# A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering
## 4 Experimental Results and Analysis
### Performance Evaluation  
In order to evaluate the performance of the proposed method, we compare the flexibility and the accuracy of the proposed method to the recent watermarking methods for 3D printing ([PERSON]'s method and [PERSON]'s method). However, to prove the superiority of the proposed method to the conventional works and the previous methods of 3D printing watermarking, we will firstly compare the proposed method to two previous methods of 3D printing watermarking. Secondly, we will compare the proposed method to the conventional works of 3D model watermarking. This comparison is to prove that our method can be applied to both 3D model watermarking and 3D printing watermarking. To compare the proposed method to the two previously-proposed methods for 3D printing watermarking, we compare the flexibility and the accuracy between the methods. This accuracy is the accuracy of the extracted watermark from the 3D-printed objects, because two previous methods also experimented with 3D-printed objects. In [PERSON]'s method [13], the watermark data are embedded in the frequency domain based on spectrum decomposition and modulation. Watermark data are extracted from the scanned 3D triangle mesh of 3D-printed objects. His experiments are divided in two parts. In the first part, he experimented on the watermark detection from the simulated scans. This experiment is performed in a virtual environment, and the length of watermark is 256 bits for all experiments. He discussed that the percentage that is precise for a casting object is improved significantly compared to the method of [PERSON] et al. [23]. The percentage of [PERSON]'s method that is precise for a casting object is 12.8% (approximate \(10^{-1}\)). In the second part, he experimented on three 3D triangle meshes: bunny, casting and hand with a 3D printer. He discussed that the percentage that is precise for a casting object is smaller than \(10^{-5}\). Therefore, we concluded that the accuracy of [PERSON]'s method is approximate 40%. In addition, the risk of [PERSON]'s method is the length of the watermark bits is fixed and equal to 256 bits for all test models. As a result, there is a limitation of watermark bits in this method, and attackers can remove the embedded watermark more easily. In [PERSON]'s method [14], the watermark data are embedded in the printed objects in the 3D printing process by a complex system of laser and halogen lights. This requires a complex hardware system, but it could not embed all expected watermark bits inside 3D-printed objects. [PERSON] experimented on his method with two 3D triangle meshes. Following the experimental results of [PERSON], the maximum length of the embedded watermark bits was 64 bits. In addition, [PERSON] did not describe how to extract the embedded watermark data from the 3D-printed objects. Therefore, we considered the accuracy of [PERSON]'s method to be approximately 0%. In our method, the length of watermark bits is flexible and can be changed by the user based on the number of facets in each model (see Table 1 and refer to Section 4, Paragraph 1, and Equation (1)). This helps the users change the content of watermark bits according to their purpose. As we explained in Section 4.2, the accuracy of our method is dependent on the quality of the 3D printer and 3D scanner. The maximum accuracy of our method is 74.42% with the test models shown in Table 2. Table 3 describes the comparison between the proposed method