  In order to extend our approach based on SU$*$(4), we were led to (real)
projective and (line) Complex geometry. So here we start from quadratic
Complexe which yield naturally the 'light cone'
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{0}^{2}=0$ when being related to (homogeneous)
point coordinates $x_{\alpha}^{2}$ and infinitesimal dynamics by tetrahedral
Complexe (or line elements). This introduces naturally projective
transformations by preserving anharmonic ratios. Referring to old work of
Pl{\"u}cker relating quadratic Complexe to optics, we discuss (linear)
symplectic symmetry and line coordinates, the main purpose and thread within
this paper, however, is the identification and discussion of special relativity
as direct invariance properties of line/Complex coordinates as well as their
relation to 'quantum field theory' by complexification of point coordinates or
Complexe. This can be established by the Lie mapping which relates
lines/Complexe to sphere geometry so that SU(2), SU(2)$\times$U(1),
SU(2)$\times$SU(2) and the Dirac spinor description emerge without additional
assumptions. We give a short outlook in that quadratic Complexe are related to
dynamics e.g.~power expressions in terms of six-vector products of Complexe,
and action principles may be applied. (Quadratic) products like
$F^{\mu\nu}F_{\mu\nu}$ or $F^{a\,\mu\nu}F^{a}_{\mu\nu}$, $1\leq a\leq 3$ are
natural quadratic Complex expressions ('invariants') which may be extended by
line constraints $\lambda k\cdot\epsilon=0$ with respect to an 'action
principle' so that we identify 'quantum field theory' with projective or
line/Complex geometry having applied the Lie mapping.
