Symmetry detection applied to B-Rep CAD models

The symmetry detection of B-Rep CAD models has also been addressed by re- searchers. Early on, Davis used candidate axes to detect symmetry properties of 2D polygons [14]. Considering two adjacent segments, an axis is their angle bisector or, considering one segment, an axis is orthogonal to this segment and located at its middle point. In case of two separated segments, there are four possible axes. The candidate symmetry axes are defined hierarchically and different symmetry axes levels contain different parts of the polygon input. This approach has been also extended to continuous curves using an average middle line so it reduces to a polygonal approach, too. Parui detected symmetry of 2D polygons [49]. The method was considering 2D loops as polygons. The location of segment extremities are the reference of symmetry axes. This work however is too basic to address the shape diversity of engineering components.

Parray-Barwick and Bowyer present a Woodwark’s Algorithm to recognize features which supports a multi-dimensional set theoretic context [47][48]. The key point is a template matching process. A partial shape or an existing shape is considered as template and then using its center of gravity, length of contour or other properties, the algorithm recognizes if the target object is the same or not. If a rotational parameter is added, this approach could detect symmetries. Woodwark’s algorithm is devoted

Figure 1.24: A one-dimensional model matching process inpx, yqplane [47][48]. This illustrates the principle of the matching mechanism where a dimension of the template intersects (matches) a range of a parameter of the input model.

to feature recognition when there is a template already available (see Figure 1.24).

In a shape analysis context, how to define a template and the matching parameters are among the major issues. The method scans every dimension defining the shape (see Figure 1.24). With an increase in the number of dimensions, the search space for the matching process increases too. Consequently, the matching process will take much more time. Indeed, reducing the extent of the area where could be located a symmetry axis (plane) is a general problem for all symmetry detection methods. As an example, Martin summarized that for a whole object, to find an axis, the object has to be translated so that its centroid is at the origin because the symmetry axes pass through the origin [38].

A different approach was proposed by Kulkarni [32]. The method reduced the 3D shape analyzed to its 2D skeleton with the help of the Medial Axis Transform. The medial axis of a 2D (3D) object is the locus of the centers of all maximal inscribed circles (spheres) [20]. Kulkarni’s work addressed 2D objects only. It is a complex task to extent to 3D because the calculation of a skeleton is expensive and it is obtained through a discretization of the object boundary, which may influence the result. In addition, a large disruption in the skeleton can happen when the object boundary changes a little. Therefore, symmetric elements of the object are not necessarily represented in the skeleton [58].

Symmetry detection methods 33 The algorithm reported by Tate is based on a B-Rep CAD model input [58][60].

Here, the boundary surface of the model is limited to five categories: plane, cylinder, cone, sphere, and torus. The symmetry detection process is broken down into five steps:

• Compute loop properties;

• Identify matching loops;

• Construct axes and planes of symmetry;

• Rationalize axes and planes of symmetry;

• Extract the primary symmetry axes from the resulting set.

The loop properties used are: surface type, loop type, loop area, loop centroid, surface normal at the loop centroid and the number of edges in the loop. If all the properties of two loops are identical, the two loops with their underlying surfaces are considered as congruent. The candidate symmetry planes and axes are located from the pair of loops centroids. By comparing the locations of candidate symmetry planes and axes, finally, the primary symmetry planes or axes can be extracted. The computational complexity of this algorithm, in the worst case scenario, isOpn⁴q, where nis the number of loops of the model. It has also to be pointed out that this algorithm requires rather complex treatments such as computing loop area and loop centroid, which increases the computational effort.

With some model shapes, the method can detect the symmetry properties. But as summarized by the author, this approach faces four limitations. The first one relates to those faces where the intersecting features have some asymmetric details that may be ignored. This problem can originate from the object boundary decomposition because of the influence of the modeling process (see Chapter 4). The second one is that there are configurations where the two properties of two loops are identical but these loops are not congruent, which is illustrated in Figure 1.25. The third one relates to loops subjected to rotations. Rotated loops have the same properties, the corresponding asymmetry created cannot be recognized. The last limitation relates to the nature of features in terms of protrusions and depressions. Also, within this work, the method uses Djinn solid modeling Application Procedural Interface (API) and the ACIS Solid Modeler from Spatial Technology to perform the extraction of loop properties. This environment raises several questions. One is how to monitor the accuracy of computations, which is limited by ACIS. The second one is that it needs to assign the status of internal or external as type to a loop but the loop type definition is ambiguous, hence not robust. The third one relates to the decomposition of surfaces and curves. Not all the surfaces of the B-Rep are maximal. In fact, it is a

Figure 1.25: Some of the limitations of Tate’s algorithm [60]: (a) asymmetry of a detail producing the same global parameters; (b) different loops with same properties;

(c) rotation problem; (d) no distinction between protrusion and depression.

general problem, because a B-Rep model always describes surfaces of revolution with two or more pieces, which breaks the object symmetry (see Chapter4).

Compared to the other approaches, Tate’s method is the one that uses effectively a B-Rep model, as an infinite point set model. As discussed in Section 1.1, B-Rep models are used in a design process as basic digital models and widely subjected to shape transformation processes. Therefore, using a B-Rep model that directly matches exactly its boundary, avoids problems of referring to its faceted representation. It is a significant advantage compared to other approaches.