## 3 Materials and Methods
### Harmonic Cross Ratio Model with Constant Ratios and Proportions  
Just as the length of a line segment is the key to metric geometry, there is a fundamental concept of projective geometry in which all the distinctive properties of projective figures can be expressed [47]. Shape constancy depends on perspective, and the visual system operates on the projective principle of perspective. For [PERSON], the equality of cross ratios implies shape constancy [48]. Followers of [PERSON] have proposed that the cross ratio, a projective invariant for four collinear points, underlies the perception of objects in motion [49]. According to the seminal discovery of projective geometry, if we have four points A, C, B, and D on a straight line and project them onto A\({}^{\prime}\), B\({}^{\prime}\), C\({}^{\prime}\), and D\({}^{\prime}\) on another line, then there is a specific property of the four points, called the _cross ratio_, which retains its value under the projection. The cross ratio is neither a length nor the ratio of two lengths but the _ratio of two such ratios_. For instance, for two ratios, _AC/BC_ and _AD/BD_, the cross ratio is given by  
\[\{A,B;C,D\}=\frac{AC\cdot BD}{BC\cdot AD} \tag{8}\]  
where the sign of a length is taken appropriately after fixing a specific orientation of the line. If a straight line CD is divided internally at B and externally at A in the same ratio \(\{A,B;C,D\}=-1\), then the segment CD is said to be harmonically divided at A and B, and each of the points A and B is called the harmonic conjugate of the other, with respect to the pair C and D. The harmonic division symmetry is a remarkable property. Namely, if C and D divide AB harmonically, then A and B also divide CD harmonically, reinforcing each other.  
The problem of intercepting a target that moves in the horizontal plane is conceived as _the cross ratio of four coplanar_ (i.e., lying in a common plane) and _concurrent straight lines_ PA, PB, PC, and PD, namely, the four points of intersection cross ratio of these lines with another straight line lying in the same plane (Figure 4). For an arbitrary point P and a harmonic range [23], the lines PA, PB, PC, PD are said to form a _harmonic pencil_, a linear line space where paths are centrally placed at the point P. Line space denotes a set where linear combinations also locate lines in the set [50].  
Equivalent to Equation (8) is the definition  
\[\{A,B;C,D\}=\frac{\sin\angle APC}{\sin\angle APD}\frac{\sin\angle BPD}{\sin \angle BPC} \tag{9}\]  
where the cross ratio depends only on the angles subtended by the connecting segments A, B, C, and D at P. Since these angles are the same for any four points A\({}^{\prime}\), B\({}^{\prime}\), C\({}^{\prime}\), and D\({}^{\prime}\) into which A, B, C, and D can be projected from P, it follows that the cross ratio remains invariant by projection. The problem of intercepting a target moving parallel to the ground plane is ubiquitous in animal locomotion. It involves at least three separable phenomenal motions, called transformations. The first (translation along the line of sight) is called expansion or contraction; the second is translation to the right or left; and the third is rotation about a vertical axis (indicated by arrows in Figure 4) [15]. The coordination of the self-motion direction and speed when intercepting a target moving parallel to the ground plane can be modeled as a plane pencil of screws, a typical line space (Figure 4).