSP.

Let us consider also a second configuration where there exists an LB-CSP attached to the vertex V1 of Γ. Ifn is even, the LB-CSP must coincide with one and only one O-CSP in Γ to propagate the symmetry property across F. If n is odd, the LB-CSP must coincide with one and only one other LB-CSP.

First level propagation and CSP chains 185 There cannot be any symmetry plane of type BS-CSP in Γ.

Figure 6.5: No more than one O-CSP in a loop coincides with an initial O-CSP.

As shown in Figure6.5,E1,E2 andE3 belong to Γ, which has no self intersection.

Let us consider an O-CSP attached toE1 as reference CSP.E1 can be used to divide the pn1q other edges and n vertices of Γ into two subsets of similar entities rE_i¹s, rE_i²s;rV_i¹s,rV_i²s. The cardinality of each setrE_i^js,rV_i^jsis identical.

Ifnis even, CardrE_i^js ⁿ2² and the edge left, Ek, is the farthest one from E1 by adjacency; CardrV_i^js ⁿ2. These sets represent the left hand side and right hand side of Γ with respect to E1 (see Figure 6.5). Then, assuming that there exists another O-CSP inrE_i¹s, ΠO1, coinciding with the reference CSP, ifF is symmetric with respect to this reference CSP then, there exists another O-CSP in rE_i²s, ΠO2, that coincides with the reference CSP and has the same adjacency position as Π_O1. If so, the edges corresponding with ΠO1 and ΠO2 must coincide, which shows that Γ self intersects along these edges. This contradicts the initial hypothesis along which Γ has no self intersection, hence there no O-CSP in rE_i¹s coinciding with the reference one. The propagation process can take place only if E_k has an O-CSP that coincides with the reference one.

Now, if n is odd, CardrE_i^js ⁿ2; CardrV_i^js ⁿ2¹ and the vertex left, V_k, is the farthest one from V1 by adjacency. Then, assuming that there exists an LB-CSP in rV_i¹s, ΠO1, coinciding with the reference CSP, if F is symmetric with respect to this reference CSP then, there exists another LB-CSP in rV_i²s, ΠO2, that coincides with the reference CSP and has the same adjacency position as ΠO1. If so, the vertices corresponding with ΠO1 and ΠO2 must coincide, which shows that Γ self intersects at these vertices. This contradicts the initial hypothesis along which Γ has no self

intersection, hence there no LB-CSP inrV_i¹scoinciding with the reference CSP. The propagation process can take place only ifVk has an LB-CSP that coincides with the reference CSP.

These two analyses show that no LB-CSP can coincide with the reference CSP when n is even and that no O-CSP can coincide with the reference CSP when n is odd.

Following a similar reasoning process, when the reference CSP is an LB-CSP and the reference entity is a vertexV1 leads to the results stated for the second configura- tion.

Now, whatever the cardinality of Γ, let us assume thatrE_i¹scontains an edge,E_k, having a BS-CSP coinciding with a reference CSP attached to an edge or a vertex, then rE_i²s must contain an edge, El, having a BS-CSP also coinciding with the reference CSP and located at the same adjacency position as inrE_i¹s. If so, it means that E_k andE_l coincide, henceF self intersects, which contradicts also the hypothesis. Ifnis odd so that there is an edge left outsiderE_i¹sandrE_i²sif the reference CSP is attached to a vertex, this edge cannot have a BS-CSP coinciding with the reference one since these extreme vertices would lie in the reference CSP, which contradicts the symmetry property of Γ that is needed to propagate fromF to adjacent faces.

This demonstrates the above property of the propagation process of CSPs over a loop.

Property 4 CSP propagation conditions through a vertexV_i: Given a vertexV_i whose neighboring faces define an LB-CSP forming three adjacent faces where F is the ref- erence face sharing the two reference edges E1, E2 with its adjacent faces. V_i as well as its surrounding faces and edges don’t self intersect. V_i has n, n ¥3, surrounding faces altogether.

Let us consider a first configuration where n is even, then the LB-CSP must co- incide with one and only one other LB-CSP to be able to propagate the symmetry property across V_i. If n is odd, the LB-CSP must coincide with one and only one BS-CSP.

Let us consider also a second configuration where there exists a BS-CSP attached to the vertexV_i withE1 the corresponding reference edge. Ifnis even, the BS-CSP must coincide with one and only one other BS-CSP to be able to propagate the symmetry property acrossVi. Ifnis odd, the BS-CSP must coincide with one and only one other LB-CSP.

As shown in Figure 6.6b,E2 and E8 are the reference edges defining an LB-CSP atV1 together withF1. Let us consider this plane as reference CSP. F1 can be used to divide thepn2q other edges and pn3q faces at V1 into two subsets of similar entitiesrE_i¹s,rE_i²s;rF_i¹s,rF_i²s. The cardinality of each setrE_i^js,rF_i^jsis identical.

First level propagation and CSP chains 187

(a) (b) (c)

(d) (e) (f)

Figure 6.6: The propagation mechanisms for the first level propagation: (a) and (d) propagation from an O-CSP to an O-CSP; (b), (e) are O-CSP to LB-CSP, and then to BS-CSP; (c) and (f) are O-CSP to LB-CSP.

Ifnis even, CardrE_i^js ⁿ2²; CardrF_i^js ⁿ2³ and the face left,Fk, is the farthest one fromF1 by adjacency. These sets represent the left hand side and right hand side of V_i with respect to F1 (see Figure 6.6b). Then, assuming that there exists another LB-CSP in rF_i¹s, ΠLB1, coinciding with the reference CSP, if F1 is symmetric with respect to this reference CSP then, there exists another LB-CSP in rF_i²s, Π_LB2, that coincides with the reference CSP and has the same adjacency position as ΠLB1. If so, the edges defining ΠLB1 and ΠLB2 must coincide, which shows that these edges including V_i self intersect. This contradicts the initial hypothesis along which the neighborhood ofV_i is self intersection free, hence there no LB-CSP inrF_i¹scoinciding with the reference one. The propagation process can take place only if Fk has an LB-CSP that coincides with the reference one.

Now, ifn is odd, CardrE_i^js ⁿ2² and the edge left, E_k, is the farthest one from F1 by adjacency; CardrF_i^js ⁿ2³. Then, assuming that there exists an BS-CSP in rE_i¹s, ΠBS1, coinciding with the reference CSP, ifF1 is symmetric with respect to this reference CSP then, there exists another BS-CSP in rE²_is, ΠBS2, that coincides with

the reference CSP and has the same adjacency position as ΠBS1. If so, the edges corresponding with ΠBS1 and ΠBS2 must coincide, which shows that these edges including V1 self intersect. This contradicts the initial hypothesis along which theV1

neighborhood has no self intersection, hence there no BS-CSP inrE_i¹scoinciding with the reference CSP. The propagation process can take place only ifEk has a BS-CSP that coincides with the reference CSP.

Changing the reference CSP into a BS-CSP and starting over the same reasoning process the property reduced to the second configuration.

Figure6.6e and f show the hypergraphG21 reduced to the neighborhood ofV1and illustrate configurations wherenis respectively odd or even aroundV1. Consequently, a new edgeE5 can be added to the symmetric elements set in Figure6.6e and a new faceF5 is added in Figure6.6f.

The above two properties show that one step of the propagation process can extend the symmetry property to:

• A new edge and its adjacent face if a new O-CSP coincides with the reference CSP;

• two new edges and their adjacent faces that bound the face of the reference CSP when a new LB-CSP coincides with the reference CSP when the propagation is performed over a loop;

• A new face and its two adjacent edges and the two faces adjacent these new face and edges when a new LB-CSP coincides with the reference CSP when the propagation is performed across a vertex;

• A new edge and its two adjacent faces when a new BS-CSP coincides with the reference CSP.

As a result of the above synthesis, propagation rules can be set up as follows:

• If the propagation takes place over a loop, the last CSP incorporated through the propagation process is either an O-CSP or an LB-CSP, then the next one, Π, can be found only in therΠOsor therΠLBslists and the next reference entity for the propagation process is:

– A loop and, more precisely, an edge in this loop if ΠP rΠOs;

– A vertex if ΠP rΠLBs;

• If the propagation takes place across a vertex, the last CSP incorporated through the propagation process is either an LB-CSP or a BS-CSP, then the next one, Π, can be found only in the rΠLBsor the rΠBSslists and the next reference entity for the propagation process is:

First level propagation and CSP chains 189 – A loop and, more precisely, an vertex in this loop if ΠP rΠLBs;

– A vertex if ΠP rΠ_BSs.

Figure6.6a through c are illustrations of some of the previous configurations.

Now that all the possible configurations of CSP propagation under the coincidence constraint have been studied when this propagation takes place:

• Over a loop;

• Across a vertex;

and the propagation rules have been set up on the CSP category basis as well as from the type of entities ofMM AX supporting each new CSP, the concept of CSP chain is introduced when BM_{M AX} solely contains faces bounded single loops.

Definition 8 CSP Chain A CSP chain contains a sequence of coinciding CSPs se- lected from the three categories: O-CSP, LB-CSP, BS-CSP. The sequence is a chain of items where each of them contains the CSP type and the corresponding entities of BM_{M AX} where the symmetry property is extended. A CSP chain can be open or closed when the propagation process stops because there is an asymmetric configura- tion encountered or when the last CSP found is the CSP having initiated the chain, respectively.

Because the propagation process scans BM_{M AX} on the basis of faces (more pre- cisely the loop bounding each face) or vertices, the CSP Chain extension needs two different geometric functions: checking coincidence of CSPs within a face; checking coincidence of CSPs around a vertex.

Based on the property of CSPs over a loop and across a vertex, a CSP chain cannot contain an element defined by a BS-CSP that is adjacent to element defined by an O-CSP.

When a CSP chain is closed, the corresponding entities ofBMM AX that define the CSPs of this chain can be used to define a loop overBM_{M AX} that intersects with the CSP having initiated this chain.

During this propagation process, each CSP added to a CSP chain is removed from the list corresponding to its category, i.e. eitherrΠOs,rΠLBsorrΠBSs, hence reducing the number CSPs available as initiators of CSP chains. Consequently, the first level propagation process stops when there no CSP left is the lists rΠ_Os,rΠ_LBsorrΠ_BSs.

Now, considering the example model of Figure 6.7 that is presently reduced to a simple cube to meet the current constraint where faces are bounded by single loops, OCSP1 andLBCSP1 are two example CSPs used to illustrate the propagation

Figure 6.7: An example model with a subset of CSPs chosen as initiators of CSP chains.

Figure 6.8: 2 closed CSP chains of the example model at Figure6.7with their 3D and unfolded representations.