# A Watermarking Method for 3D Printing Based on Menger Curvature and K-Mean Clustering
## 1 Introduction  
Three-dimensional (3D) printing, also known as additive manufacturing, makes directly physical solid objects from digital models by the layer adding process [1; 2]. Due to the flexibility and ease of production, 3D printing is applied to many areas, such as healthcare systems, industry, aerospace and automotive production [3; 4; 5]. Due to the fact that the benefit of 3D printing is great, the products of 3D printing are widely used. However, the products of 3D printing are often illegally used in many cases. This means manufacturers cannot protect their copyright and cannot receive fees from users. Besides, manufacturers also desire to track the ownership of their products in commercial transactions. Thus, a watermarking method is suitable and necessary to protect the ownership and copyright of 3D printing [6].  
Previously, there were many watermarking methods for 3D models that have been extensively researched. 3D watermarking schemes generally focus on both the geospatial and frequency domains [7; 8; 9; 10; 11; 12]. Overall, the main content of 3D model watermarking is to embed watermark data into the 3D model to obtain the watermarked 3D model. The watermark data are then extracted fromthe watermarked 3D model. This means 3D model watermarking methods only extract the embedded watermark from the watermarked 3D models, which is not the output of 3D printing, while the target of 3D printing watermarking is to extract the embedded watermark from the 3D-printed object of the watermarked 3D printing model. Thus, these methods could not extract the embedded watermark from physical 3D-printed objects. Consequently, the watermarking methods for the 3D model could not be applied to the purpose of 3D printing watermarking because the output of 3D printing is a physical 3D-printed object. Moreover, the accuracy of current watermarking methods for 3D printing [13; 14] is very low and not flexible. There is even a method [14] that requires a complex system for the experiments. Therefore, the watermarking methods for 3D printing should have high accuracy, be flexible and reduce the complexity of the experiments.  
To replace the unsuitability of 3D model watermarking methods for the purpose of 3D printing watermarking and to respond to the risks of the previous watermarking methods for 3D printing, we would like to propose a novel watermarking method for 3D printing in this paper. The main content of the proposed method is to classify the facets of the 3D printing model into groups based on the Menger facet curvature before embedding watermark data. The watermark data are embedded by changing the value of the mean Menger curvature of each group based on the reference of a special value. After the watermark embedding process, the watermarked 3D printing model will be used to print the physical 3D object by a 3D printer. The watermark data are then extracted from the scanned 3D triangle mesh of the 3D-printed object. The proposed method is valid for the copyright protection of 3D printing, because it could extract the embedded watermark data into a 3D printing model from a physical 3D-printed object. That is what 3D model watermarking methods could not do. Moreover, the proposed method is more flexible than the two previously proposed methods for 3D printing watermarking because the length of the watermark bits is flexible, while the length of watermark bits in the two previous methods was fixed. It does not require a complex system, as did one of the previous method [14]. Finally, the accuracy of the proposed method is also higher than the two previously proposed methods. To clarify the proposed method, in Section 2, we look into previous watermarking techniques for the 3D model and 3D printing and explain the relation of the Menger curvature to the proposed method. In Section 3, we show the proposed method in detail. Experimental results and the evaluation of the proposed method will be shown in Section 4. Section 5 provides the conclusion.