Lorentz Transformation as a Two-Dimensional Perspective Structure and Its Direction-Selective Manifestation in Three-Dimensional Space

This paper begins from the fact that the nontrivial geometric action of a Lorentz transformation lies in the 1+1-dimensional plane formed by the time axis and the spatial axis along the direction of motion. On the observer's simultaneity section in this active plane, a proper length L₀ is measured as L=L₀/γ. In two-dimensional central perspective, a length L₀ located at distance R from the observation point is projected onto an image line at reference distance f as L=(f/R)L₀. Defining the two scale factors as pE=f/R and pL=1/γ, the condition R/f=γ gives pE=pL=p, so both minimal-dimensional problems induce the same one-dimensional scaling operator L↦pL. In 3+1 dimensions, this two-dimensional core structure does not disappear or become a different structure. The same p continues to act along the spatial direction coupled to the time axis, while two transverse spatial axes are added and remain invariant. The observer-space operator is Cp=pΠ∥+Π⊥, or Cp≅[p]⊕I₂ in a basis aligned with the direction of motion. Relative to the isotropic extension pI of central perspective, the rank of the matrix difference is exactly the number of added transverse spatial degrees of freedom. Thus, the two phenomena appear different in three dimensions not because their two-dimensional core structures differ, but because additional spatial axes are attached to the same core in different ways. The paper formulates the equality of the two-dimensional induced length operators and their direction-selective three-dimensional manifestation within a single dimension-to-dimension framework.