# On Hyperbolic Complex Numbers
## 3 Spherical Hyperbolic Complex Numbers  
### Three Dimensional Space-Time Structure  
Let now  
\[C_{h}^{+}(r)=\{(w,x,y)^{T}\in\mathbb{R}^{3}:w^{2}-x^{2}-y^{2}=r^{2},\ w>0\},r>0\]  
be a hyperbolic half-sphere of hyperbolic radius \(r>0\) and assume that the event space is the cone  
\[\mathfrak{X}=\bigcup_{r>0}C_{h}^{+}(r)=\{(w,x,y)^{T}\in\mathbb{R}^{3}:w^{2}-x^ {2}-y^{2}>0,\ w>0\}.\]  
We recall that a class of new analytical vector products in \(\mathbb{R}^{3}\) is introduced in [23]. Here, we consider a hyperbolic analog as follows. Let \(\varrho_{k}=\sqrt{x_{k}^{2}+y_{k}^{2}},k=1,2\). Unless for \(\varrho_{k}=0\) for at least one value of \(k\in\{1,2\}\) the analytical hyperbolic vector product of \(\mathrm{r}_{1}=(w_{1},x_{1},y_{1})^{T}\) and \(\mathrm{r}_{2}=(w_{2},x_{2},y_{2})^{T}\) is defined as  
\[\left( \begin{array}{c}w_{1}\\ x_{1}\\ y_{1}\end{array} \right)\odot_{h}\left( \begin{array}{c}w_{2}\\ x_{2}\\ y_{2}\end{array} \right)=\left( \begin{array}{c}w_{1}w_{2}+\varrho_{1}\varrho_{2 }\\ (\frac{w_{1}}{\varrho_{1}}+\frac{w_{2}}{\varrho_{2}})\left(\begin{array}{c}x_ {1}x_{2}-y_{1}y_{2}\\ y_{1}x_{2}+x_{1}y_{2}\end{array}\right)\end{array} \right). \tag{16}\]  
Moreover, we put  
\[(w,x,y)^{T}\odot_{h}(\lambda,0,0)^{T}=(\lambda w,\lambda x,\lambda y)^{T}. \tag{17}\]  
It should be noted that this product differs only slightly from (12) and (13) in [23]. To better understand this vector product, we first introduce a new coordinate system. **Definition 9**. : _With the notation \(\Theta=\mathbb{R}^{+}\times[0,2\pi)\times\mathbb{R}^{+}\), we define the hyperbolic coordinate transformation \(Hyp:\Theta\to\mathfrak{X},(w,x,y)^{T}=Hyp(r,\varphi,\psi)\), by_  
\[w=r\cosh\psi,\ x=r\cos\varphi\sinh\psi\ \ \text{and}\ \ y=r\sin\varphi\sinh\psi. \tag{18}\]  
With \(\varrho^{2}=x^{2}+y^{2}\), the a.e. uniquely defined inverse map \(Hyp^{-1}:\mathfrak{X}\to\Theta\) is given by  
\[r=\sqrt{w^{2}-\varrho^{2}},\ \cos\varphi=\frac{x}{\varrho},\ \sin\varphi=\frac{y}{ \varrho}\ \ \text{and}\ \ \psi=\ln\sqrt{\frac{w+\varrho}{w-\varrho}}.\]  
The event space \(\mathfrak{X}\) is a subset of the vector space \(\mathbb{R}^{3}\). The latter has the set of vectors \(\mathfrak{e}=(1,0,0)^{T},j_{1}=(0,1,0)^{T}\) and \(j_{2}=(0,0,1)^{T}\) as a basis. The subspaces spanned by \(j_{1}\) and \(j_{2}\) or \(\mathfrak{e}\) are, respectively, denoted by \(\mathfrak{M}\) and \(\mathfrak{M}\) and the orthogonal projections of the vector \(\mathfrak{x}=(w,x,y)^{T}\) onto the spaces \(\mathfrak{M}\) and \(\mathfrak{M}\) by \(\Pi_{1}\) and \(\Pi_{2}\), respectively. It is recalled that in [23], it was discussed what technical implications the choice of a different basis of the vector space \(\mathbb{R}^{3}\) or the choice of another three-dimensional space may have. Such a choice obviously depends on the specific application at hand and also applies to the hyperbolic model. The variable \(\varphi\) describes the angle between the vectors \(j_{1}\) and \(\Pi_{1}\), which one can imagine in Figure 8.  
If the angle \(\varphi\) runs through the interval \([0,2\pi)\), then the point \(\mathfrak{x}\) runs through an Euclidean circle of radius \(\varrho\), centered at the point \(w\bullet\epsilon\) and belonging to the two-dimensional plane being parallel to the subspace \(\mathfrak{M}\). If the intersection point of \(C_{h}^{+}(r)\) and \(\mathfrak{M}\) is denoted \(A\) and \(S=S(\varrho,x,A)\) denotes the hyperbolic sector built by the points \(\varrho,\mathfrak{x}=(w,x,y)^{T}\in C_{h}^{+}(r)\) and \(A\) then \(\frac{\Phi}{2}r^{2}\) equals the area content of \(S\), see Figure 9.  
The meaning of \(r\) and \(\varphi\) lacking as a hyperbolic radius and an angle, and \(\psi\) as a non-negative area content motivates the following geometric definition of a vector product in \(\mathbb{R}^{3}\). Figure 8: Three\(-\)dimensional space\(-\)time: the hyperbola rotates around the \(y-\)axis. **Definition 10**.