Projection Invariants in Rotating Curve Families: Universality of the Diametral Abel Measure and Variational Characterization of the Circular Family

We study a one-parameter family of circles {C_θ}, each of radius n/4, whose centers rotate on a circle of radius n/4 and that pass through a common point. Using envelope theory and the co-area formula (Federer, 1969), we establish three results. First, the envelope of the family is the circle of diameter n, and the radial superposition density is ρ(r) = 1/√((n/2)² − r²), whose surface integral over the limit disk equals M(n) = nπ. Second, the diametral Abel projection of ρ satisfies P(0) = π independently of n, giving the factorization M(n) = n · P(0), which we identify as the Cauchy–Crofton formula for this circular geometry. Third, and centrally, we prove a universality theorem: for any admissible incidence function g : [0, π/2] → [0, R] (continuous, strictly decreasing, with g(0) = R and g(π/2) = 0), the diametral Abel projection of the associated superposition density is always π, regardless of g. The universal value arises as the total measure of the half-orbit [0, π/2] of the rotation group SO(2) under the Haar measure, and distinguishes from the total mass M_g = n_g · π, where n_g = 2∫₀π/2 g(u) du encodes the geometry of the specific family. We further show that g(u) = R cos u satisfies the Euler–Lagrange equation of the harmonic-oscillator action, a variational observation that connects the circular family to a natural extremal problem. Open questions on envelope structure and singularity classification for general admissible families are stated precisely.