5.4 Self symmetry planes of a boundary loop
5.4.1 Symmetries defined through a Candidate Symmetry Axis (CSA)125
cone or a sphere) or a set of circles having centers lying on a straight lineLorthogonal
to their planes. In these configurations, the number of symmetry plane is infinite because the bounded faces involved are rotational symmetric. Indeed, L coincides with the reference face axis (for cylinders, cones, tori) or center (for spheres) and any plane containing L is a reflective symmetry plane and there is an infinite number of such planes. Such a configuration is denominated axisymmetry.
Among the five reference surfaces, four are surfaces of revolution. Consequently, when a plane is orthogonal to an axis of revolution of either of the four revolving surfaces or, alternatively, if two axes of revolving surfaces coincide, if the intersection curves producing a reference face form a set of full circles, these faces are axisymmet- ric. Each of their corresponding reflective symmetry planes is aCandidate Symmetry Plane. In this case, the concept of CSP is turned into the concept of Candidate Sym- metry Axis (CSA): a means to characterize the fact that an infinite number of CSPs is attached to these axisymmetric faces.
Axisymmetry is a strong property contributing to the symmetry analysis algo- rithm. In other symmetry detection algorithms [36], [35], [40], [44], [61], [62], axisym- metry is impossible to detect rigorously with point set models or mesh based ones.
They are restricted to axisymmetry approximation only.
5.4.2 O-CSP of a maximal edge through the analysis its two adjacent faces
Let us consider the maximal edgeE, the intersection curve between two adjacent reference facesF and Fa;F and Fa are maximal faces. F and Fa are simple analytic surfaces but E has no simple analytic properties, in general. F and Fa interact commutatively with each other, i.e. considering the intersection ofF with Fa, or the opposite, produces the same edge E. Hence, F can be arbitrarily taken as target surface for the current analysis. F and Fa define the smallest possible interaction producing an intersection curve E. In a general configuration of a B-Rep model, intersection curves between faces are bounded by other surrounding faces. The effect of these faces will be addressed later at section5.4.4.
Symmetry properties ofE take place at locations defined with curvature and tor- sion extreme and reflective symmetry planes are contained in the Frenet reference plane defined by the normal and bi-normal vectors at these extreme. As a result, these symmetry planes can be said ‘orthogonal’ to E, hence the designation of O- CSP, which the main focus of this section.
The symmetry properties of the maximal curve E can be either addressed as a stand alone entity or characterized by the symmetry properties of its adjacent facesF andFa. On the one hand, ifEis analyzed as a stand alone entity, one of the purposes of this analysis is to detect E self reflective symmetry planes. However, E has no simple equation and must be discretized to extract some symmetry properties, which
Self symmetry planes of a boundary loop 127 requires discretization parameters and will be less robust than analytical treatments.
In addition, studying E as a stand alone entity does not take into account the sym- metry constraints deriving fromF and Fa (see the beginning of section5.4). Another limitation in studyingE on its own holds in the fact that under specific locations ofF andFa,E becomes a planar curve and can be the locus of a bisector plane betweenF andFa: such a global property would be difficult to extract through a digitized repre- sentation ofE. On the other hand, if the symmetry properties ofE are derived from that of F and Fa and their relative position, the resulting properties are compatible withE and F,Fa both, including the identification of bisector planes.
Indeed, all the loops bounding F result from an intersection between F and Fa. Here, only one loop is analyzed since the focus is set on single intersection curves.
Finally, studying the different categories of couplespF, Faq help defining the sym- metry properties of the intersection curves E forming BF. This, leads to a combi- natorial study, as a first approach, and the commutative interaction between F and Fa reduces the combinations studied to those listed in Table 5.2. In a first place, this combinatorial study is reduced to O-CSPs only; bisector symmetry planes will be studied in detail at section 5.6.
F / Fa Plane Cylinder Cone Sphere Torus
Plane 1:pP1, P2q 2:pP, Cyq 3:pP, Coq 4:pP, Spq 5:pP, T oq Cyliner 6:pCy, Cyq 7:pCy, Coq 8:pCy, Spq 9:pCy, T oq Cone 10:pCo, Coq 11:pCo, Spq 12:pCo, T oq
Sphere 13:pSp, Spq 14:pSp, T oq
Torus 15:pT o, T oq
Table 5.2: Combinations of two reference surfaces.
Unless stated otherwise, symmetry planes are valid for any intrinsic parameter of F (radius, angle,. . . ) and Fa.
Symmetry planes of pF, Faq intersections
Plane/PlanepP1, P2q: Because of the existence of an intersection curveE,P1and P2 are not parallel to each other. P1 andP2 are bounded byE only, their intersection ΓPP reduces to a unique straight line. Since P1 (F) and P2 (Fa) belong to MM AX, ΓPP is necessarily bounded to form E. There exists always a symmetry plane Π1, orthogonal to P1 and P2 and located at the midpoint of the extreme points ofE (see Figure 5.11).
Plane/Cylinder pP, Cyq: Configurations with P and Cy subdivide into three categories (see Table 5.3):
• P is orthogonal to the axis ofCy,Ac (see Figure 5.12a). There, Π1 coinciding with P could be a symmetry plane, even if P and Cy are infinite, but the
Figure 5.11: Symmetry plane derived from the intersection between two planes.
pF, Faq pP, Cyq pP, Cyq pP, Cyq Geometric constraint PKAc P Ac P notKAc,
P notAc Π 8: axisymmetry Π1: AcKΠ1 Π1:AcΠ1
Table 5.3: Configurations of symmetry planes for P and Cy.
neighborhood of ΓPCy must be topologically equivalent to three half disks at least. This configuration is non-manifold and cannot be included in a CAD volume. Hence, the figure represents only the relation between the two surfaces:
when Cy is effectively bounded by P at E. In this case, E is a circle and it is an edge without vertex to express axisymmetry. Ac is the symmetry axis.
Axisymmetry is highlighted on Figure 5.12a with Π8;
• Now, if P is parallel to Ac, Figure 5.12b refers to only one intersection curve, as in Table 5.3 to conform to the content of the hypergraphs describing the object boundary, i.e. every edge in G21 and G10 can be associated with an O-CSP. Indeed, extendingP, bounded byE, up to infinity can produce another intersection line (see Figure 5.12d). Consequently, another symmetry plane Π2
appears. Indeed, Π2 is not missing, it belongs to the loop symmetry CSP cate- gory introduced at section 5.3.2that will be detailed later on;
• The most general configuration ofP generatesE as an ellipse (see Figure5.12c).
Only one O-CSP is valid for this configuration.
These configurations are summarized in Table5.3where the first linepF, Faqdesig- nates the type of face, then lineGeometric constraint expresses the geometric location
Self symmetry planes of a boundary loop 129
(a) (b) (c) (d)
Figure 5.12: (a) P is orthogonal to Cy axisAc; (b) P is parallel toAc; (c) P has an arbitrary orientation with respect to Cydiffering from (a) and (b); (d) ifP is infinite it produces a two segments configuration.
ofFawith respect to F. Π states the maximum number and relative position of sym- metry planes. An illustration of each configuration is given in Figure5.12.
The enumeration of all the possible intersections only considers a full intersection curve. The original surfaces are infinite, though bounded byE. In Figure 5.12a,E is a full circle but to conform to the content of the hypergraphs,E may be bounded by vertices. Likewise the intersection between two planes, the location of these vertices will be taken into account to eliminate some of the CSPs as described at section5.4.4.
(a) (b)
Figure 5.13: (a) P is orthogonal toCoaxis, (b)P is not parallel to Co axis.
pF, Faq pP, Coq pP, Coq Geometric constraint PKAc P notKAc,
Π 8: axisymmetry Π1: AcΠ1
Table 5.4: Configurations of symmetry planes for P and Co.
Plane/Cone pP, Coq: Compared to Cy, the symmetry planes intrinsically at- tached toComisses symmetry planes orthogonal to its axis Ac when P goes through
Coapex (see Figure 5.13 and Table 5.4), which explains the difference compared to Cy. If P goes through Co apex, each straight line segment forming the intersection contributes to a unique maximal edge E. The corresponding symmetry plane is still of type O-CSP and can be seen as the limit configuration of P parallel to Co axis.
Indeed, it is not attached to a vertex (Coapex) but it is not lost.
The thirteen combinatorial cases left are illustrated in Appendix A for sake of conciseness.
The tables and figures above synthesize all the configurations of the intersection betweenpF, Faq. However, they only address the complete intersection curves obtained with infinite surfaces (planes, cylinders, cones), full spheres or full tori. In this case, most intersection curves are closed and form one or more loops. Section 5.4.4 will take care of the reduction of intersection curves and the corresponding effect over the existence of CSPs.