Symmetry constraints derived from loops over spherical faces . 121

sphere O_s. If BS_p is the boundary of S_p and Γi P t1, . . . , nLu its loops, all the areas bounded by Γi are finite and so are their complement, i.e. AreapSpq AreapSp|Γ_iq. This property shows that the configurations 1 and 2 set for cylinders and cones are no longer applicable here since the loop status over S_p does not refer to the concept of external/internal. Differing from cylinders and cones, a sphere has no particular direction for symmetry planes. Hence, symmetry planes overS_p rely more strongly on the symmetry properties ofBS_p, which can be characterized from symmetry and loca- tion properties of facesFa adjacent toSp. These properties are detailed in section5.4 where the influence of references surfaces will be detailed.

5.3.6 Synthesis about loops and loop symmetry CSP

From the analysis of symmetry constraints related to face boundaries (see sec- tion 5.3), the importance of loops and the diversity of CSPs ΠO,ΠLB,ΠLS becomes clear in a surface symmetry analysis process. Whether a face F has an external loop or not, the number of loops and their relative position over F, all these informations are key elements influencing the symmetry properties of F and give bounds on the maximum number of symmetry planes depending on loop configurations.

Each loop Γi of F has its own symmetry plane set, namely self symmetry planes:

rΠ^slf_Γ

i s. These symmetry planes cross Γi, which indicates that they can be either of type orthogonal rΠ_Os or loop bisector rΠ_LBs only. Now, when considering different loops Γi,Γj PF, their common self symmetry planes may come from the intersection betweenrΠ^slf_Γ

i sandrΠ^slf_Γ

js. Because loops overF must be either disconnected or touch each other at isolated points only to stay consistent with B-Rep model fundamentals, rΠ^slfΓ_i sXrΠ^slfΓ_jsmay not produce all the symmetry planes set by the constraints ΓiYΓj. The relative position of Γi and Γj, as two disconnected infinite point sets, may also satisfy the symmetry condition5.1whereF1 andF2 are substituted by Γi and Γj. Consequently, the corresponding symmetry plane Π is calledloop symmetrycandidate symmetry plane and noted ΠLS. Indeed, ΠLS being derived from loops Γi and Γj, it can be noted more precisely ΠLS,Γ_i,j. Regarding their relative position in M_{M I},

Γi and Γj cannot touch each other more than once, otherwise they define a face F_k, homologous toF, lying in between the facesFi andFj that could be associated to Γi

and Γ_j respectively. Each contact point being surrounded by more than three faces F_i,F_j,F_kandF, and no more than two loops in contact can be attached to two faces (see section 5.4.2), e.g. Fi Fj Fij, then Fij crosses Fk (Fk F). Hence, other loops (faces) touchingF_i or F_j or F_k are not homologous with these ones, therefore these faces are not crossingF andF_ij and the manifold vertex splitting operator will duplicate these contact points but the edge merging one will remove them, leaving no more than one contact point representing the crossing configuration inM_{M AX}. The CSP attached to that point is necessarily of type ΠLB not ΠLS. All the contact points associated to non crossing configurations will be removed and represented by ΠLS not Π_LB.

This analysis shows that ΠLS are strictly attached to loop entities and, hence, disconnected from the hypergraphs Gij, which contain vertices and maximal edges only. This justifies the existence of loop datastructures derived from the hypergraphs to complement the faces, edges and vertices.

Let rΓ^exs, rΓⁱⁿs and rΓ^uks be the sets of loops representing the external loops, internal loopsandloops of unknown statusofF, respectively. The loops Γi, Γj and Γk

belong to either set rΓ^exs orrΓⁱⁿs orrΓ^uks without influencing the existence of ΠLS. However, ΠLS cannot be defined by loops Γi and Γjbelonging to either set: Γi P rΓ^exs, Γj P rΓⁱⁿs, for example. Consequently to the above analysis of connected/disconnected loops, one pair of loops only has a unique ΠLS plane. Because Π^LS_Γ

pi,jq is not crossing any other loop, Π^LS_Γ

pi,jq R rΠ^slf_Γ

i XΠ^slf_Γ

js. Therefore, the set of symmetry planes of Γi

and Γ_j satisfy:

rΠΓ_pi,jqs prΠ^slf_Γ

i s X rΠ^slf_Γ

jsq YΠ^LSΓ_pi,jq. (5.2) In case a faceF is bounded by external loops rΓ^exsas well as internal ones rΓⁱⁿs, because rΓ^exs can be said as the mandatory limitation of F, i.e. the loops that avoid havingF unbounded, and they are distinct from rΓⁱⁿs, whatever the symmetry properties ofrΓⁱⁿs, ifrΓ^exsis not symmetric,F is not symmetric. As a result,rΠ_BFs € rΠΓ^exs. Now, consideringrΓⁱⁿs, the symmetry planes ofF must be contained inrΠΓⁱⁿs: rΠ_BFs € rΠΓⁱⁿs. Finally, we have:

rΠ_BFs rΠΓ^exs X rΠΓⁱⁿs. (5.3) To unify eq. (5.3), when Γ^ex does not exist, rΠ^Γexs means all possible symmetry planes ofF inIR³. Similarly, if Γⁱⁿdoes not exist,rΠΓⁱⁿsmeans all possible symmetry planes of F in IR³, too. In these cases, symmetry plane sets are infinite. When F contains loops of unknown status, it means thatrΠ^Γexsdoes not exist. Indeed, rΓ^uks

Symmetry constraints originated from face boundary loops 123 behaves likerΠΓⁱⁿsand can be included in it.

As an example, regarding cylinders, their maximal number of external loops is two. Then, with the eq. (5.2), rΠ^Γexsis expressed as:

rΠΓ^exs rΠΓ^ex

p1,2qs prΠ^slf_Γex

1 s X rΠ^slf_Γex

2 sq YΠ^LSΓ^ex

p1,2q. (5.4) If the number of internal loops is smaller than two, eq. (5.4) has to be filtered with the set of symmetry planes for the internal loop but there is no Π_LS plane involved in rΠΓⁱⁿs. However, if the number of internal loops is greater than or equal to two, there is no criterion for filtering the symmetry planes. rΠΓⁱⁿs can be bounded by the union of the symmetry properties of each pair of internal loops. As a result, eq. (5.4) changes to:

rΠΓⁱⁿs € ¤

i1...n j1...n i j

rΠΓⁱⁿ

pi,jqs. (5.5)

Finally, for a surface having two external loops and several internal loops, the symmetry planes combine as follows:

rΠ_BFs € prΠΓ^exs X ¤

i1...n j1...n i j

rΠΓⁱⁿ

pi,jqs

q, (5.6)

rΠ_BFs € r

prΠ^slf_Γex

1 s X rΠ^slf_Γex

2 sq YΠ^LSΓ^ex

p1,2q

X ¤

i1...n j1...n i j

prΠ^slf_Γin

i s X rΠ^slf_Γin

j sq YΠ^LS_Γin pi,jq

s. (5.7)

The boundary loops ofF are strong symmetry constraints. There are two different boundary loop configurations: external ones Γ^ex and internal ones Γⁱⁿ. Reference surfaces having at least one external loop, such as planes, cylinders and cones, the CSPs ofF are constrained by the external loops first. If the number of internal loops is less than two, symmetry planes are defined by the intersection between external and internal symmetry plane groups only, i.e. there is no ΠLS plane. Otherwise, the symmetry planes are defined by the intersection between external loops symmetry planes and the union of internal loops symmetry planes including the ΠLS planes belonging to both sets. If F has no external loop, the internal loops are the only constraints.

Right now, the symmetry planes associated with loops stand for symmetry infor- mation attached to a face boundary as well as neighboring faces adjacent to edges and vertices of the loop. As a result, it incorporates symmetry properties of these faces:

ΠO and ΠLB symmetry planes. For example in Figure 5.9, a boundary loop of the cylinder is a planar curve and it is an ellipse. The ellipse has two symmetry planes orthogonal to the plane containing this curve. Back to the symmetry properties of a cylinder, valid symmetry planes either pass through the axis or are orthogonal to it. In this case, the symmetry plane of the ellipse coinciding with its minor axis is invalid. Consequently, the combination of symmetry constraints of faces with loop structures reduces the number of CSPs compared to independent processing of loops and surfaces.

Figure 5.9: Combination of face and its boundary loop symmetry properties.