M ⟶ R 5 ( t , r , θ , ϕ ) ⟼ ( τ , W , X , Y , Z ) = ( arctan ( r + t ) + arctan ( − r + t ) , − r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 ( r 2 − t 2 − 1 ) r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 , 2 r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 r cos ( ϕ ) sin ( θ ) r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 , 2 r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 r sin ( ϕ ) sin ( θ ) r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 , 2 r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 r cos ( θ ) r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 ) ( u , v , θ , ϕ ) ⟼ ( τ , W , X , Y , Z ) = ( arctan ( u ) + arctan ( v ) , u 2 + 1 ( u v + 1 ) v 2 + 1 ( u 2 + 1 ) v 2 + u 2 + 1 , − ( u cos ( ϕ ) sin ( θ ) − v cos ( ϕ ) sin ( θ ) ) u 2 + 1 v 2 + 1 ( u 2 + 1 ) v 2 + u 2 + 1 , − ( u sin ( ϕ ) sin ( θ ) − v sin ( ϕ ) sin ( θ ) ) u 2 + 1 v 2 + 1 ( u 2 + 1 ) v 2 + u 2 + 1 , − u 2 + 1 v 2 + 1 ( u cos ( θ ) − v cos ( θ ) ) ( u 2 + 1 ) v 2 + u 2 + 1 ) ( U , V , θ , ϕ ) ⟼ ( τ , W , X , Y , Z ) = ( U + V , cos ( U ) cos ( V ) + sin ( U ) sin ( V ) , − ( cos ( V ) sin ( U ) − cos ( U ) sin ( V ) ) cos ( ϕ ) sin ( θ ) , − ( cos ( V ) sin ( U ) − cos ( U ) sin ( V ) ) sin ( ϕ ) sin ( θ ) , − ( cos ( V ) sin ( U ) − cos ( U ) sin ( V ) ) cos ( θ ) ) ( τ , χ , θ , ϕ ) ⟼ ( τ , W , X , Y , Z ) = ( τ , cos ( χ ) , cos ( ϕ ) sin ( χ ) sin ( θ ) , sin ( χ ) sin ( ϕ ) sin ( θ ) , cos ( θ ) sin ( χ ) ) ( t , ρ , θ , ϕ ) ⟼ ( τ , W , X , Y , Z ) = ( arctan ( t + e ρ ) + arctan ( t − e ρ ) , t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 ( t 2 − e ( 2 ρ ) + 1 ) t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 , 2 t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 cos ( ϕ ) e ρ sin ( θ ) t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 , 2 t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 e ρ sin ( ϕ ) sin ( θ ) t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 , 2 t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 cos ( θ ) e ρ t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 ) \displaystyle \begin{array}{llcl} & M & \longrightarrow & \mathbb{R}^5 \\ & \left(t, r, {\theta}, {\phi}\right) & \longmapsto & \left({\tau}, W, X, Y, Z\right) = \left(\arctan\left(r + t\right) + \arctan\left(-r + t\right), -\frac{\sqrt{r^{2} + 2 \, r t + t^{2} + 1} \sqrt{r^{2} - 2 \, r t + t^{2} + 1} {\left(r^{2} - t^{2} - 1\right)}}{r^{4} + t^{4} - 2 \, {\left(r^{2} - 1\right)} t^{2} + 2 \, r^{2} + 1}, \frac{2 \, \sqrt{r^{2} + 2 \, r t + t^{2} + 1} \sqrt{r^{2} - 2 \, r t + t^{2} + 1} r \cos\left({\phi}\right) \sin\left({\theta}\right)}{r^{4} + t^{4} - 2 \, {\left(r^{2} - 1\right)} t^{2} + 2 \, r^{2} + 1}, \frac{2 \, \sqrt{r^{2} + 2 \, r t + t^{2} + 1} \sqrt{r^{2} - 2 \, r t + t^{2} + 1} r \sin\left({\phi}\right) \sin\left({\theta}\right)}{r^{4} + t^{4} - 2 \, {\left(r^{2} - 1\right)} t^{2} + 2 \, r^{2} + 1}, \frac{2 \, \sqrt{r^{2} + 2 \, r t + t^{2} + 1} \sqrt{r^{2} - 2 \, r t + t^{2} + 1} r \cos\left({\theta}\right)}{r^{4} + t^{4} - 2 \, {\left(r^{2} - 1\right)} t^{2} + 2 \, r^{2} + 1}\right) \\ & \left(u, v, {\theta}, {\phi}\right) & \longmapsto & \left({\tau}, W, X, Y, Z\right) = \left(\arctan\left(u\right) + \arctan\left(v\right), \frac{\sqrt{u^{2} + 1} {\left(u v + 1\right)} \sqrt{v^{2} + 1}}{{\left(u^{2} + 1\right)} v^{2} + u^{2} + 1}, -\frac{{\left(u \cos\left({\phi}\right) \sin\left({\theta}\right) - v \cos\left({\phi}\right) \sin\left({\theta}\right)\right)} \sqrt{u^{2} + 1} \sqrt{v^{2} + 1}}{{\left(u^{2} + 1\right)} v^{2} + u^{2} + 1}, -\frac{{\left(u \sin\left({\phi}\right) \sin\left({\theta}\right) - v \sin\left({\phi}\right) \sin\left({\theta}\right)\right)} \sqrt{u^{2} + 1} \sqrt{v^{2} + 1}}{{\left(u^{2} + 1\right)} v^{2} + u^{2} + 1}, -\frac{\sqrt{u^{2} + 1} \sqrt{v^{2} + 1} {\left(u \cos\left({\theta}\right) - v \cos\left({\theta}\right)\right)}}{{\left(u^{2} + 1\right)} v^{2} + u^{2} + 1}\right) \\ & \left(U, V, {\theta}, {\phi}\right) & \longmapsto & \left({\tau}, W, X, Y, Z\right) = \left(U + V, \cos\left(U\right) \cos\left(V\right) + \sin\left(U\right) \sin\left(V\right), -{\left(\cos\left(V\right) \sin\left(U\right) - \cos\left(U\right) \sin\left(V\right)\right)} \cos\left({\phi}\right) \sin\left({\theta}\right), -{\left(\cos\left(V\right) \sin\left(U\right) - \cos\left(U\right) \sin\left(V\right)\right)} \sin\left({\phi}\right) \sin\left({\theta}\right), -{\left(\cos\left(V\right) \sin\left(U\right) - \cos\left(U\right) \sin\left(V\right)\right)} \cos\left({\theta}\right)\right) \\ & \left({\tau}, {\chi}, {\theta}, {\phi}\right) & \longmapsto & \left({\tau}, W, X, Y, Z\right) = \left({\tau}, \cos\left({\chi}\right), \cos\left({\phi}\right) \sin\left({\chi}\right) \sin\left({\theta}\right), \sin\left({\chi}\right) \sin\left({\phi}\right) \sin\left({\theta}\right), \cos\left({\theta}\right) \sin\left({\chi}\right)\right) \\ & \left(t, {\rho}, {\theta}, {\phi}\right) & \longmapsto & \left({\tau}, W, X, Y, Z\right) = \left(\arctan\left(t + e^{{\rho}}\right) + \arctan\left(t - e^{{\rho}}\right), \frac{\sqrt{t^{2} + 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} \sqrt{t^{2} - 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} {\left(t^{2} - e^{\left(2 \, {\rho}\right)} + 1\right)}}{t^{4} - 2 \, t^{2} {\left(e^{\left(2 \, {\rho}\right)} - 1\right)} + e^{\left(4 \, {\rho}\right)} + 2 \, e^{\left(2 \, {\rho}\right)} + 1}, \frac{2 \, \sqrt{t^{2} + 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} \sqrt{t^{2} - 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} \cos\left({\phi}\right) e^{{\rho}} \sin\left({\theta}\right)}{t^{4} - 2 \, t^{2} {\left(e^{\left(2 \, {\rho}\right)} - 1\right)} + e^{\left(4 \, {\rho}\right)} + 2 \, e^{\left(2 \, {\rho}\right)} + 1}, \frac{2 \, \sqrt{t^{2} + 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} \sqrt{t^{2} - 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} e^{{\rho}} \sin\left({\phi}\right) \sin\left({\theta}\right)}{t^{4} - 2 \, t^{2} {\left(e^{\left(2 \, {\rho}\right)} - 1\right)} + e^{\left(4 \, {\rho}\right)} + 2 \, e^{\left(2 \, {\rho}\right)} + 1}, \frac{2 \, \sqrt{t^{2} + 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} \sqrt{t^{2} - 2 \, t e^{{\rho}} + e^{\left(2 \, {\rho}\right)} + 1} \cos\left({\theta}\right) e^{{\rho}}}{t^{4} - 2 \, t^{2} {\left(e^{\left(2 \, {\rho}\right)} - 1\right)} + e^{\left(4 \, {\rho}\right)} + 2 \, e^{\left(2 \, {\rho}\right)} + 1}\right) \end{array} M ( t , r , θ , ϕ ) ( u , v , θ , ϕ ) ( U , V , θ , ϕ ) ( τ , χ , θ , ϕ ) ( t , ρ , θ , ϕ ) ⟶ ⟼ ⟼ ⟼ ⟼ ⟼ R 5 ( τ , W , X , Y , Z ) = ( arctan ( r + t ) + arctan ( − r + t ) , − r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 ( r 2 − t 2 − 1 ) , r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 2 r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 r c o s ( ϕ ) s i n ( θ ) , r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 2 r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 r s i n ( ϕ ) s i n ( θ ) , r 4 + t 4 − 2 ( r 2 − 1 ) t 2 + 2 r 2 + 1 2 r 2 + 2 r t + t 2 + 1 r 2 − 2 r t + t 2 + 1 r c o s ( θ ) ) ( τ , W , X , Y , Z ) = ( arctan ( u ) + arctan ( v ) , ( u 2 + 1 ) v 2 + u 2 + 1 u 2 + 1 ( uv + 1 ) v 2 + 1 , − ( u 2 + 1 ) v 2 + u 2 + 1 ( u c o s ( ϕ ) s i n ( θ ) − v c o s ( ϕ ) s i n ( θ ) ) u 2 + 1 v 2 + 1 , − ( u 2 + 1 ) v 2 + u 2 + 1 ( u s i n ( ϕ ) s i n ( θ ) − v s i n ( ϕ ) s i n ( θ ) ) u 2 + 1 v 2 + 1 , − ( u 2 + 1 ) v 2 + u 2 + 1 u 2 + 1 v 2 + 1 ( u c o s ( θ ) − v c o s ( θ ) ) ) ( τ , W , X , Y , Z ) = ( U + V , cos ( U ) cos ( V ) + sin ( U ) sin ( V ) , − ( cos ( V ) sin ( U ) − cos ( U ) sin ( V ) ) cos ( ϕ ) sin ( θ ) , − ( cos ( V ) sin ( U ) − cos ( U ) sin ( V ) ) sin ( ϕ ) sin ( θ ) , − ( cos ( V ) sin ( U ) − cos ( U ) sin ( V ) ) cos ( θ ) ) ( τ , W , X , Y , Z ) = ( τ , cos ( χ ) , cos ( ϕ ) sin ( χ ) sin ( θ ) , sin ( χ ) sin ( ϕ ) sin ( θ ) , cos ( θ ) sin ( χ ) ) ( τ , W , X , Y , Z ) = ( arctan ( t + e ρ ) + arctan ( t − e ρ ) , t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 ( t 2 − e ( 2 ρ ) + 1 ) , t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 2 t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 c o s ( ϕ ) e ρ s i n ( θ ) , t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 2 t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 e ρ s i n ( ϕ ) s i n ( θ ) , t 4 − 2 t 2 ( e ( 2 ρ ) − 1 ) + e ( 4 ρ ) + 2 e ( 2 ρ ) + 1 2 t 2 + 2 t e ρ + e ( 2 ρ ) + 1 t 2 − 2 t e ρ + e ( 2 ρ ) + 1 c o s ( θ ) e ρ )