1.4.3 The Tangent Space of the Cone

In conic sec­tions, the cone is typ­i­cal­ly not given much thought and is con­sid­ered as null, but when viewed as being a group in com­plex space, we start to see its im­por­tance, given other geome­tries arise from it. When talk­ing about the $\mathcal{SU}(2)$ set in the pre­vi­ous sec­tion, the tan­gent spaces of the in­di­vid­ual ex­po­nen­tial func­tion points on the cone com­prise the chord ma­tri­ces of the form $\Delta X_{j}$. The reader can recall that a tan­gent space is de­ter­mined by the fol­low­ing cal­cu­la­tion

and using our pre­vi­ous chord points $B,B',C,C'$, we denote them as unit equa­tions

Cal­cu­lat­ing the tan­gent spaces of the above equa­tions, we find

and we can see that the tan­gent spaces for our unit func­tions in­ter­sect the $U(3)$ sphere ex­act­ly at the points $B,B',C,C'$. The tan­gent spaces meet at a con­flu­ence which is the focal point of the pos­i­tive unit hy­per­bo­lae in the $X_{1}$ and $X_{3}$-planes, and this con­flu­ence also forms the center of the in­scribed $SU(2)$ sphere. In Figure 1.4.8, blue lines rep­re­sent the tan­gent spaces for points $B,B',C,C'$in­ter­sect­ing at the point $O'$, which is the hy­per­bol­ic focal point and center of the in­scribed $SU(2)$ sphere. The tan­gent points $B,B',C,C'$are in­ter­sec­tion points on the $U(3)$ sphere, seen in the figure for the in­scribed $SU(2)$ sphere, and the cone ori­ent­ed along the pos­i­tive $X_{1}$-axis.

point for the pos­i­tive unit hy­per­bo­lae in the $X_{1},X_{3}$-plane, and the center of the in­scribed $SU(2)$ sphere.

The tan­gent spaces for our matrix equa­tions will con­tin­ue to orient along points as­ymp­tot­ic to the unit hy­per­bo­la, based on in­creas­es in the mag­ni­tude of their ar­gu­ments, form­ing the cones that bound the hy­per­bo­lae and the in­scribed spheres. To show this, we take the matrix equa­tion at the point $E$, with a scalar con­stant $q$, in the ar­gu­ment

for $\alpha$, the angle in ra­di­ans, and cal­cu­late the tan­gent space of $E$, $\left. \ \frac{\partial}{\partial\alpha}E \right|_{\alpha = 0}$, find­ing

re­mark­ing that its tan­gent spaces simply scale by the con­stant,$q$. Chord ma­tri­ces for the scaled tan­gent spaces would then take the form for our orig­i­nal $\Delta X_{1},\ \Delta X_{3}$ ma­tri­ces, but would be scaled by a factor of $\cosh(q)$, defin­ing the radial dis­tance to the points $qE,qG$ from the origin

The $u_{1}$ matrix el­e­ments cal­cu­late to

and the scaled $SU(2)$ group simply be­comes

In Figure 1.4.9, we scale the in­scribed sphere by the value $q = \sqrt{2}$, and take a slice of this along the $X_{1}$-plane. The radial dis­tance to the points $E^{'},G^{'}$ is $\left| e^{q}e^{\left( \frac{\pm i\pi}{4} \right)} \right| \approx 2.91,$the new center point of the scaled in­scribed $SU(2)$ sphere is lo­cat­ed at $\mathcal{O}^{''} = \left( \left| e^{q}e^{\left( \frac{- i\pi}{4} \right)} \right|^{2} + \left| e^{q}e^{\left( \frac{i\pi}{4} \right)} \right|^{2} \right)^{\frac{1}{2}} \approx 5.82$, and the scaled in­scribed sphere meets the cone at points $E',G'$. As the in­scribed sphere in­creas­es in size, the unit hy­per­bo­la and the $U(3)$ sphere scale by the same factor of $e^{q}$ (Figure 1.4.10).

with origin at $\mathcal{O}$, and scaled unit hy­per­bo­la with focal point at $O''$, and seg­ment, $\overline{OJ} = e^{q}\cosh(\alpha = 0)$.

The in­scribed sphere of the scaled $SU(2)$ group will always have its origin at the co­or­di­nates, ($\sqrt{2}e^{q},0,0)$, for those in­scribed spheres cen­tered along the $X_{1}^{+}$-axis, with sim­i­lar axis-based po­si­tions for spheres ori­ent­ed about the other axes.