A Geometric Reparameterization of Ultra-Relativistic Kinematics and Its Implications for Numerical Stability

We present a geometric angular reparameterization of ultra-relativistic kinematics in which the energy–momentum invariant is expressed through bounded trigonometric projections on the unit circle. In dimensionless form, this representation makes explicit the circular structure underlying standard Lorentz transformations, velocity relations, and rapidity, while remaining fully equivalent to the conventional formulation of special relativity. Beyond its geometric interpretation, the parametrization provides a structured framework for analyzing subtraction-sensitive expressions that arise in ultra-relativistic regimes. In finite-precision arithmetic, quantities such as E−p, which appear in high-energy expansions and mass-dependent observables, may suffer from significant loss of significance when the Lorentz factor becomes large. Within the angular representation, algebraically equivalent forms can be derived in which subtractive cancellation is avoided, leading to improved numerical conditioning without introducing additional approximations or extended precision. Double-precision numerical tests demonstrate that the stabilized formulation preserves the correct asymptotic behavior up to Lorentz factors of order 10¹², whereas direct subtraction degrades substantially at much lower ultra-relativistic scales. The proposed reparameterization does not modify the underlying physical theory; rather, it reorganizes the computational structure in a manner that may be relevant for precision-oriented ultra-relativistic calculations in high-energy physics.