We present "φ-geometry," a visual re-parameterization of familiar relations by a single angle on a unit (or scaled) circle: β = sinφ = v/c, γu207b¹ = cosφ, and γβ = tanφ. In this language, key results of special relativity—time dilation, length contraction, the energy—momentum invariant—and collinear velocity addition reduce to elementary trigonometric identities. We also give compact sin/cos forms for basic electromagnetic transformations and the de Broglie pair, introduce an atomic scale φ(Z,n) from v/c ≈ Zα/n, and read the string momentum—winding plane as (cosφ, sinφ). The aim is clarity and pedagogy: one diagram organizes diverse formulas and helps trace transitions across domains. A practical benefit is numerical robustness: switching to the complementary branch (sin ↔ cos) mitigates division-by-zero and precision loss near limiting angles. We do not propose new dynamics. Rather, we present a re-parameterization aimed at clarity and pedagogy. The physical content of SR/GR remains unchanged; what we provide is an alternative geometric language. The accuracy for specific processes and regimes requires further study, analysis, and refinement, which we leave to future work.