3.2.1. Extruded Shape
First, we present the results and discuss the variations in the coefficient of determination concerning the number of segmented surfaces.
Figure 6 illustrates the relationship between the proposed index and the sensory evaluations of the order for different numbers of bins (integers from 2 to 20) for the extruded shapes. In this study, the center of gravity of the shape was set at the origin of the
-space, and the shape was oriented such that the extrusion direction was aligned with the
-axis. We computed seven proposed indices,
,
,
,
,
,
, and
, where the subscripts
,
, and
mean
-plane,
-plane, and
-plane, respectively. From
Figure 6, it can be observed that the maximum coefficient of determination with the sensory evaluation values of order is achieved when the number of bins is 20, and the proposed index
is computed, resulting in a coefficient of determination of 0.36. Furthermore,
Figure 6 shows that considering the proposed indices related to symmetry in the
-plane (
,
, and
) and not considering them (
,
, and
) produces similar results. The extruded shapes exhibited complete mirror symmetry with respect to planes orthogonal to the extrusion direction (
-plane). Therefore, the
values were nearly zero for all shapes and had little impact on the index. Furthermore,
Figure 6 shows that the coefficient of determination with the sensory evaluation values of order is greater compared to the proposed index
considering mirror symmetry about the
-plane and
-planes compared to the proposed index
or
considering mirror symmetry about the
-plane or
-planes. This could be because humans evaluate the mirror symmetry of shapes from multiple angles and not just from a single perspective [
38,
39,
40,
54]. Therefore, considering mirror symmetry from multiple facets during segmentation likely contributed to the increased coefficient of determination between the proposed indices and the sensory evaluation values of the order.
Figure 6.
Relationship between the number of bins and coefficient of determination in extruded shapes.
Figure 6.
Relationship between the number of bins and coefficient of determination in extruded shapes.
Next, we examined the proposed index
, which exhibited the highest coefficient of determination among the seven proposed indices. We examined the reasons for the coefficient of determination’s variations with bin count, particularly for different bin counts.
Figure 6 indicates that the coefficient of determination for
is the smallest when the number of bins is four.
Figure 7 shows a scatter plot of the proposed index and sensory evaluation values of the order for this case. The red-colored plots (Shapes A, B, and C) are discussed below. This figure shows that the proposed index takes values close to zero for almost all the shapes, indicating that the index values vary little between the shapes. To investigate further, we consider multiple shapes (A, B, and C) as examples and provide contour plots of their curvature as well as histograms of the curvature of each segment when the shape is divided into two, as shown in
Figure 8,
Figure 9 and
Figure 10. These figures include contour plots of curvature from four viewpoints (top-down perspective and perspectives along the
-axis,
-axis, and
-axis) and histograms of curvature for each part when segmented along the
-plane and
-plane. These figures show that in bin 4, all curvatures of the side portions, except the curvature of the edges, are assigned to the same bins, and there is almost no difference in the curvature distribution in each part of the divided geometry. Consequently, the values of the proposed index increased (became closer to 0) for almost all shapes, and the differences in index values between shapes diminished, leading to a smaller coefficient of determination between the proposed index and the sensory evaluation values of order. Furthermore, we examined the contour plots and histograms of curvature for bins 2, 3, and 5, which showed similar behavior to that observed for bin 4. When the number of bins is small, the curvatures assigned to them become biased, causing the correlation coefficient of determination to become unstable and fluctuate significantly. In contrast, for bin numbers 6 and above, the coefficient of determination remained consistently high, suggesting that six or more bins were necessary in this experiment. Even with six or more bins, there is still a slight variation in the correlation coefficient of determination between even and odd numbers of bins. This may be because when the number of bins is even, the values around the most common curvature of 0 are divided into two bins, leading to an overestimation of the index and a decrease in the correlation coefficient of determination.
Figure 7.
Relationship between the proposed index and sensory evaluation values of ‘order’ in bin number 4.
Figure 7.
Relationship between the proposed index and sensory evaluation values of ‘order’ in bin number 4.
Figure 8.
Contour plots and curvature distributions in Shape A: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 8.
Contour plots and curvature distributions in Shape A: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 9.
Contour plots and curvature distributions in Shape B: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 9.
Contour plots and curvature distributions in Shape B: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 10.
Contour plots and curvature distributions in Shape C: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 10.
Contour plots and curvature distributions in Shape C: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 11a,b shows the scatter plots of the proposed index and sensory evaluation values for the cases with the lowest coefficient of determination at bin number 15 and the highest coefficient of determination at bin number 20. The blue-colored plots (Shapes B and D) are discussed below. The two scatter plots show that as the bin number changed from 15 to 20, the values of the proposed index for Shapes B and D became smaller than those of the other shapes, approaching the regression line. This suggests that the coefficient of determination increases significantly at bin number 20. The contour plots of the curvature and histograms of the curvature distributions for Shapes B and D are shown in
Figure 12,
Figure 13,
Figure 14 and
Figure 15.
Figure 12 and
Figure 14 show contour plots and histograms in bin 15 for Shapes B and D, respectively. In addition,
Figure 13 and
Figure 15 show contour plots and histograms at bin 20 for Shapes B and D, respectively. These figures reveal that at bin 20, the large curvature values (red regions in the contour plots) found in the side cavities, which were indistinguishable at bin 15, could now be distinguished from the other side curvatures. This is confirmed by the fact that in the histograms of
Figure 12 and
Figure 13, when the number of bins changes from 15 to 20, a new distribution appears in the fourth bin from the left, and in the histograms of
Figure 14 and
Figure 15, when the number of bins changes from 15 to 20, a new distribution appears in the third bin from the left. Consequently, the differences in the curvature distributions for each part of the shapes divided on the
-plane become significant in bin 20, causing the proposed index values to become smaller than those at bin 15 (i.e., the proposed index values become closer to the sensory evaluation values); therefore, the coefficient of determination increases at bin 20.
Figure 11.
Relationship between the proposed index and sensory evaluation values of ‘order’ in (a) bin number 15 and (b) bin number 20.
Figure 11.
Relationship between the proposed index and sensory evaluation values of ‘order’ in (a) bin number 15 and (b) bin number 20.
Figure 12.
Contour plots and curvature distributions in Shape B in bin number 15: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 12.
Contour plots and curvature distributions in Shape B in bin number 15: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 13.
Contour plots and curvature distributions in Shape B in bin number 20: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 13.
Contour plots and curvature distributions in Shape B in bin number 20: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 14.
Contour plots and curvature distributions in Shape D in bin number 15: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 14.
Contour plots and curvature distributions in Shape D in bin number 15: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 15.
Contour plots and curvature distributions in Shape D in bin number 20: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 15.
Contour plots and curvature distributions in Shape D in bin number 20: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Finally, we discuss the proposed index
(bin number 20), which exhibited the highest coefficient of determination with the order sensory evaluation value.
Figure 16 shows a scatter plot of
(bin number 20) and the order of the sensory evaluation value. The yellow-colored plots (Shapes B, C, D, E, F, G, and H) are discussed below.
Figure 16 shows that the coefficient of determination was 0.36, indicating a moderate level of correlation between
(bin number 20) and the order of sensory evaluation values in the case of extruded shapes. Now, we examine shapes for which the proposed index and sensory evaluation values differ significantly.
Figure 16.
Relationship between the proposed index and sensory evaluation values of ‘order’ in bin number 20.
Figure 16.
Relationship between the proposed index and sensory evaluation values of ‘order’ in bin number 20.
- ▪
Shapes for which the proposed index overestimates order
In
Figure 16, it is evident that the proposed index tends to overestimate the order for shapes resembling Shapes B and D.
Figure 17 displays the histogram of the shape’s curvature of each part for Shape B as well as the contour plots of the shape’s curvature. Similarly, for Shapes E and D, the contour plots of the shape’s curvature and the histogram of the shape’s curvature are shown in
Figure 18 and
Figure 19. The shape is viewed from multiple angles, as shown in
Figure 20.
First, we consider Shape B.
Figure 17 shows that Shape B is the only sample shape that lacks mirror symmetry in the plane parallel to the extrusion direction.
Figure 16 shows that it received the lowest-order sensory evaluation values. However, as observed from the contour maps and histograms in
Figure 17, irrespective of whether the shape is divided along the
-plane or
-plane, there is little difference in the curvature distribution. Therefore, the value of the proposed index is probably close to the average (rather than the minimum). For comparison,
Figure 18 displays the contour maps and histograms of the curvature for Shape E, where the index value is at its lowest. It is evident from
Figure 17 and
Figure 18 that there is no discernible difference between the curvature distribution of each part in Shape B when we compare the histogram of Shape E with that of Shape B, where the difference in curvature distribution is noticeably larger and the index value is the smallest when divided in the
-plane. Furthermore,
Figure 17 shows that although Shape B does not have mirror symmetry with respect to the plane parallel to the extrusion direction, it does not have sharp convex areas, and almost all the existing convex areas have moderately large curvatures (pink areas in the contour plots) that are distributed in the same area. Therefore, when the shape is divided into the
- and
-planes, the large curvature (the red portion in the contour plots) that is present in the concave portion on the side appears only on one side, but the magnitude of the curvature in the other portions is not significantly different. This led us to believe that the shape did not exhibit a large difference in curvature distribution. Therefore, the proposed index overestimates the order.
Shape D, shown in
Figure 19, has sharp convexity in only one part of the shape, and it has the second-lowest sensory evaluation value of order, as shown in
Figure 16. However, as can be seen from the contour plots and histograms in
Figure 19, there is no significant difference in the distribution of the curvature in each of the divided sections, suggesting that the value of the proposed index is close to the average (rather than close to the minimum).
Figure 21 shows how the shape is viewed from multiple angles for Shape F, which has the same level of value for the proposed index as Shape D and is on the regression line as a comparison object to illustrate why the sensory evaluation value of the order for Shape D is low. In
Figure 20 and
Figure 21, the two shapes are similar in that only one part of the shape has a sharp convexity. However, when Shape D was presented to the subject in the experiment, it was challenging for the subject to recognize the mirror symmetry of the shape when the shape was viewed from the angle shown in
Figure 22. The subject judged the shape to have no mirror symmetry about the plane parallel to the extrusion direction and evaluated the order instead. The reason why Shape F does not exhibit this phenomenon is that Shapes D and F have similar shapes, but Shape F has a sharper and more pointed shape. When the shape is presented in the experiment, the subject may judge it to have a shape that does not have mirror symmetry with respect to a plane parallel to the extrusion direction. This may have facilitated the recognition of the mirror symmetry with respect to the plane parallel to the extrusion direction when the shapes were presented in the experiment (
Figure 21). Therefore, we believe that the subjects underestimated the order of Shape D and overestimated the order of the proposed index.
Figure 17.
Contour plots and curvature distributions in Shape B: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 17.
Contour plots and curvature distributions in Shape B: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 18.
Contour plots and curvature distributions in Shape E: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 18.
Contour plots and curvature distributions in Shape E: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 19.
Contour plots and curvature distributions in Shape D: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 19.
Contour plots and curvature distributions in Shape D: (a) contour plots from different perspectives; (b) curvature distribution of each part (blue and orange) obtained for the -plane; (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 20.
Appearance of Shape D when viewed from multiple perspectives.
Figure 20.
Appearance of Shape D when viewed from multiple perspectives.
Figure 21.
Appearance of Shape F when viewed from multiple perspectives.
Figure 21.
Appearance of Shape F when viewed from multiple perspectives.
Figure 22.
Perspective from which Shape D appears irregular.
Figure 22.
Perspective from which Shape D appears irregular.
- ▪
Shapes for which the proposed index underestimates order
Figure 16 shows that the proposed index underestimated the order of Shapes C, E, G, and H.
Figure 23 shows that these shapes share common characteristics, such as familiar shapes and mirror asymmetry in the
-plane. Shapes such as Shapes G and H, which are basic geometric shapes such as equilateral triangles, and shapes such as Shape C, which has star-like configurations, are probably familiar and recognizable by the participants. Furthermore, Shape E, with its regular protrusions and a shape reminiscent of the letter ‘W’, is probably a familiar shape to the participants. Therefore, even if these shapes had mirror asymmetry in the
-plane, the order of the sensory evaluation values was considered to be high. Consequently, it is suggested that the difference between the proposed index and sensory evaluation values is significant for these shapes and that the proposed index underestimates their order.
Figure 23.
Shapes for which order is underestimated by the proposed index.
Figure 23.
Shapes for which order is underestimated by the proposed index.
3.2.3. Vase Shape
First, we detail the results, and then we discuss our analysis of how the coefficient of determination vary with the number of segmented surfaces.
Figure 25 illustrates the relationship between the proposed index and the sensory evaluation of the order in vase shapes for varying numbers of bins (integers ranging from 2 to 20). In this study, we set the center of mass of the shapes at the origin in
-space and oriented the shape such that the bottom of the vase lies parallel to the
-plane, and we computed seven proposed indices, such as the extruded shapes.
Figure 25 shows that the maximum coefficient of determination with the sensory evaluation values of order was achieved when the proposed index
was calculated for bin number 14, resulting in a coefficient of determination of 0.66. Additionally,
Figure 25 shows that the proposed indices considering mirror symmetry in the
-plane (
,
, and
) have a lower coefficient of determination with the sensory evaluation values of the order. This is because humans perceive mirror symmetry in objects. Humans are more likely to perceive and prioritize mirror symmetry in planes that include a vertical axis among all symmetrical planes [
38,
40,
62,
63]. Therefore, when evaluating the order of the vase shapes, the participants largely ignored mirror symmetry concerning the planes parallel to the bottom of the vase. Consequently, disregarding mirror symmetry in the
-plane led to higher coefficient of determination with the order sensory evaluation scores. Furthermore, compared with the proposed indices considering mirror symmetry in the
-plane or
-plane (
or
), those considering mirror symmetry in both the
- and
-planes (
) achieved a larger coefficient of determination with sensory evaluation values of order. This is likely because humans evaluate the mirror symmetry of shapes from multiple angles, not just one [
38,
39,
40,
54]. Therefore, calculating the proposed indices by dividing the shapes into multiple planes resulted in a larger coefficient of determination with the sensory evaluation values of the order.
Figure 25.
Relationship between the number of bins and coefficient of determination in vase shapes.
Figure 25.
Relationship between the number of bins and coefficient of determination in vase shapes.
Next, we considered the index
, which exhibited the highest coefficient of determination among the seven proposed indices. We examined why the coefficient of determination varied with bin count, specifically for larger and smaller numbers of bins. In
Figure 25, it is evident that the proposed index
exhibits the smallest coefficient of determination when two bins are used. This decline shares the same causes as the proposed
indices for the extruded shapes.
Figure 25 also reveals that, similar to extruded shapes, the coefficient of determination for vase shapes gets consistently larger for bin numbers greater than or equal to 6. Therefore, it can be inferred that six or more bins are necessary in this experiment.
Therefore, from bin numbers 6–20,
Figure 26a,b presents scatter plots of the proposed index and sensory evaluation values for the cases with the minimum coefficient of determination in bin number 9 and maximum coefficient of determination in bin number 14, respectively. It is evident from the two scatter plots that as the bin number changes from 9 to 14, the proposed index values for Shapes I and J become smaller compared to other shapes, approaching the regression line. This suggests that the coefficient of determination increased significantly for bin 14. The contour plots of the curvature and histograms of the curvature distributions for Shapes I and J are shown in
Figure 27,
Figure 28,
Figure 29,
Figure 30,
Figure 31 and
Figure 32.
Figure 27 and
Figure 30 show contour plots and histograms in bin 9 for Shapes I and J, respectively. In addition,
Figure 28 and
Figure 31 show contour plots and histograms at bin 14 for Shapes I and J, respectively. In
Figure 27 and
Figure 28, as well as
Figure 30 and
Figure 31, it can be observed that for bin 14, there is a significant curvature (highlighted in red in the contour plots) in the part of the shape corresponding to the handle, which is not distinguishable at bin 9. This observation is further supported by the histograms in
Figure 27 and
Figure 28, as well as
Figure 30 and
Figure 31, where for both shapes, a new distribution appears in the second bin from the left when the bin number changes from 9 to 14. This suggests that for bin 14, there is a significant difference in the curvature distribution across various parts of the shape when compared to bin 9, resulting in the proposed index values becoming smaller (i.e., closer to the sensory evaluation values). Therefore, it is believed that the coefficient of determination is larger for bin 14.
Furthermore, scatter plots of the sensory evaluation values of the proposed index and order for the number of bins with the largest coefficient of determination (14) and the number of bins with the largest coefficient of determination (20) in the extruded shape are shown in
Figure 26b and
Figure 26c, respectively. The blue-colored plots (Shapes I and J) are discussed below. The two scatter plots show that as the bin number changes from 14 to 20, the values of the proposed index for Shapes I and J become smaller compared with the other shapes and move farther away from the regression line. Therefore, it is believed that the coefficient of determination decreases in the case of 20 bins. Contour plots of the curvature in shape and histograms for each part are shown in
Figure 27,
Figure 28,
Figure 29,
Figure 30,
Figure 31 and
Figure 32 for each number of bins.
Figure 28 and
Figure 31 show contour plots and histograms at bin 14 for Shapes I and J, respectively. In addition,
Figure 29 and
Figure 32 show contour plots and histograms at bin 20 for Shapes I and J, respectively. These figures indicate that for 14 bins, the curvature around the handle area is not distinguishable from the curvature of the other surfaces, whereas for 20 bins, the curvatures of the other surfaces become more distinguishable. Consequently, for 20 bins, there is a larger difference in the curvature distribution across various parts of the shape, leading to higher values for the proposed index. However, Shapes I and J are asymmetric shapes with a ‘handle’ on one side, which can be recognized as a single function in a vase. Therefore, the subjects did not significantly reflect the asymmetry of the shape due to the presence of the ‘handle’ in the evaluation of the order. This resulted in the order of sensory evaluation values of these shapes being moderate. When the number of bins was increased to 20, the correlation between the proposed index and the sensory evaluation values of the order weakened, leading to a smaller overall coefficient of determination for the vase shape.
Figure 26.
Relationship between the proposed index and sensory evaluation values of ‘order’ in (a) bin number 9, (b) bin number 14, and (c) bin number 20.
Figure 26.
Relationship between the proposed index and sensory evaluation values of ‘order’ in (a) bin number 9, (b) bin number 14, and (c) bin number 20.
Figure 27.
Contour plots and curvature distributions in Shape I in bin number 9: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 27.
Contour plots and curvature distributions in Shape I in bin number 9: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 28.
Contour plots and curvature distributions in Shape I in bin number 14: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 28.
Contour plots and curvature distributions in Shape I in bin number 14: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 29.
Contour plots and curvature distributions in Shape I in bin number 20: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 29.
Contour plots and curvature distributions in Shape I in bin number 20: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 30.
Contour plots and curvature distributions in Shape J in bin number 9: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 30.
Contour plots and curvature distributions in Shape J in bin number 9: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 31.
Contour plots and curvature distributions in Shape J in bin number 14: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 31.
Contour plots and curvature distributions in Shape J in bin number 14: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 32.
Contour plots and curvature distributions in Shape J in bin number 20: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 32.
Contour plots and curvature distributions in Shape J in bin number 20: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Finally, we discuss the proposed index
(bin number 14), which exhibits the highest coefficient of determination with the order of sensory evaluation values.
Figure 33 shows the scatter plot of
(bin number 14) and the order of the sensory evaluation values. The yellow-colored plots (Shapes J, K, L, M and N) are discussed below.
Figure 33 shows a strong correlation, with a coefficient of determination of 0.66 for the vase shape, indicating a robust relationship between
(bin number 14) and the sensory evaluation values of the order. We consider shapes for which there is a significant difference between the proposed index and the sensory evaluation values.
Figure 33.
Relationship between the proposed index and sensory evaluation values of ‘order’ in bin number 14.
Figure 33.
Relationship between the proposed index and sensory evaluation values of ‘order’ in bin number 14.
- ▪
Shapes for which the proposed index overestimates order
Figure 33 shows that the proposed index overestimates the order for Shapes K, L, and M.
Figure 34,
Figure 35 and
Figure 36 depict the contour plots and histograms of the curvature for each shape. Among these, Shape K exhibits irregular ‘undulations’ on its surface, while both Shapes L and M have left-right asymmetric ‘openings’.
Figure 33 indicates that the sensory evaluation values for the order of these shapes were low. However, as is evident from the contour plots and histograms in
Figure 34,
Figure 35 and
Figure 36, when the shapes were divided in the
-plane, the curvature distributions in each divided part were nearly identical. Similarly, when divided in the
-plane, there was little variation in the curvature distribution among the divided parts. Consequently, it is safe to conclude that the proposed index values are approximately the average (rather than near the minimum) in these cases. Therefore, we suggest that the proposed index overestimates order.
Figure 34.
Contour plots and curvature distributions in Shape K: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 34.
Contour plots and curvature distributions in Shape K: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 35.
Contour plots and curvature distributions in Shape L: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 35.
Contour plots and curvature distributions in Shape L: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 36.
Contour plots and curvature distributions in Shape M: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 36.
Contour plots and curvature distributions in Shape M: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
- ▪
Shapes for which the proposed index underestimates order
Figure 33 shows that the proposed index underestimates the order for Shapes N and J. For each shape, the contour plots of the shape curvature and histograms of the curvature for each part are shown in
Figure 37 and
Figure 38. Shape N is a shape where the’ openings’ part functions convincingly as a ‘pouring spout’. For Shape J, the presence of a ‘handle’ was functionally acceptable for a vase. However, these shapes exhibit significant changes in curvature distribution among the divided parts due to the asymmetry caused by the ‘mouth’ and ‘handle’ when divided in the
-plane, as evident from the contour plots and histograms in
Figure 37 and
Figure 38. Consequently, the proposed index values for the same shapes increase. Therefore, we suggest that the proposed index underestimates the order.
Figure 37.
Contour plots and curvature distributions in Shape N: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 37.
Contour plots and curvature distributions in Shape N: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 38.
Contour plots and curvature distributions in Shape J: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.
Figure 38.
Contour plots and curvature distributions in Shape J: (a) contour plots from different perspectives, (b) curvature distribution of each part (blue and orange) obtained for the -plane, and (c) curvature distribution of each part (blue and orange) obtained for the -plane.