## 6 Coordinate and Time Transformations  
We will derive time transformations and position coordinates parameterized by two parameters: longitudinal contraction \(\xi(v)\) and transverse contraction \(\psi(v)\). These parameters meet the dependencies already given in Equation (15) and in Figure 5, i.e.,  
\[D_{x}(v)=D_{x}(0)\;\xi(v) \tag{36}\]  
\[D_{y}(v)=D_{y}(0)\;\psi(v) \tag{37}\]  
We accept the markings shown in Figure 7.  
We want the body moving relative to the universal frame of reference to be contracted according to patterns (36) and (37). At the same time, we want the time dilation to be in line with Equation (12) when the arm of the light clock is parallel to velocity \(v\). On the basis of (36) dilation (12), we can save it in the form of  
\[t_{||}=t_{||}^{\prime}\frac{\xi(v)}{1-(v/c)^{2}} \tag{38}\]  
Figure 7: Inertial system moves in relation to the ether with speed \(v\), while the ether moves in relation to the inertial system with speed \(v^{\prime}\) (\(v\cdot v^{\prime}\leq 0\)). The symbol \(\|\) indicates that the equation refers to time, measured by clocks, in which arms are parallel to velocity \(v\). On the basis of accepted requirements, the following equations are obtained  
\[x=\xi(v)\ x^{\prime}+v\ t_{||}=\xi(v)\ x^{\prime}+v\frac{\xi(v)}{1-\left(v/c \right)^{2}}t^{\prime}_{||} \tag{39}\]  
\[x^{\prime}=\frac{1}{\xi(v)}x+v^{\prime}\ t^{\prime}_{||}=\frac{1}{\xi(v)}x+v^{ \prime}\frac{1-\left(v/c\right)^{2}}{\xi(v)}t_{||} \tag{40}\]  
Based on (39), we obtain  
\[\xi(v)x^{\prime}=x-v\ \frac{\xi(v)}{1-\left(v/c\right)^{2}}t^{\prime}_{||} \tag{41}\]  
\[x^{\prime}=\frac{1}{\xi(v)}x-v\ \frac{1}{1-\left(v/c\right)^{2}}t^{\prime}_{||} \tag{42}\]  
When comparing (42) and Equation (40), the relation between velocities \(v\) and \(v^{\prime}\) is obtained in the form of  
\[v^{\prime}=-v\ \frac{1}{1-\left(v/c\right)^{2}} \tag{43}\]  
On the basis of (37)-(43), it is possible to save the transformations from the inertial system \(U^{\prime}\) to the ether \(U\)  
\[ \begin{cases}t_{||}=\frac{\xi(v)}{1-\left(v/c\right)^{2}}t^{\prime}_{||}&y= \psi(v)y^{\prime}\\ x=\xi(v)\ x^{\prime}+v\frac{\xi(v)}{1-\left(v/c\right)^{2}}t^{\prime}_{||}&z= \psi(v)z^{\prime}\end{cases}  \tag{44}\]  
The transformation from ether \(U\) to inertial system \(U^{\prime}\) has the following form  
\[ \begin{cases}t^{\prime}_{||}=\frac{1-\left(v/c\right)^{2}}{\xi(v)}t_{||}&y^{ \prime}=\frac{1}{\psi(v)}y\\ x^{\prime}=\frac{1}{\xi(v)}(x-v\ t_{||})&z^{\prime}=\frac{1}{\psi(v)}z\end{cases}  \tag{45}\]  
In the paper [2], all linear transformations (without revolutions) have been derived for which the average speed of light in a vacuum on the road back and forth is always \(c\). Those transformations are parameterized by two parameters, \(e(v)\) and \(\psi(v)\). The paper [2] shows that parameter \(e(v)\) does not create new kinematics but desynchronizes the clocks in inertial systems. Therefore, all kinematics with a constant average speed of light are parameterized with one transverse contraction \(\psi(v)\) parameter. In the paper [1], all such transformations were derived using a different method, based on a geometric analysis of [PERSON]'s experiment. Transformations (44) and (45) derived in this paper are more general than those parametrized only by parameter \(\psi(v)\). The additional parameter \(\xi(v)\) creates new kinematics for which longitudinal contraction can be independent of transverse contraction. In these kinematics, it is possible to model clocks for which the ticking speed depends on the inclination angle of their arms. Based on the transformation (44) and (45), it can be shown (calculations are omitted) that the average speed of light in the inertial system \(U^{\prime}\) depends on the direction of emission.