The Derivation of 2D+1D Relativistic Kinematics in Projective Space with Time Cones

This article presents a foundational, bottom-up derivation of 2D+1D relativistic kinematics from within the basic primal projective space. Standard projective models of spacetime typically require six-dimensional Grassmannian or Plücker coordinates to track 1-dimensional particle worldlines, an algebraic complexity that obscures the direct geometric interaction with the causal horizon represented by the dual absolute quadric which are represented by 4x4 matrices. To resolve this, we introduce the Time Cone—a geometric device that substitutes 1-dimensional worldlines with localized, codimension-1 tangent envelopes (Velocity Hyperplanes). By modeling the invariant speed of light as an absolute dual quadric, this framework allows kinematic trajectories to be evaluated directly via algebraic incidence. We demonstrate that the Minkowski spacetime metric, the plane of absolute simultaneity, and the canonical Lorentz transformations emerge organically from these pure projective boundary interactions.