## 3 Materials and Methods  
### Mathematical Solutions to Terrestrial Interception Based on the Geometrical Constraints  
First, researchers have exhibited that human interception behavior is best accounted for by the constant bearing angle (CBA), which refers to the angle between the direction ofthe heading and the agent's line of sight [23; 24; 25]. According to this model, an agent adopts a straight path to the target to keep the angle \(\beta\) constant, as illustrated in Figure 3a. It has been claimed that the agent does not need to adjust explicitly or compute the bearing angle \(\beta\) to null the change in \(\beta\), but simply steering a straight path will result in a successful interception at a constant angle \(\beta\)[26]. To understand these fundamental abilities, [PERSON] and [PERSON] proposed a mathematical model of \(\beta\), the behavioral dynamics of locomotion, i.e., the time evolution of behavior as an agent interacts with the environment [27]. This model is consistent with the view that the decision-making process, as an integral part of goal-directed behavior, is influenced by functional constraints at the scale of the environment-performer relationship. Thus, decision-making behavior emerges from the individual's interactions with environmental constraints over time toward specific goals [28]. We argue here that a simple perception of the approaching constant rate of change and bearing of the target informs the catcher whether they are moving at the right speed and in the right direction for the interception [11]. At this point, it is proper to recall the Chapman problem to determine \(\beta\): A ballplayer (point A) is running at the right speed in the right direction to make the catch (or point B) [11]. In which direction should he set off to catch the ball? We have tried to show here that a ballplayer solves trigonometric equations to catch a ball. The first step of the solution is to construct a pair of harmonic poles C and D [29] such that their ratio is given by the ratio of the player's speed \(\lambda_{1}\) to the ball's speed \(\lambda_{2}\), as shown in Figure 3b. Note that the agent has two control variables: direction and running speed. Adopting a different running speed would result in a different angle \(\beta\). This is shown by computing a solution in the harmonic ratio in interceptive walking. The first step of the solution is to construct harmonic poles C and D so that their length ratio is the ratio of the player's speed to the ball's speed (Figure 3):  
\[\frac{AC}{BC}=\frac{AD}{BD}=\frac{1}{g} \tag{1}\]  
where \(g\equiv\lambda_{2}/\lambda_{1}\) is the ratio of the speed of the target to the speed of the agent. Starting at points A and B, the next step is to construct point P as follows: Draw a circle of radius PA around A and draw a circle of radius PB around B so that the intersection P of these two circles forms the correct ratio with or without any scaling, as needed. Next, dropping the internal and external _angle bisectors_, a line segment dividing the angle APB into two equal parts gives the harmonic conjugates C and D. If the velocities \(\lambda_{1}\) and \(\lambda_{2}\) had opposite signs, then the point would have to divide the angle APB externally into the given ratio so that they are centered in the system. The coordinates of C and D are treated as a column vector of homogeneous coordinates by augmenting one and are given by  
\[C =gA+B,\] \[D =\frac{-gA+B}{1-g} \tag{2}\]  
At the point P, the ray from B (target) intersects a circle with a diameter CD (Figure 3). A straight line drawn to this point will intersect it. For example, the Matlab function 'linescirc' (MathWorks, Boston, MA, USA) can be used to find the intersection point of a circle defined by its center and radius in \(x\)-\(y\) coordinates and a line defined by its slope and \(y\)-intercept (Appendix A). This raises a new question: How could an effective bearing angle somehow be determined by a locomoting agent [30]?