{
  "abstract": "Special Relativity is traditionally taught through hyperbolic spacetime geometry, where coordinate speed is asymptotically bounded by c and Lorentz boosts correspond to hyperbolic rotations parameterized by rapidity φ. While mathematically standard, the non-compact nature of hyperbolic functions introduces well-documented pedagogical obstacles for introductory and intermediate physics students. In this work, we develop a comprehensive pedagogical framework for single-frame relativistic kinematics based on the exact velocity identity:\n\nv² + 𝕋² = c²\n\nwhere 𝕋 ≡ c(dτ/dt) = c/γ represents the chronokinetic velocity—the coordinate rate of proper-time progression within an inertial frame. Building upon the intuitive spacetime heuristics of Epstein and the trigonometric parameterizations of Bartlett, we formalize the Gudermannian bridge (θ = gd(φ)) into a unified instructional curriculum.\n\nThe kinematic pitch angle θ = arcsin(v/c), measured as a zenith angle from the vertical proper-time axis, maps the velocity domain onto a compact interval θ ∈ [0, π/2), unifying: • The Lorentz factor: γ = sec θ • Relativistic spatial momentum: p = mc tan θ • Total relativistic energy: E = mc² sec θ • The metric dispersion invariant: (mc²)² + (pc)² = E² ⟺ 1 + tan²θ = sec²θ\n\nThis formulation eliminates the problematic historical construct of \"relativistic mass\" by attributing the nonlinear growth of momentum to the geometric depletion of the orthogonal proper-time rate 𝕋. Furthermore, we demonstrate that the longitudinal relativistic optical Doppler factor simplifies from a radical expression into an elementary stereographic half-angle projection: D = sec θ + tan θ = tan(π/4 + θ/2). Inelastic collision thresholds are similarly solved via straightforward right-triangle bookkeeping without frame transformations. We explicitly delineate the operational boundaries of this representation: while multi-frame spacetime transformations remain fundamentally governed by the pseudo-Riemannian Minkowski metric (+, -, -, -), the circular ledger provides an intuitive, mathematically exact tool for single-frame state analysis, energy-momentum bookkeeping, and optical Doppler phenomena.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Brett A. Bray",
      "orcid": "0009-0008-6926-8378"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/16cdd4xb9npdm.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23671047",
  "fulltext_url": null,
  "id": "16cdd4xb9npdm",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23671047",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23671047/files/Bray_2025_Circular_Relativistic_Kinematics.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/16cdd4xb9npdm.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:20:47Z",
      "native_id": "23671047",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23671047"
    }
  ],
  "subjects": [],
  "title": "A Circular Geometric Framework for Teaching Relativistic Kinematics and Dynamics",
  "title_text": "A Circular Geometric Framework for Teaching Relativistic Kinematics and Dynamics",
  "type": "preprint",
  "updated_at": "2026-11-09T12:20:47Z",
  "url": "https://research.draconitic.com/paper/16cdd4xb9npdm",
  "venue": "Zenodo",
  "versions": []
}