# Gaussian Curvature Entropy for Curved Surface Shape Generation

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## Abstract

**:**

## 1. Introduction

## 2. Gaussian Curvature Entropy

#### 2.1. Construction of Gaussian Curvature Entropy

_{max}κ

_{min}

_{max}and κ

_{min}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:

_{i}is the occurrence probability of state s

_{i}. 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:

_{i}

_{,j}is the transition probability of state s

_{i}to state s

_{j}. 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.

_{t}(t = 1, 2, …, m) is a triangle neighboring v, A

_{t}is area of f

_{t}, and α

_{t}is angle of f

_{t}at v (Figure 3). To get rid of difference in K because of difference in A

_{t}between shapes, the dimensionless Gaussian curvature K’ is calculated by using the maximum diameter D of the curved surface shape, as (Figure 2a):

^{2}

_{2}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.

_{i}is calculated, as:

_{i}is the number of them allotted to the i-th (i ∈ {1, 2, …, V}) state s

_{i}. Afterwards, transition probability p

_{i}

_{,j}is calculated as:

_{i}

_{,neighbor}is the number of points neighboring on a point allotted to state s

_{i}and N

_{i}

_{,j}is the number of transitions from s

_{i}to j-th (j ∈{1, 2, …, V}) state s

_{j}.

_{G}is calculated while using p

_{i}and p

_{i,j}. At first, information entropy in one-dimensional Markov chain is calculated using Equation (2). Afterwards, H

_{G}is obtained by dividing the information entropy by the maximum entropy, as:

_{G}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.

_{G}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 H

_{G}is 0 on shape A, because the value of K is constant on the shape, while the value of H

_{G}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

#### 2.2.1. Experimental Methods

- 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

^{2}of logarithmic approximation between the sensory evaluation values about “complexity” and the value of H

_{G}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. R

^{2}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 R

^{2}becomes larger as V gets larger. Figure 9 shows the relationship between the sensory evaluation values and H

_{G}when R

^{2}is the largest value ({E, V} = {200, 11}). This figure shows the R

^{2}is 0.85 and the correspondence between H

_{G}and the sensory evaluation values. However, there are shapes whose values of H

_{G}differs while their sensory evaluation values are close, such as shapes 8 and 14 (Figure 9).

^{2}of sample shapes B. R

^{2}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 R

^{2}becomes larger as V gets larger. Figure 11 show the relationship between the sensory evaluation values and H

_{G}when R

^{2}is the largest value ({E, V} = {50, 19}). The R

^{2}is 0.83 and the correspondence between H

_{G}and the sensory evaluation values is confirmed, as shown in Figure 11. However, there are shapes whose relationship between the sensory evaluation values and H

_{G}is reversed, such as shapes 9 and 11 (Figure 11).

_{G}differs, while their sensory evaluation values are close

_{G}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 H

_{G}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 H

_{G}.

^{2}. First, the setting of E is discussed, as follows. In shape 11, points at sharp convexities are allotted to s

_{11}while points at concavities are allotted to s

_{1}. In shape 13, points in gradual convexities are allotted to s

_{9}or s

_{8}, while points in gradual concavities are allotted to s

_{4}or s

_{3}. In shape 6, points located out of convexities are allotted to s

_{6}. 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}.

_{G}is 0.071. When V = 4, the points on planes are allotted to two states because of minute unevenness and the value of H

_{G}is 0.520. Therefore, it is appropriate to set V at an odd number to prevent overestimating minute unevenness.

^{2}becomes larger as V gets larger is discussed. For example, the value of R

^{2}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.

_{G}is reversed

_{G}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 H

_{G}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.

^{2}. In shape 8, points at sharp convexities are allotted between s

_{19}and s

_{17}, while points at sharp concavities are allotted to s

_{1}. In shape 5, the points at gradual convexities are allotted between s

_{15}and s

_{12}, while points at gradual concavities are allotted to s

_{9}. In shape 6, the points at plane surfaces are allotted to s

_{10}. 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.

_{G}and total absolute Gaussian curvature I is discussed. Figure 17 shows shapes 10 and 13 of sample shapes B as shapes corresponding to H

_{G}rather than I. In these shapes, sensory evaluation values are 3.38 and 3 and the values of H

_{G}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 H

_{G}differ because the values of K’ differ in concavities with different size. Therefore, it seems that H

_{G}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 R

^{2}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.

_{G}. 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 H

_{G}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 s

_{6}, 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 s

_{5}, while the state close to the corner is between s

_{7}and s

_{11}. Consequently, the variance of states is larger and the value of H

_{G}is higher in shape 10. Therefore, it seems that H

_{G}sometimes overestimates a slight difference in a value of K’.

## 3. Shape Generation Method

#### 3.1. Construction of Shape Generation Method

**Q**

_{ab}(a = 1, 2, …, k, b = 1, 2, …, l) weight w

_{ab}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.

**Q**

_{ab}and w

_{ab}of NURBS surface based on the value of H

_{G}. Note that PSO is capable of solving problems, such as minimization of a multimodal function and optimization of a combination [23].

- (i)
- Set the initial shape (Figure 19a). Subsequently, the initial shape is expressed while using the NURBS surface and
**Q**_{ab}and w_{ab}of the surface is defined as the position vectors of particles in PSO (Figure 19b). Note that**Q**_{ab}is expressed by polar coordinates whose origin is the position of the controlling point at the initial shape. - (ii)
- Set H
_{G,target}(targeted value of H_{G}) and f_{u}(allowable difference between H_{G}and H_{G,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, H
_{G}and fitness f of the shapes is calculated. Note that the movable range of r_{ab}during movement is the half of the distance to the closest controlling point. In addition, this research set the range of w_{ab}to 0.5 ≤ w_{ab}≤ 2.0, since the shape transforms drastically by changing the value in the range. Finally, if f of a generated shape is lower than f_{u}, 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.

#### 3.2. Experiment for Validation of Shape Generation Method

#### 3.2.1. Experimental Methods

_{G}at the shape was 0.322 and shapes are generated by setting five levels of H

_{G,target}at 0.30, 0.38. 0.46, 0.54, and 0.62. Note that f

_{u}was set at 0.004 and 15 shapes (three shapes at each level) were generated. The reason of setting f

_{u}at 0.004 was because of the difference between the levels of H

_{G,target}was 0.08. Under this setting, the H

_{G}of the generated shape should be between [H

_{G,target}– 0.04, H

_{G,target}+ 0.04]. Subsequently, f

_{u}was set at 0.004, which was 1/10 of 0.04. In addition, the parameters of H

_{G}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 H

_{G}of shapes generated by random transformation of the initial shape is examined and the R

^{2}of logarithmic approximation on the relationship is maximized when {E, V} = {2, 3}. Figure 21 shows the generated shape examples.

- 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

_{G}. The R

^{2}of logarithmic approximation on the relationship is 0.71, and the correspondence between H

_{G}and the sensory evaluation values is confirmed. However, there are shapes that deviate from the approximate curve, such as shape 10, 11, and 15.

_{1}, 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, H

_{G}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 R

^{2}of logarithmic approximation is 0.89. Therefore, in calculating H

_{G}, 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

_{G}and the sensory evaluation values. Moreover, knowledge about the parameters of Gaussian curvature entropy was obtained.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

_{t}is angle of t-th triangle neighboring v (Figure 3). Then, I can be calculated approximately using Equation (9) and (A1) as:

## Appendix B

- (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 R
^{2}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 R
^{2}in combination of 2 groups.

^{2}in combinations of groups in sample shapes A and B, respectively.

**Figure A1.**Relationship between parameters (E and V) and coefficient of determination R

^{2}in sample shapes A during cross-validation; (

**a**) Group A1 and A2; (

**b**) Group A1 and A3; (

**c**) Group A2 and A3.

**Figure A2.**Relationship between parameters (E and V) and coefficient of determination R

^{2}in sample shapes A during cross-validation; (

**a**) Group B1 and B2; (

**b**) Group B1 and B3; (

**c**) Group B2 and B3.

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**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 4.**Quantization of dimensionless Gaussian curvature at each point: (

**a**) {E, V} = {30, 3}; and, (

**b**) {E, V} = {15, 15}.

**Figure 5.**Quantization of dimensionless Gaussian curvature on shapes without concavities when {E, V} = {20, 15}: (

**a**) sphere; (

**b**) distorted sphere.

**Figure 8.**Relationship between parameters (E and V) and coefficient of determination R

^{2}in sample shapes A.

**Figure 9.**Relationship between Gaussian curvature entropy H

_{G}and sensory evaluation values about “complexity” in sample shapes A.

**Figure 10.**Relationship between parameters (E and V) and coefficient of determination R

^{2}in sample shapes B.

**Figure 11.**Relationship between Gaussian curvature entropy H

_{G}and sensory evaluation values about “complexity” in sample shapes B.

**Figure 12.**Shapes whose values of H

_{G}differs while their sensory evaluation values are close: (

**a**) shape 8; (

**b**) shape 14.

**Figure 13.**Examples of quantization when {E, V} = {200, 11}: (

**a**) shape 11; (

**b**) shape 13; and, (

**c**) shape 6.

**Figure 15.**Shapes whose relationship between the sensory evaluation values and H

_{G}is reversed: (

**a**) shape 9; (

**b**) shape 11.

**Figure 16.**Examples of quantization when {E, V} = {50, 19}: (

**a**) shape 8; (

**b**) shape 5; and, (

**c**) shape 6.

**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 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 19.**Shape generation method based on Gaussian curvature entropy: (

**a**) Setting of an initial shape, H

_{G,target}and f

_{u}; (

**b**) Generation of position vectors; and, (

**c**) Generation and output of shapes.

**Figure 23.**Relationship between Gaussian curvature entropy H

_{G}and sensory evaluation values about “complexity” in generated shapes.

**Figure 24.**Quantization of dimensionless Gaussian curvature in shapes deviating from approximate curve: (

**a**) shape 10; (

**b**) shape 11; and, (

**c**) shape 15.

**Figure 25.**Relationship between Gaussian curvature entropy H

_{G}considering front part of shapes and sensory evaluation values about “complexity” in generated shapes.

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Okano, A.; Matsumoto, T.; Kato, T.
Gaussian Curvature Entropy for Curved Surface Shape Generation. *Entropy* **2020**, *22*, 353.
https://doi.org/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://doi.org/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://doi.org/10.3390/e22030353