## 2 Methodology
### Zero-Watermark Based on Delaunay Triangle Mesh  
Given that topological relation and angles in vector maps are resistant to geometric attacks [PERSON] (1999); [PERSON] (1999), this study first constructs Delaunay Triangulation Mesh (DTM) based on the feature vertices and then produces zero-watermark of high capacity and strong robustness by using angles in the DTM as feature information the feature vertices [PERSON] (1999). The specific steps are as follows:  
The DTM is first generated from the feature vertices of each block and the distances between adjacent vertices are derived based on (2). Angles of the triangles in DTM in blocks are then obtained using inverse cosine functions (3) to form an angle sequence \(A_{n}\) (4), of which the length is greater than that of the watermark sequence \(W_{n}\) in each block (Figure 1).  
\[\left\{\begin{array}{l}D_{12}=\sqrt{\left(p_{1}^{x}-p_{2}^{x}\right)^{2}+ \left(p_{1}^{y}-p_{2}^{y}\right)^{2}}\\ D_{23}=\sqrt{\left(p_{2}  
\[\left\{\begin{array}{l}\theta_{1}=arc\cos\big{[}\big{(}D_{12}^{2}+D_{31}^{2}-D_{ 23}^{2}\big{)}/~{}(2\cdot D_{12}\cdot D_{31})\big{]}\\ \theta_{2}=arc\cos\big{[}\big{(}D_{23}^{2}+D_{12}^{2}-D_{31}^{2}\big{)}/~{}(2 \cdot D_{23}\cdot D_{12})\big{]}\\ \theta_{3}=arc\cos\big{[}\big{(}D_{12}^{2}+D_{31}^{2}-D_{23}^{2}\big{)}/~{}(2 \cdot D_{12}\cdot D_{31})\big{]}\end{array}\right. \tag{3}\]  
\[A_{n}=\{~{}\theta_{1},~{}\theta_{2},~{}\theta_{3}\cdots,~{}\theta_{k}\} \tag{4}\]  
where \(p_{1}^{x},~{}p_{2}^{x},~{}p_{3}^{x}\) are the X-axis values of the first, second and third vertex, \(p_{1}^{y},p_{2}^{y},p_{3}^{y}\) are the Y-axis values of the first, second and third vertex, \(D_{12}\), \(D_{31}\), \(D_{23}\) are distances between two different vertex in a same triangle, \(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\) are three angle values of the same triangle and \(A_{n}\) is a set of angle values of each block.  
If the length of \(A_{n}\) is several times longer than the watermark sequence \(W_{n},d\) watermark sequences would be supplemented at the end of \(W_{n}\) to extend the watermark length. For example, if the length of \(A_{n}\) is 3300 and the length of \(W_{n}\) is 800, three message bit string of \(W_{n}\) would be extended after \(W_{n}\), and the watermark length becomes 3200. Applying the operator of exclusive disjunction (XOR) to \(A_{n}\) and \(W_{n}\) (5) would then generate zero-watermarking images \(Z_{n}\) (i.e., the watermark secret key) for the corresponding blocks. Both the original watermarking images and zero-watermarking images are then saved to an authoritative third-party agency IPR (e.g., the Intellectual Property Rights center) with added time stamps of legal copyrights to defend against interpretation attacks [52].  
\[Z_{n}=A_{n}~{}\oplus~{}W_{n},~{}n=1,~{}2,~{}3,\cdots,d \tag{5}\]  
When detecting watermarks, it only requires applying the same compression method to extract feature vertices for constructing DTM such that the same watermarks can be obtained by applying the same operations. As long as the feature vertices are unchanged, the constructed DTM is unique for a vector map no matter where to start constructing the DTM of the vector map. The characteristic of DTM greatly reduces the complexity of constructing triangle networks and avoids losing synchronicity when experiencing geometric attacks. Because attacks such as addition or deletion to feature vertices would only affect adjacent triangles and have no impacts on other triangles, the use of DTM could maintain local stability well.