## 2 Related Works  
One of the common methods for creating a robust and blind watermarking scheme for protecting digital and printed 3D models is based on embedding the watermark along the pre-defined axis and printing the 3D model in the same direction because the layer thickness and printing direction affect the accuracy and detail of the printed model [59; 60]. In this scheme, the layers of the 3D model are treated as a template that provides information about the orientation of the watermark in the procedure of watermark detection. [PERSON] et al. [61] created and presented an algorithm for copyright protection of 3D models based on embedding watermark data in the characteristic points of 3D models. The main goal of the algorithm is to cut the 3D digital model into slices along the \(Z\)-axis. Authors pointed out that the methods presented earlier for watermarking 3D models based on geospatial domain (changing topology, length, area, or value of the vertices or geometric features) or frequency domain (such as discrete Fourier transform, discrete wavelet transform [62], or discrete cosine transform) are useful for copyrights protection of the digital 3D models but not for physical and printed 3D models. The same authors created an algorithm for copyright protection of 3D printed models based on Menger facet curvature and K-mean clustering [20]. Another algorithm based on slices of 3D mesh, which are embedded into watermark data, was also presented [21]. The coordinates of each point in the obtained 2D slices are changed according to the changed mean distance value of the 2D slice. For constructing a robust watermark on the 3D model (mesh) and creating a set of scalar basis functions over the mesh vertices, it is necessary to use multiresolution analysis [22]. The watermark perturbs vertices along the direction of the surface normal, and it was weighted by the different basis functions such as hat, derby, and sombrero. The watermark is reliable in the case of a wide variety of attacks such as vertex reordering, noise addition, similarity transforms, cropping, smoothing, simplification, and insertion of a second watermark. Methods that are based on readjustment of the distances from the mesh faces to the mesh center calculate the distance from the face center to the mesh center as Euclidean distance [23; 24]. There are also methods for creating a watermark based on determining perceptually conspicuous regions over the given 3D mesh based on mesh saliency [25]. The mesh saliency is based on Gaussian function and depends on the parameter that is set arbitrarily [63]. After extracting salient regions from the 3D mesh, the authors embedded a watermark into these regions, and this is the second phase. For this purpose, they used the statistical approach [26]: for each vertex from the extracted salient region, the authors calculated vertex norm as the distance between the vertex and barycenter of the 3D mesh. According to the size of a given vertex norms in comparison with other vertices, a watermark is added to the observed vertex. The presented method takes into account that the visual appearance of the 3D shape is undisturbed, which is a special quality of this work. Every blind, non-reversible fragile watermarking algorithm is created to protect the integrity of 3D models [27]. An algorithm that does not require the existence of a host object is blind; otherwise, it is non-blind [28]. An algorithm that can recover the original data is reversible; otherwise it is named lossy or non-reversible. A digital watermark can be robust, semi-fragile, and fragile. A fragile watermark creates a minimal modification of the digital object, and its fields of application are integrity protection and authentication. The authors systematized paper [61], showing robust watermarking and fragile watermarking. They also presented the discrete Karhunen-Loeve transform (KLT), which is a linear transformation that maps vectors from one n-dimensional vector space to another vector space of the same dimension.