## 3 Materials and Methods
### Cross Ratios at Infinity
If A, B, and C are three ordinary points on \(l\), we can assign a value to their ratio in the following way: Choose a point D on \(l\); then {_A,B;C,so} should be the limit approached by {_A,B;C,D_} as D recedes to infinity along \(l\). If {_A,B;C,so} = -1, then C is the midpoint of the segment AB (Figure 5). The midpoint and the infinity point in the direction of a segment divide the segment harmonically [10]. Figure 5: Cross ratio with elements at infinity, where one pole reaches infinity. The interceptor uses two sources of information: optical flow generated by taking the path \(AD^{\infty}\) and optical flow generated by taking the route C\({}_{M}\)P. The perception with covariation is different from either of the two information sources without covariation [51]. To refer to the flow pattern that informs about a stable environment, take the point at infinity as a point of the cross ratio reference. The arrangement shown in Figure 6, where elements such as \(CP\) and the line \(l\) form the principal axes of a system, illustrates how dynamic adjustments based on cross ratios can facilitate complex interactions within a controlled environment. These configurations allow for precise control over the movement and interaction of objects in virtual spaces, optimizing both the aesthetic and functional aspects of the environment. In essence, the role of cross ratios at infinity is not merely theoretical, but of substantial practical application in enhancing how we interact with and perceive VR and AR. By exploiting these projective invariants, we can create more immersive and navigable environments that closely mimic the real world or offer new ways of experiencing space that are not possible in physical settings. The interception paths are synthesized by a four-bar link connected by a slider-pin joint model. The dynamical link (equations of constraint) selectively reduces the number of independently controlled DOFs, allowing a rich set of trajectories [52]. The configuration allows link \(l\) to move only parallel to the agent's line of sight, since it is equal to the line \(l\) defined in Figure 5. The configuration shows a trapezoid AA\({}^{\prime}\)B\({}^{\prime}\)B, where AB is parallel to A\({}^{\prime}\)B\({}^{\prime}\). If C and C\({}^{\prime}\) are the midpoints of AB and A\({}^{\prime}\)B\({}^{\prime}\), respectively, the lines AA\({}^{\prime}\), CC\({}^{\prime}\), and BB\({}^{\prime}\) are all coincident at P. It is necessary to displace the agent along with the desired path CC\({}^{\prime}\) without disturbing the path AB approach mapping (Figure 6). Displacements around the two rays of the principal screws achieve this. Perception with covariation is different from either source of information without covariation [51]. As noted already, CP and the line \(l\) are called the principal screws (rays). Inspection of Equation (5) reveals that there is a linearly dependent screw along each of the x- and y-axes (Figure 6), corresponding, respectively, to  
\[ \begin{array}{l}\lambda_{2}=\lambda_{1}\\ \lambda_{2}=-\lambda_{1}\end{array}  \tag{10}\]  
Figure 6: Virtual heading chains in a four-bar linkage system. This diagram displays the configuration of virtual heading chains featuring links and a slider pin joint model, with CP and line \(l\) centrally positioned at point P. The arrangement demonstrates how dynamic motion and heading are controlled within a virtual environment. The diagram highlights the paths of rays CP and AB, representing the principal screws of the system, and shows how they meet at a common interception point, P. The paths are designed to illustrate how adjustments are made to the agent’s trajectory to maintain a constant bearing angle \(\beta\), which is crucial for precise object interception in simulated environments. This setup exemplifies the practical application of cross ratios at infinity, showing how geometric and kinematic constraints are managed to ensure stable and accurate navigational outcomes in virtual reality systems.