Next Article in Journal
Radiative MHD Nanofluid Flow over a Moving Thin Needle with Entropy Generation in a Porous Medium with Dust Particles and Hall Current
Next Article in Special Issue
Particle Swarm Optimisation: A Historical Review Up to the Current Developments
Previous Article in Journal
Stability Analysis of the Explicit Difference Scheme for Richards Equation
Previous Article in Special Issue
Path Planning of Pattern Transfer Based on Dual-Operator and a Dual-Population Ant Colony Algorithm for Digital Mask Projection Lithography
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Gaussian Curvature Entropy for Curved Surface Shape Generation

1
School of Integrated Design Engineering, Graduate School of Keio University, Yokohama 223-8522, Japan
2
Department of Mechanical Engineering, Keio University, Yokohama 223-8522, Japan
*
Author to whom correspondence should be addressed.
Submission received: 31 January 2020 / Revised: 15 March 2020 / Accepted: 16 March 2020 / Published: 18 March 2020
(This article belongs to the Special Issue Intelligent Tools and Applications in Engineering and Mathematics)

Abstract

:
The overall shape features that emerge from combinations of shape elements, such as “complexity” and “order”, are important in designing shapes of industrial products. However, controlling the features of shapes is difficult and depends on the experience and intuition of designers. Among these features, “complexity” is said to have an influence on the “beauty” and “preference” of shapes. This research proposed a Gaussian curvature entropy as a “complexity” index of a curved surface shape. The proposed index is calculated based on Gaussian curvature, which is obtained by the sampling and quantization of a curved surface shape and validated by the sensory evaluation experiment while using two types of sample shapes. The result indicates the correspondence of the index to perceived “complexity” (the determination coefficient is greater than 0.8). Additionally, this research constructed a shape generation method that was based on the index as a car design supporting apparatus, in which the designers can refer many shapes generated by controlling “complexity”. The applicability of the proposed method was confirmed by the experiment while using the generated shapes.

1. Introduction

In designing shapes of products, the overall shape features that emerge due to combinations of shape elements, such as points and lines, are important [1]. These features include “order” and “complexity”, and their control/adjustment is difficult and it depends on the experience and intuition of designers. There are some studies on the relationship between these features and the “beauty” or “preference” of shapes. Birkhoff carried out sensory evaluation while using polygon shapes and argued that the “beauty” of shapes is the ratio of “order” to “complexity” [2]. Eysenck conducted regression analysis for the beauty of polygon shapes and confirmed positive correlation between “complexity”, “order”, and “beauty” [3]. Berlyne insisted that the relationship between “complexity” and “preference” is expressed as an inverted U-shape [4]. Some researchers have confirmed this relationship. Munsinger et al. carried out sensory evaluation about “preference” of 2D shapes and indicated that the shapes having moderate number of vertices are the most preferable [5]. Hung et al. analyzed relationship between “complexity” and “aesthetic preference” and insisted that the relationship is expressed as an inverted U-shape [6].
From the above, since “complexity” is said to have an influence on “beauty” and “preference” of shapes, there are many studies regarding the quantification of “complexity” of curves and curved surface shapes, which are important in designing shapes. For example, Vitz defined the number of lines as “complexity” and analyzed the relationship between “preference” and the number of lines in black line drawings that are generated by the random-walk method [7]. Backes et al. proposed a “complexity” index by employing pixel intensity values of grayscale images and fractal dimension, and proposed a method for image analysis and retrieval [8]. Wang et al. defined the “complexity” of 3D shapes as the difference between outlines of shapes viewed from various angles and proposed an index based on this definition [9,10]. In addition, since Farin argued that curvatures on shapes enables detection of slight changes in shape [11], researches about “beauty” of curves and automatic generation of beautiful curves employed curvature [12,13,14,15]. Similarly, some of the researches applied curvature to quantify “complexity” of curves and curved surface shapes. Ujiie et al. proposed the total absolute curvature and curvature entropy as “complexity” indices of curves and constructed a curve generation method that was based on the indices [1,16]. In addition, Matsumoto et al. proposed total absolute Gaussian curvature as a “complexity” index of curved surface shapes and confirmed the correspondence to the sensory evaluation values about “complexity” [17]. However, the index cannot evaluate the “complexity” of shapes with the same number of concavities, since the index evaluates the number of concavities as “complexity”. Hence, the index sometimes fails to evaluate “complexity” of shapes having small concavities that are difficult to recognize, such as automobiles. Expanding curvature entropy to evaluate curved surface shapes might solve this.
This research aims to propose a “complexity” index of curved surface shapes while using Gaussian curvature entropy and construct a curved surface shape generation method that is based on the index to support the product design. This paper is organized, as follows. Chapter 2 presents the proposal of Gaussian curvature entropy as a “complexity” index of curved surface shapes. Chapter 3 describes the construction of a curved surface shape generation method that is based on the index and its validation experiment. Chapter 4 summarizes the achievements and future tasks.

2. Gaussian Curvature Entropy

A conventional “complexity” index of curved surface shapes fails to evaluate shapes with small concavities, as is mentioned in the previous section. This research aims to solve the problem by adopting the information entropy, which evaluates “complexity” to evaluate the “complexity” of curved surface shapes.

2.1. Construction of Gaussian Curvature Entropy

This section describes the definition and calculation method of Gaussian curvature entropy. Gaussian curvature entropy is the information entropy in one-dimensional Markov chain of Gaussian curvature. Gaussian curvature K expresses the curve at arbitrary point p on a curved surface, and it is calculated as:
K = κmaxκmin
where κmax and κmin are principal curvatures (i.e., the maximum and minimum normal curvature) at p (Figure 1). Information entropy H expresses the randomness of information [18], and it is calculated as:
H = i = 1 n p i log 2 p i
where n is the number of states and pi is the occurrence probability of state si. H becomes higher when the occurrence probability of each state is similar. Note that the occurrence probabilities usually depend on the surrounding states, since there tends to be correlation between each state. This stochastic process considering surrounding states is called the Markov chain. Especially when occurrence probabilities depend on neighboring states, the process is called first order Markov chain. The information entropy H’ of the process is calculated as:
H = i = 1 n j = 1 n p i p i , j log 2 p i , j
where pi,j is the transition probability of state si to state sj. H’ becomes higher when the occurrence probability of each state and transition probability between each state are similar. This research employed H’ of Gaussian curvature, just as the previous research [1]. Note that, in this research, the transition of Gaussian curvature is expressed as a transition of states in order to calculate H’ of Gaussian curvature. Calculation method of the index is explained, as follows. Note that, sampling and quantization are required to allot Gaussian curvature to the states.
(i) Sample a curved surface shape by dividing the entire shape with equilateral triangles with the same area using the Advancing Front Method (Figure 2a) [19]. Note that the triangles generated by the method tend to have similar areas and be close to equilateral triangles. The vertices of the triangles are called sampling points and the Gaussian curvature K at point v is approximately calculated, as [20]:
K 2 π t = 1 m α t 1 3 t = 1 m A t
where, ft (t = 1, 2, …, m) is a triangle neighboring v, At is area of ft, and αt is angle of ft at v (Figure 3). To get rid of difference in K because of difference in At between shapes, the dimensionless Gaussian curvature K’ is calculated by using the maximum diameter D of the curved surface shape, as (Figure 2a):
K’ = KD2
(ii) Set the parameters for deviation E and the state number V to define the states of K’. Subsequently, quantize K’ (i.e., allot K’ to each state, as in Figure 2b). K’ is quantized based on Equation (6), as:
s 1   ( K   < E ) ,   s 2   ( E     K   < E +   Δ E ) ,   s 3   ( E +   Δ E     K   < E + 2 Δ E ) ,   , s V 1   ( E     Δ E     K   < E ) ,   s V ( K E ) Δ E = 2 E V 2
For example, the results of the quantization when {E, V} = {30, 3} and {E, V} = {15, 15} are shown in Figure 4a,b, respectively. In Figure 4a, the points at a concavity or convexity are allotted to the second state s2 as same as points on a plane. Whereas, Figure 4b shows that those points are allotted to different states based on their values of K’. Note that, methods to set parameters, which enables calculating entropy without bias, is proposed [21]. However, it is necessary to carry out a sensory evaluation to set parameters since it is important to consider human perceived “complexity” in this research. The setting of parameters without carrying out a sensory evaluation is a future task.
(iii) Calculate the occurrence probability and transition probability of each state (Figure 2c). Occurrence probability pi is calculated, as:
p i = N i N
where, N is the number of sampling points on the curved surface shapes and Ni is the number of them allotted to the i-th (i ∈ {1, 2, …, V}) state si. Afterwards, transition probability pi,j is calculated as:
p i , j = N i , j N i , neighbor
where, Ni,neighbor is the number of points neighboring on a point allotted to state si and Ni,j is the number of transitions from si to j-th (j ∈{1, 2, …, V}) state sj.
(iv) Gaussian curvature entropy HG is calculated while using pi and pi,j. At first, information entropy in one-dimensional Markov chain is calculated using Equation (2). Afterwards, HG is obtained by dividing the information entropy by the maximum entropy, as:
H G = 1 log 2 V i = 1 V j = 1 V p i p i , j log 2 p i , j
HG ranges between 0 and 1 and becomes 0 when every sampling point on a curved surface shape is allotted to the same state and increases according to the variety of the states that the points are allotted to.
Figure 5 shows comparison of Gaussian curvature entropy HG and total absolute Gaussian curvature I that was proposed by Matsumoto et al. Note that, calculation method of I is described in Appendix A. The values of I are 1 on both shapes since the shapes have no concavities. On the other hand, the value of HG is 0 on shape A, because the value of K is constant on the shape, while the value of HG is 0.23 on shape B when {E, V} = {20, 15} since the value of K is various on the shape. Consequently, it seems that Gaussian curvature entropy can evaluate “complexity” due to changes in Gaussian curvature, which is not expressed as the number of concavities.

2.2. Experiment for Validation of Gaussian Curvature Entropy

This section illustrates the experiment to examine the correspondence of Gaussian curvature entropy to human perceived “complexity”.

2.2.1. Experimental Methods

1. Sample Shapes
The experiment utilized to types of samples shapes. The shapes are obtained by extrusion and rotation, which are common operations for generating shapes in 3DCAD.
a. Sample Shapes A (Shapes obtained by extrusion)
Sample shapes A are created from 2D shapes while using extrusion. In the experiment, 15 shapes (Figure 6a) are randomly extracted from 25 shapes used in the conventional research [1] and extruded in a direction perpendicular to the shapes. Note that the extruded distance is set to twice as long as the maximum diameter of each shape (Figure 6b).
b. Sample Shapes B (Shapes Obtained by Rotation)
Sample shapes B are created using rotation. 15 shapes are randomly extracted in the same manner, and the right half of the shapes (Figure 7a) are rotated 360 degrees around the axis passing through the center of gravity on each shape (Figure 7b).
2. Experimental Conditions
  • Evaluation method: This experiment adopts a five-point Likert scale (1: “not complex”, 2: “slightly complex”, 3: “fairly complex”, 4: “complex”, and 5: “very complex”) to obtain sensory evaluation values about complexity in samples shapes.
  • Sample shapes displaying method: White sample shapes on a black background are simultaneously displayed on a seven-inch tablet device. During the experiment, these shapes are rotated at a constant speed (x axis: 0 rpm, y axis: 0 rpm, z axis: 12 rpm) with the center of gravity as the axis, so that the appearance of all concavities can be observed.
  • Presentation method: The distance between the eyeball of a participant and the device was set to 500 mm. 15 sample shapes on the display simultaneously enter the field of view.
  • Participant: 30 participants (23 men and seven women), ranging in age from 18 to 54 years (M = 25.5, SD = 7.78).

2.2.2. Experimental Results and Discussion

1. Results
a. Results of Sample Shapes A
Figure 8 shows the relationship between parameters (E and V) and the coefficient of determination R2 of logarithmic approximation between the sensory evaluation values about “complexity” and the value of HG of sample shapes A. Note that logarithmic approximation is applied based on Fechner’s law, which indicates the relationship between human sensitivity and stimuli using logarithmic function. R2 is higher when E is between [20, 300] and V is an odd number, as shown in Figure 8. Moreover, the value of E with high R2 becomes larger as V gets larger. Figure 9 shows the relationship between the sensory evaluation values and HG when R2 is the largest value ({E, V} = {200, 11}). This figure shows the R2 is 0.85 and the correspondence between HG and the sensory evaluation values. However, there are shapes whose values of HG differs while their sensory evaluation values are close, such as shapes 8 and 14 (Figure 9).
b. Result of Sample Shapes B
Just the same as sample shapes A, Figure 10 shows the relationship between parameters (E and V) and R2 of sample shapes B. R2 is higher when E is between [20, 70] and V is an odd number, as shown in Figure 10. As same as sample shapes A, the value of E with high R2 becomes larger as V gets larger. Figure 11 show the relationship between the sensory evaluation values and HG when R2 is the largest value ({E, V} = {50, 19}). The R2 is 0.83 and the correspondence between HG and the sensory evaluation values is confirmed, as shown in Figure 11. However, there are shapes whose relationship between the sensory evaluation values and HG is reversed, such as shapes 9 and 11 (Figure 11).
2. Discussion
a. Discussion about Sample Shapes A
(i) Shapes whose values of HG differs, while their sensory evaluation values are close
Figure 12 shows shapes 8 and 14. Between these shapes, values of HG are higher in shape 8, while sensory evaluation values are close. Shape 8 has a concavity where values of K’ are low and this makes the value of HG lower than shape 14, which has no concavities, as shown in Figure 12. However, the concavity of shape 8 is bent at a right angle, which is often seen in general industrial products. As a result, it seems that the concavity does not affect human perceived “complexity” and the sensory evaluation value of shape 8 become low and closer to that of shape 14. This might be improved by considering right angles in the calculation of HG.
(ii) Setting of Parameters
Figure 13 shows an example of quantization when {E, V} = {200, 11}, which presents the highest value of R2. First, the setting of E is discussed, as follows. In shape 11, points at sharp convexities are allotted to s11 while points at concavities are allotted to s1. In shape 13, points in gradual convexities are allotted to s9 or s8, while points in gradual concavities are allotted to s4 or s3. In shape 6, points located out of convexities are allotted to s6. Therefore, it seems that various convexities and concavities of sample shapes A are allotted to different states that are based on their sharpness by setting parameters at {E, V} = {200, 11}.
Next, setting of V is discussed, as follows. Figure 14 shows the difference in quantization of K’ when V = 3 or 4 in shape 6 of sample shapes A. When V = 3, the points on planes (surface where value of K’ is nearly 0) are allotted to the same state, while points on corners are allotted to other states and the value of HG is 0.071. When V = 4, the points on planes are allotted to two states because of minute unevenness and the value of HG is 0.520. Therefore, it is appropriate to set V at an odd number to prevent overestimating minute unevenness.
Afterwards, the tendency that the value of E with high R2 becomes larger as V gets larger is discussed. For example, the value of R2 becomes high not only when {E, V} = {200, 11} but also when {E, V} = {300, 15} or {300, 17}, as shown in Figure 8. This is because boundaries between states are similar among these settings. For example, there are boundaries at K’ = 22, 67, 111, 166, and 200 when {E, V} = {200, 11}. On the other hand, boundaries are at K’ = 23, 69, 115, 162, and 208 when {E, V} = {300, 15} and K’ = 20, 60, 100, 140, and 180 when {E, V} = {300, 17}. Therefore, the tendency occurred because boundaries of K’ are similar among those settings of parameters. Further, cross-validation is carried out in Appendix B to confirm the tendency.
b. Discussion about Sample Shapes B
(i) Shapes whose relationship between the sensory evaluation values and HG is reversed
Figure 15 shows shapes 9 and 11. Between these shapes, the values of HG were higher in shape 9 while the sensory evaluation values were higher in shape 11. Shape 9 has a sharp edge and points neighboring the edge are allotted to various states, as shown in Figure 15. As a result, the value of HG rises. It seems that this is because the sampling points are not distributed along the edge and the values of K’ at the points are varied. Therefore, this can be solved by distributing sampling points along the edge during sampling.
(ii) Setting Parameters
Figure 16 shows the examples of quantization when {E, V} = {50, 19}, which presents the highest value of R2. In shape 8, points at sharp convexities are allotted between s19 and s17, while points at sharp concavities are allotted to s1. In shape 5, the points at gradual convexities are allotted between s15 and s12, while points at gradual concavities are allotted to s9. In shape 6, the points at plane surfaces are allotted to s10. This suggests that the convexities and concavities of sample shapes B are allotted to different states based on their sharpness by setting parameters at {E, V} = {50, 19}. Note that the reason why V becomes an odd number is to prevent overestimating minute unevenness, the same as sample shapes A.
At last, the difference between HG and total absolute Gaussian curvature I is discussed. Figure 17 shows shapes 10 and 13 of sample shapes B as shapes corresponding to HG rather than I. In these shapes, sensory evaluation values are 3.38 and 3 and the values of HG are 0.210 and 0.152, respectively. However, values of I are 1.46 and 1.54, which are not corresponding to the sensory evaluation values. It seems that this is because I evaluates the number of concavities as “complexity” and I cannot evaluate the size of concavities. On the other hand, values of HG differ because the values of K’ differ in concavities with different size. Therefore, it seems that HG overcomes a shortcoming of I, which is unable evaluate “complexity” between shapes having the same number of concavities. Note that the tendency that the value of E with high R2 becomes larger as V gets larger is because the boundaries of K’ are similar among those settings of parameters, as same as sample shapes A. Further, cross-validation is carried out in Appendix B to confirm the tendency.
Figure 18 shows shapes 4 and 10 of sample shapes A as shapes corresponding to I rather than HG. In these shapes, sensory evaluation values are 2.71 and 2.79 and values of I are 1.59 and 1.66, respectively. However, values of HG are 0.044 and 0.057, which do not correspond to sensory evaluation values. It seems that this is because states of sampling points on edges are allotted to different states. On an edge of shape 4 (Edge A in Figure 18a), the edge is straight,, except for a bent and the state of the points are s6, since the value of K’ is close to 0. On the other hand, on an edge of shape 10 (Edge B in Figure 18b), the edge is slightly bent and the state around the center of is s5, while the state close to the corner is between s7 and s11. Consequently, the variance of states is larger and the value of HG is higher in shape 10. Therefore, it seems that HG sometimes overestimates a slight difference in a value of K’.

3. Shape Generation Method

3.1. Construction of Shape Generation Method

This section describes the construction of curved surface generation method that is based on Gaussian curvature entropy.
At first, an expression method of a curved surface shape and an optimization method of a shape based on Gaussian curvature entropy is explained. This research employed Non-Uniform Rational B-Spline (NURBS) surface as the expression method. NURBS surface is generated based on position Qab (a = 1, 2, …, k, b = 1, 2, …, l) weight wab of controlling points distributed on the k × l grid [22]. Note that the NURBS surface has higher flexibility and it does not require constraining positions of controlling points in order to connect multiple surfaces with their tangents continuous. Therefore, it might be possible to generate curved surface shapes effectively by applying NURBS surface.
In addition, this research employed Particle Swarm Optimization (PSO) as a method to optimize Qab and wab of NURBS surface based on the value of HG. Note that PSO is capable of solving problems, such as minimization of a multimodal function and optimization of a combination [23].
Figure 19 shows the summary of the proposed method. The procedure of the method is explained, as follows.
(i)
Set the initial shape (Figure 19a). Subsequently, the initial shape is expressed while using the NURBS surface and Qab and wab of the surface is defined as the position vectors of particles in PSO (Figure 19b). Note that Qab is expressed by polar coordinates whose origin is the position of the controlling point at the initial shape.
(ii)
Set HG,target (targeted value of HG) and fu (allowable difference between HG and HG,target) as a condition of a candidate for solution.
(iii)
Generate shapes that are based on a position vector of particles renewed by movement of particles. Subsequently, HG and fitness f of the shapes is calculated. Note that the movable range of rab during movement is the half of the distance to the closest controlling point. In addition, this research set the range of wab to 0.5 ≤ wab ≤ 2.0, since the shape transforms drastically by changing the value in the range. Finally, if f of a generated shape is lower than fu, the shape is output as a candidate for solution. On the other hand, the movement of particles generates other shapes if there are no shapes meeting the condition.
Note that the proposed method cannot generate a shape that is based on a grid whose controlling point positions are inverted to prevent the generation of shapes with crossed surface. This means that there is a possibility that the generated shapes lack diversity.

3.2. Experiment for Validation of Shape Generation Method

This section explains the experiment to examine human perceived “complexity” of shapes that are generated by the proposed method.

3.2.1. Experimental Methods

1. Generated Shapes
In this research, the method was applied to generate the shapes of Coupe-typed automobiles. At first, the NURBS surface was generated as the initial shape based on controlling points (Figure 20). Note that the movement of the points was constrained, as shown in Figure 20. The value of HG at the shape was 0.322 and shapes are generated by setting five levels of HG,target at 0.30, 0.38. 0.46, 0.54, and 0.62. Note that fu was set at 0.004 and 15 shapes (three shapes at each level) were generated. The reason of setting fu at 0.004 was because of the difference between the levels of HG,target was 0.08. Under this setting, the HG of the generated shape should be between [HG,target – 0.04, HG,target + 0.04]. Subsequently, fu was set at 0.004, which was 1/10 of 0.04. In addition, the parameters of HG were set at {E, V} = {2, 3}. This is because, in the preliminary experiment, the relationship between the sensory evaluation values about “complexity” and the value of HG of shapes generated by random transformation of the initial shape is examined and the R2 of logarithmic approximation on the relationship is maximized when {E, V} = {2, 3}. Figure 21 shows the generated shape examples.
2. Experimental Conditions
  • Evaluation method: Just the same as Section 2.2.1.
  • Sample shapes displaying method: White generated shapes on a black background were simultaneously displayed on a 13.3-inch laptop computer (Figure 22). During the experiment, each shape is rotated at a constant speed (x axis: 0 rpm, y axis: 0 rpm, z axis: 10 rpm), so that the appearance of all concavities can be observed.
  • Presentation method: The distance between the eyeball of a participant and the computer was set to 500 mm. 15 sample shapes on the display simultaneously enter the field of view.
  • Participants: 40 participants (36 men and four women) that ranged in age from 16 to 61 years (M = 23.9, SD = 7.67).

3.2.2. Experimental Results and Discussion

1. Results
Figure 23 shows the relationship between the sensory evaluation values about “complexity” and the value of HG. The R2 of logarithmic approximation on the relationship is 0.71, and the correspondence between HG and the sensory evaluation values is confirmed. However, there are shapes that deviate from the approximate curve, such as shape 10, 11, and 15.
2. Discussion
Figure 24 shows the quantization of K’ on shapes that deviate from the approximate curve. These shapes have concavities at rear section and sampling points at the concavities are allotted to s1, as shown in this figure. It seems that the sensory evaluation value of the shape become lower, since it is difficult to recognize concavities at rear section. This is because it is natural to focus on the front section of the shape of an automobile rather than rear section. To examine this, HG considering only the front section is calculated and the correspondence to the sensory evaluation value is verified (Figure 25). Consequently, the deviation of shape 10, 11, and 15 from the approximate curve is mitigated and the R2 of logarithmic approximation is 0.89. Therefore, in calculating HG, it seems that it is necessary to consider the position of states and this can be achieved by weighing states based on their position.

4. Conclusions

This research proposed Gaussian curvature entropy as a “complexity” index of curved surface shapes by expanding curvature entropy, which is a “complexity” index of curves. Subsequently, correspondence between the index and human perceived “complexity” was examined by an experiment while using samples shapes generated by extrusion and rotation. Consequently, the coefficient of determinations between the value of the index and the sensory evaluation values about “complexity” were 0.85 and 0.83, and we confirmed the correspondence between HG and the sensory evaluation values. Moreover, knowledge about the parameters of Gaussian curvature entropy was obtained.
Using the proposed index, this research constructed a curved surface shape generation method. The method was applied to generate the shapes of Coupe-typed automobiles and correspondence between the index and human perceived “complexity” of generated shapes was examined. The result shows the coefficient of determination between the value of the index and the sensory evaluation values regarding “complexity” was 0.71, and the usability of the method was confirmed. The result also shows the difficulty to recognize concavities at rear section of the shapes, and this decreases the sensory evaluation values. This means that the human bias focusing on the front section rather than rear section suggested the need to define the weight based on the section of shapes.
The future study should consider the construction of a method for defining the weight based on their position during the calculation of Gaussian curvature entropy and carry out the validation experiment of the curved surface design by designers.

Author Contributions

Investigation, A.O.; Methodology, T.M.; Software, A.O. and T.M.; Supervision, T.K. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

Total absolute Gaussian curvature I is a “complexity” index of a curved surface shape calculated by integrating Gaussian curvature K on the entire shape as [17]:
I = 1 4 π A | K | d A
where, A is the area of the entire shape. The index evaluates the number of concavities as “complexity”. For example, the index becomes the minimum value (1) when there are no concavities such as a sphere, and the value becomes larger as it changes from the sphere due to the concavity.
Further, I can be calculated approximately from a curved surface shape divided by triangles. At first, the following equation about K (Gauss-Bonnet formula) holds for the minute area dA around vertice v [20]:
A K d A 2 π t = 1 m α t
where, αt is angle of t-th triangle neighboring v (Figure 3). Then, I can be calculated approximately using Equation (9) and (A1) as:
I 1 4 π r = 1 N | 2 π t = 1 m α t |
where, N is the number of points on a curved surface shape.

Appendix B

To examine that the parameters selected in Chapter 2.2.2. are not overfitting. A 3-folds cross-validation was carried out as follows.
(i)
Divide sample shapes A into 3 groups (Group A1, A2 and A3). In order to ensure that each group includes shapes with various sensory evaluation values about “complexity”, the shapes are assigned to the group based on the sensory evaluation values. In particular, 15 shapes were sorted in descending order of the sensory evaluation values and 5 categories were made by selecting 3 shapes in the order. Then, one shape was randomly selected from each category and 3 groups were made for the cross-validation.
(ii)
Combine 2 groups and examine the relationship between parameters (E and V) and coefficient of determination R2 about 10 shapes in the combined groups.
(iii)
As same as sample shapes A, divide samples shapes B into 3 groups (Group B1, B2 and B3) and examine the relationship between parameters and R2 in combination of 2 groups.
Figure A1 and Figure A2 show relationship between parameters and R2 in combinations of groups in sample shapes A and B, respectively.
Figure A1. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes A during cross-validation; (a) Group A1 and A2; (b) Group A1 and A3; (c) Group A2 and A3.
Figure A1. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes A during cross-validation; (a) Group A1 and A2; (b) Group A1 and A3; (c) Group A2 and A3.
Entropy 22 00353 g0a1aEntropy 22 00353 g0a1b
Figure A2. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes A during cross-validation; (a) Group B1 and B2; (b) Group B1 and B3; (c) Group B2 and B3.
Figure A2. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes A during cross-validation; (a) Group B1 and B2; (b) Group B1 and B3; (c) Group B2 and B3.
Entropy 22 00353 g0a2aEntropy 22 00353 g0a2b

References

  1. Ujiie, Y.; Kato, T.; Sato, K.; Matsuoka, Y. Curvature entropy for curved profile generation. Entropy 2012, 14, 533–558. [Google Scholar] [CrossRef]
  2. Birkhoff, G.D. Aesthetic Measure; Harvard University Press: Cambridge, UK, 1933. [Google Scholar]
  3. Eysenck, H.J. The empirical determination of an aesthetic formula. Psychol. Rev. 1941, 48, 83–92. [Google Scholar] [CrossRef]
  4. Berlyne, D.E. Aesthetics and psychobiology; Appleton Century Crofts: New York City, NY, USA, 1971. [Google Scholar]
  5. Munsinger, H.; Kessen, W. Uncertainty, structure, and preference. Psychol. Monogr. Gen. Appl. 1964, 78, 1–24. [Google Scholar] [CrossRef]
  6. Hung, W.K.; Chen, L. Effects of novelty and its dimensions on aesthetic preference in product design. Int. J. Des. 2012, 6, 81–90. [Google Scholar]
  7. Vitz, P.C. Preference for different amounts of visual complexity. Behav. Sci. 1966, 11, 105–114. [Google Scholar] [CrossRef] [PubMed]
  8. Backes, A.R.; Bruno, O.M. Medical image retrieval based on complexity analysis. Mach. Vis. Appl. 2010, 21, 217–227. [Google Scholar] [CrossRef]
  9. Wang, D.; Belyaev, A.; Saleem, W.; Seidel, H. Shape Complexity from Image Similarity; Max-Planck-Institut für Informatik: Saarbrücken, Germany, 2008. [Google Scholar]
  10. Saleem, W.; Belyaev, A.; Wang, D.; Seidel, H. On visual complexity of 3d shapes. Comput. Graph. 2011, 35, 580–585. [Google Scholar] [CrossRef]
  11. Farin, G.; Rein, G.; Sapidis, N.; Worsey, A.J. Fairing cubic B-spline curves. Comput. Aided Geom. Des. 1987, 4, 91–103. [Google Scholar] [CrossRef]
  12. Harada, T.; Yoshimoto, F.; Moriyama, M. An aesthetic curve in industrial design. In Proceedings of the 1999 IEEE Symposium on Visual Languages, Tokyo, Japan, 13–16 September 1999. [Google Scholar]
  13. Harada, T.; Yoshimoto, F. Automatic curve fairing system using visual languages. In Geometric Modeling: Techniques, Applications, Systems and Tools; Sarfraz, M., Ed.; Springer: Berlin/Heidelberg, Germany, 2001; pp. 301–327. [Google Scholar]
  14. Miura, T. A general equation o aesthetic curves and its self-affinity. Comput. Aided Des. Appl. 2006, 3, 457–464. [Google Scholar] [CrossRef]
  15. Inoguchi, J.; Kajiwara, K.; Miura, K.T.; Sato, M.; Schiefe, S.K.; Shimizu, Y. Log-aesthetic curves as similarity geometric analogue of Euler’s elasticae. Comput. Aided Geom. Des. 2018, 61, 1–5. [Google Scholar] [CrossRef] [Green Version]
  16. Ujiie, Y.; Matsuoka, Y. Total absolute curvature to represent the complexity of diverse curved profiles. In Proceedings of the 6th Asian Design Conference, Tsukuba, Ibaraki, Japan, 14–17 October 2003. [Google Scholar]
  17. Matsumoto, T.; Sato, K.; Matsuoka, Y.; Kato, T. Quantification of “complexity” in curved surface shape using total absolute curvature. Comput. Graph. 2019, 78, 108–115. [Google Scholar] [CrossRef]
  18. Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J. 1948, 27, 379–423, 623–656. [Google Scholar] [CrossRef] [Green Version]
  19. Owen, J.S. A Survey of Unstructured Mesh Generation Technology. In Proceedings of the 7th International Meshing Roundtable, Livermore, CA, USA, 26–28 October 1998. [Google Scholar]
  20. Surazhsky, T.; Magid, E.; Soldea, O.; Elber, G.; Rivlin, E. A comparison of Gaussian and mean curvatures estimation methods on triangular meshes. In Proceedings of the 2003 IEEE International Conference on Robotics and Automation, Taipei, Taiwan, 14–19 September 2003. [Google Scholar]
  21. Endres, D.; Földiák, P. Bayesian Bin Distribution Inference and Mutual Information. IEEE Trans. Inf. Theory 2005, 51, 3766–3779. [Google Scholar] [CrossRef]
  22. Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Method Appl. Mech. Eng. 2005, 194, 4135–4195. [Google Scholar] [CrossRef] [Green Version]
  23. Kennedy, J.; Eberhart, R.C. Particle swarm optimization. In Proceedings of the 1995 IEEE Conference of Neural Networks, Perth, Western Australia, Australia, 27 November–1 December 1995. [Google Scholar]
Figure 1. Principal curvatures κmax and κmin at point p.
Figure 1. Principal curvatures κmax and κmin at point p.
Entropy 22 00353 g001
Figure 2. Calculation of Gaussian curvature entropy: (a) Division of curved surface and calculation of dimensionless Gaussian curvature; (b) Quantization of dimensionless Gaussian curvature; and, (c) Calculation of occurrence probability and transition probability.
Figure 2. Calculation of Gaussian curvature entropy: (a) Division of curved surface and calculation of dimensionless Gaussian curvature; (b) Quantization of dimensionless Gaussian curvature; and, (c) Calculation of occurrence probability and transition probability.
Entropy 22 00353 g002aEntropy 22 00353 g002b
Figure 3. Calculation of Gaussian curvature at sample point v.
Figure 3. Calculation of Gaussian curvature at sample point v.
Entropy 22 00353 g003
Figure 4. Quantization of dimensionless Gaussian curvature at each point: (a) {E, V} = {30, 3}; and, (b) {E, V} = {15, 15}.
Figure 4. Quantization of dimensionless Gaussian curvature at each point: (a) {E, V} = {30, 3}; and, (b) {E, V} = {15, 15}.
Entropy 22 00353 g004
Figure 5. Quantization of dimensionless Gaussian curvature on shapes without concavities when {E, V} = {20, 15}: (a) sphere; (b) distorted sphere.
Figure 5. Quantization of dimensionless Gaussian curvature on shapes without concavities when {E, V} = {20, 15}: (a) sphere; (b) distorted sphere.
Entropy 22 00353 g005
Figure 6. Generation of sample shapes A: (a) Curved shapes to extrude; (b) Sample shapes A.
Figure 6. Generation of sample shapes A: (a) Curved shapes to extrude; (b) Sample shapes A.
Entropy 22 00353 g006
Figure 7. Generation of sample shapes B: (a) Curved shapes to rotate; (b) Sample shapes B.
Figure 7. Generation of sample shapes B: (a) Curved shapes to rotate; (b) Sample shapes B.
Entropy 22 00353 g007
Figure 8. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes A.
Figure 8. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes A.
Entropy 22 00353 g008
Figure 9. Relationship between Gaussian curvature entropy HG and sensory evaluation values about “complexity” in sample shapes A.
Figure 9. Relationship between Gaussian curvature entropy HG and sensory evaluation values about “complexity” in sample shapes A.
Entropy 22 00353 g009
Figure 10. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes B.
Figure 10. Relationship between parameters (E and V) and coefficient of determination R2 in sample shapes B.
Entropy 22 00353 g010
Figure 11. Relationship between Gaussian curvature entropy HG and sensory evaluation values about “complexity” in sample shapes B.
Figure 11. Relationship between Gaussian curvature entropy HG and sensory evaluation values about “complexity” in sample shapes B.
Entropy 22 00353 g011
Figure 12. Shapes whose values of HG differs while their sensory evaluation values are close: (a) shape 8; (b) shape 14.
Figure 12. Shapes whose values of HG differs while their sensory evaluation values are close: (a) shape 8; (b) shape 14.
Entropy 22 00353 g012
Figure 13. Examples of quantization when {E, V} = {200, 11}: (a) shape 11; (b) shape 13; and, (c) shape 6.
Figure 13. Examples of quantization when {E, V} = {200, 11}: (a) shape 11; (b) shape 13; and, (c) shape 6.
Entropy 22 00353 g013
Figure 14. Difference in quantization of dimensionless Gaussian curvature: (a) V = 3; (b) V = 4.
Figure 14. Difference in quantization of dimensionless Gaussian curvature: (a) V = 3; (b) V = 4.
Entropy 22 00353 g014
Figure 15. Shapes whose relationship between the sensory evaluation values and HG is reversed: (a) shape 9; (b) shape 11.
Figure 15. Shapes whose relationship between the sensory evaluation values and HG is reversed: (a) shape 9; (b) shape 11.
Entropy 22 00353 g015
Figure 16. Examples of quantization when {E, V} = {50, 19}: (a) shape 8; (b) shape 5; and, (c) shape 6.
Figure 16. Examples of quantization when {E, V} = {50, 19}: (a) shape 8; (b) shape 5; and, (c) shape 6.
Entropy 22 00353 g016
Figure 17. Shapes whose sensory evaluation values correspond to Gaussian curvature entropy rather than total absolute Gaussian curvature: (a) shape 10; (b) shape 13 in samples shapes B.
Figure 17. Shapes whose sensory evaluation values correspond to Gaussian curvature entropy rather than total absolute Gaussian curvature: (a) shape 10; (b) shape 13 in samples shapes B.
Entropy 22 00353 g017
Figure 18. Shapes whose sensory evaluation values correspond to total absolute Gaussian curvature rather than Gaussian curvature entropy: (a) shape 4; (b) shape 10 in sample shapes A.
Figure 18. Shapes whose sensory evaluation values correspond to total absolute Gaussian curvature rather than Gaussian curvature entropy: (a) shape 4; (b) shape 10 in sample shapes A.
Entropy 22 00353 g018
Figure 19. Shape generation method based on Gaussian curvature entropy: (a) Setting of an initial shape, HG,target and fu; (b) Generation of position vectors; and, (c) Generation and output of shapes.
Figure 19. Shape generation method based on Gaussian curvature entropy: (a) Setting of an initial shape, HG,target and fu; (b) Generation of position vectors; and, (c) Generation and output of shapes.
Entropy 22 00353 g019
Figure 20. Initial shape of shape generation.
Figure 20. Initial shape of shape generation.
Entropy 22 00353 g020
Figure 21. Examples of generated shapes: (a) HG = 0.30; (b) HG = 0.46; and, (c) HG = 0.62.
Figure 21. Examples of generated shapes: (a) HG = 0.30; (b) HG = 0.46; and, (c) HG = 0.62.
Entropy 22 00353 g021
Figure 22. Displaying method of generated shapes.
Figure 22. Displaying method of generated shapes.
Entropy 22 00353 g022
Figure 23. Relationship between Gaussian curvature entropy HG and sensory evaluation values about “complexity” in generated shapes.
Figure 23. Relationship between Gaussian curvature entropy HG and sensory evaluation values about “complexity” in generated shapes.
Entropy 22 00353 g023
Figure 24. Quantization of dimensionless Gaussian curvature in shapes deviating from approximate curve: (a) shape 10; (b) shape 11; and, (c) shape 15.
Figure 24. Quantization of dimensionless Gaussian curvature in shapes deviating from approximate curve: (a) shape 10; (b) shape 11; and, (c) shape 15.
Entropy 22 00353 g024
Figure 25. Relationship between Gaussian curvature entropy HG considering front part of shapes and sensory evaluation values about “complexity” in generated shapes.
Figure 25. Relationship between Gaussian curvature entropy HG considering front part of shapes and sensory evaluation values about “complexity” in generated shapes.
Entropy 22 00353 g025

Share and Cite

MDPI and ACS Style

Okano, A.; Matsumoto, T.; Kato, T. Gaussian Curvature Entropy for Curved Surface Shape Generation. Entropy 2020, 22, 353. https://0-doi-org.brum.beds.ac.uk/10.3390/e22030353

AMA Style

Okano A, Matsumoto T, Kato T. Gaussian Curvature Entropy for Curved Surface Shape Generation. Entropy. 2020; 22(3):353. https://0-doi-org.brum.beds.ac.uk/10.3390/e22030353

Chicago/Turabian Style

Okano, Akihiro, Taishi Matsumoto, and Takeo Kato. 2020. "Gaussian Curvature Entropy for Curved Surface Shape Generation" Entropy 22, no. 3: 353. https://0-doi-org.brum.beds.ac.uk/10.3390/e22030353

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop