Projective Parameterization of Moving Diagonal Triangles on the Null Conic via the Cayley Transform

This paper develops a finitist and fully algebraic framework for the projective parameterization of moving diagonal triangles generated by a null quadrangle in Universal Hyperbolic Geometry (UHG). Starting from a fixed nil triangle and a fourth moving null point on the absolute null conic, the paper derives explicit rational coordinate trajectories for the associated diagonal triangle vertices.

The principal result establishes that the moving diagonal vertices are rational projective representations of the scalar Cayley transform

C(t) = (1 - t)/(1 + t).

Crucially, the paper demonstrates that this Cayley parameterization acts as a computationally optimal coordinate system projectively linked to the intrinsic cross-ratio (lambda = 1 - t) of the ordered null quadrangle. This mathematical bridge connects the Cayley coordinate representation of the moving diagonal vertices to the classical anharmonic projective action on the projective line P¹(F), canonically identifying the sixfold reciprocal metrical orbits discovered in prior papers with the classical six-element anharmonic group S3.

The paper proves that the moving diagonal triangle remains fully-right (mutually orthogonal) for all admissible parameter values over arbitrary fields of characteristic not equal to two. The coordinate trajectories are shown to arise from linear projective mappings from the projective line P¹(F) into the projective plane P²(F), forming a rational curve in the moduli space of fully-right triangles in UHG.

Three distinct layers of the Cayley transform are analyzed:

• the scalar 1×1 fractional-linear Cayley transform,

• the rational 2×2 orthogonal Cayley transform generating SO(2,F),

• and the complex Cayley conjugation relating SL(2,R) and SU(1,1).

The paper further studies the algebraic degeneracy structure of the configuration, identifying universally rational degeneration loci at t = 0 and t = 1, contrasting with earlier irrational degeneration structures found in related reciprocal quadrance orbit constructions.

Within the broader finitist program initiated by N. J. Wildberger, the work demonstrates how Lorentzian metric structures, orthogonal projective frames, and parameterized null configurations can be modeled exactly over the rationals or finite fields without reliance on real analysis, transcendental functions, or smooth manifold methods.

This manuscript forms the fourth paper in an ongoing research series investigating reciprocal quadrance structures, null quadrangles, and algebraic dynamical systems in Universal Hyperbolic Geometry.