Planar Development of Free-Form Surfaces: Quality Evaluation and Visual Inspection

Phillip N. Azariadis   and   Nickolas S. Sapidis                 

Department of Product & Systems Design Engineering

University of the Aegean

Ermoupolis, Syros, 84100 Greece



1          Test cases


1.1       Color Maps

All local, geometric accuracy-indices (Section 3.3.1) are presented to the user through appropriate color maps. More specifically, application of a particular quality criterion commences with calculation of the corresponding accuracy index at each vertex of the planar development through averaging of values produced by all triangles sharing this vertex. This value is then  projected onto the  interval and, using an appropriate color model, it is eventually associated to a color value expressed as a triple of Red, Green and Blue components defined on . In the present implementation, the mapping  is accomplished using a 1st degree B-spline with control points (0,0,1), (0,1,1), (0,1,0), (1,1,0), (1,0,0), (1,0,1) and with a chord-length parameterization. Thus, the minimum value (zero) corresponds to pure Blue, while the maximum (one) to Magenta. All examples, presented in this section, are based on this color-map model.


1.2       Examples

The quality-control criteria and visualization methods, introduced in the previous sections of this paper, are applied to planar developments of varying quality to confirm/demonstrate their effectiveness in identifying inaccuracies. All numerical results related to the present examples are assembled in a single table (Table 1) to facilitate comparisons. Color images are presented separately for each example.

The first example is the classical surface used for benchmarking flattening methods: a torus part (Fig. 3). We shall perform quality control on two planar developments of this surface derived using the method [19]. In order to illustrate the effectiveness of the proposed tools in detecting accuracy problems, we produce (a) a suboptimal development by forcing the method to terminate prematurely, and (b) a high-quality development using the method appropriately. The resulting planar developments are shown in Fig.4a and Fig.4b respectively.

Fig.3. Part of a torus surface.



For the first planar development (Fig.4a): Since the local isometry index  takes values up to 0.76113, meaning that there exist triangles distorted up to 76%, it is evident that this planar development is not accurate. However, distortion is dispread within this planar development quite uniformly, except for the two areas in Fig. 5 colored in orange/red where  reaches local maxima, i.e., there is a high concentration of deformation. Figure 6 depicts a color map of the local aspect ratio index  (Eq. (19)) together with the nodal forces (Section 4). It is obvious that significant distortion exists near the upper-right corner of the planar development, where also excessive stretching forces appear. In general, high   appears in areas where shear stress exists, as it is shown at the upper-right corner of Fig. 6(right). Conclusion: All criteria agree that the development in Fig. 4(a) is not accurate and thus inappropriate for applications. To confirm this, we use this planar development to map a texture pattern onto the corresponding toroidal surface using the method [2]. The texture pattern is monochrome and consists of uniformly distributed black circular-disks. The texture mapped image (Fig. 7) identifies vividly the parts of the surface corresponding to distorted areas of the planar development. Indeed, in the heavily distorted areas where  is high, the circles of the pattern are also distorted into elliptical shapes. It is interesting to note that the orientation of the ‘‘ellipses’’ agrees with that of the calculated nodal forces.


                                      a                                                                                 b

Fig.4. Planar developments of the surface in Fig. 3 using the method [19] : (a) suboptimal development, (b)  high-quality development.


Fig.5. Color map of the  index.

Fig.6. Color map of the  index.


Fig.7. Texture mapped image on the torus part using the planar development of Fig.5.



Applying the same quality-control methods on the second planar development (Fig. 4b) reveals its high quality/accuracy, indicated, e.g., by all indices in Table 1. Figure 8 agrees with this observation as the maximum  value is significantly lower than that of Fig. 6. Furthermore, nodal forces along vertical boundaries appear almost symmetrical as one would expect based on the symmetry of the toroidal surface. However, there is an area near the middle of the horizontal boundaries where  is quite high, indicating a deformation of the aspect ratio. This deformation is confirmed by the texture mapped image (Fig. 9), generated using this second development. Indeed, in Fig. 9 the result is quite satisfactory everywhere except for the indicated part corresponding to the area discussed above.


Fig.8. Color map of the  index.

Fig.9. Texture mapped image of the torus part using the second planar development.



As a second test case we use an example from the footwear industry: flattening of the surface of a last (Fig. 10). We flatten the last using the global optimization method under constrains [3] making sure that the method terminates prematurely, so that the result is not accurate; see Fig. 10. The numerical indicators in Table 1 agree with that; e.g.,  indicates up to 58% distortion of triangle-edges. Furthermore, the color maps in Figs. 11 and 12 also identify inaccuracies at the forepart and at the perimeter of the surface, respectively. These results are empirically confirmed through the texture mapped image of the last illustrated in Fig. 12, where significant deformation of the pattern is shown at the forepart.



Fig.10. Surface of a last and the corresponding planar development.

Fig.11. Color map of the  index.


Fig.12. Color map of the  index.


Fig.13. Texture mapped image last surface.



The third example tests our proposals on a highly complex surface, the human-head model in Fig. 14. The corresponding planar development is derived using the method [13]. All indices in Table 1 indicate significant inaccuracies in this development. The problematic areas are highlighted by the color maps of Figs. 15 and 16: The former points out inaccuracies at the area of the nose, while the latter indicates non-uniform deformations near the nose and in the neck area. These observations are empirically confirmed by texture mapping; see the two marked areas in Fig. 17.



Fig.14. Human-head surface and the corresponding planar development.


Fig.15. Color map of the  quality index.

 Fig.16. Color map of the  quality index.


Fig.17. Texture mapped image on the human head with areas of maximum deformation colored yellow.


The final example is the revolved surface in Fig. 18. All numerical indices in Table 1 agree that this is an accurate development. The color map of  indicates a relatively uniform distribution of the distortion, while the  color map points out two areas with a larger deformation of the aspect ratio; see Fig. 19 and 20. Figure 19 shows also the normalized force-vectors derived according to Section 4, while Fig. 20 shows them in a non-normalized form. In the latter figure we note that shear stresses exist in the vicinity of the areas with the largest deformations. Again, we observe that there is a close relation between the aspect ratio  and the nodal forces as it was also observed in the first example of this section. The above observations are also confirmed by the texture map in Fig. 21, which indicates minor problems in the aforementioned areas.



Fig.18. A revolved surface and its planar development.


Fig.19. Color map of the  quality index.


Fig.20. Color map of the  quality index.


Fig.21. Texture mapped image on the revolved surface.



Table 1. Quality control results for the presented test cases


Global homogeneity of distortion h

Global aspect ratio r













Torus (Fig.4a)









Torus (Fig.4b)



























Revolved Surface











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