## 4 Angle Selection and Ray Categorization
### Defining a Common Coordinate System  
As a result of the node orientation stage discussed in Section 4.2, nodes \(n_{T}\) and \(n_{R}\) obtain a common \(z\)-axis. Moreover, nodes visualize a beam as an infinite number of rays with an infinitesimal angular deviation that spans the entire beam width. Using such a definition, nodes attempts to identify the \"connecting-rays\" within a beam. As shown in Figure 8, a connecting-ray is the shortest ray responsible for establishing a link and carrying information between a node pair, i.e., \(n_{T}\) and \(n_{R}\). Recall from Section 3.2 that refraction in the horizontal domain is ignored; thus, if a transmitted ray spans some vertical plane, it can never leave that plane due to refraction. Therefore, it is safe to conclude that a set of transmitted connecting-rays will only span the subspace of a vertical plane \((\Omega)\) in their journey from the transmitter to a receiver. Specifically, \(\Omega\) must satisfy the following two properties: (1) \(\Omega\) passes through both nodes \(n_{T}\) and \(n_{R}\), (2) \(\Omega\) incorporates the transmitting node's \(z\)-axis as a proper subset (or equivalently the \(z\)-axis of \(n_{R}\) since the orientation stage, discussed in Section 4.2, yields the \(z\)-axis in the same direction for all nodes). Nodes then selects \(\Omega\) and forks an \(x\)-axis with the following three properties: (1) \(x\)-axis is perpendicular to the \(z\)-axis of \(n_{R}\), (2) positive \(x\)-axis direction gases at the direction of the received \"_hello_\" message, and (3) the \(x\)-axis is a proper subset of \(\Omega\). Figure 8 shows examples of \(x\)-\(z\) planes for two connecting-rays \((n_{2},n_{1})\) and \((n_{3},n_{1})\). Finally, the receiver node, \(n_{R}\), regards itself as the origin of the coordinate system it has just formed, i.e., the receiver lies at the intersection of the \(z\)- and \(x\)-axis. In the example of Figure 8, \(n_{1}\) will be the origin for the two coordinate systems with respect to \(n_{2}\) and \(n_{3}\).  
Once \(n_{R}\) identifies the vertical subspace \(\Omega_{i}\) for each established link \(k_{i}\), it creates a projection of all these subspaces in 2D. Since the underwater environment is modeled as stacked disks, each subspace \(\Omega_{i}\) must have corresponding layer boundaries defined at similar depths. Moreover, due to ignoring refraction in the horizontal directions, a node projects the 3D space into an equivalent 2D by simply superimposing the subspaces \(\Omega_{i}\) formed by different links. Specifically, superimposing is achieved by selecting an \(x\)-axis as a reference and then rotating each vertical plane \(\Omega_{i}\) along the \(z\)-axis until all other \(x\)-axis coincide to the chosen one as shown on the right on Figure 8. In Figure 8, nodes \(n_{2}\) and \(n_{3}\) located at depth \(z_{2}\) and \(z_{3}\) are establishing links with node \(n_{1}\) via the connecting-rays. Since both connecting-rays span distinct subspaces, node \(n_{1}\) preserves the transmitters' depth as well as the behavior of the connecting-rays along the depth by rotating both subspaces along its \(z\)-axis until both  
Figure 7: Representation of the spherical coordinate system where the angle \(\theta\) lies in \(xy\) plane, the angle \(\phi\) lies in the \(yz\) plane and the length of the vector connecting the nodes is denoted by \(r\).  
subspaces overlap. Specifically, the subspace \(\Omega_{1}\) is rotated by an angle of \(\theta_{1}\) while \(\Omega_{2}\) is rotated by an angle of \(\theta_{2}\). Thus, the transmitted signals at different subspaces in 3D are combined into 2D while preserving the acoustic signal behavior along the depth. Nodes utilizes this 2D plane to estimate the range of the connecting-rays and to find the SSP. Such a 2D plane suffices for ranging and for identifying reflected rays, as we show later.