arXiv:2610.11814v1 Announce Type: new Abstract: We study the relation between equal-weight circle quadrature, Gauss--Chebyshev quadrature, and planar Minkowski reconstruction. The regular $N$-point grid is an $\mathbb S^1$-design of strength $N-1$; a half-step rotation of the $2M$-point grid projects to the classical $M$-point Gauss--Chebyshev rule. A common family of quadrature-error functionals gives exact identities for integration error, mixed-area error, and support-function error. For $N\ge3$, the centered Minkowski polygon $P_N$ with perimeter $2\pi$ has the exact Hausdorff distance from the unit disk $B^2$ \[ d_H(P_N,B^2)=1-\frac{\pi}{N}\cot\frac{\pi}{N} =\frac{\pi^2}{3N^2}+O(N^{-4}). \] With Steiner centering, we obtain $d_H(K_\mu,K_\nu)\le\pi W_1(\mu,\nu)$ for balanced, nondegenerate probability measures, where $W_1$ uses geodesic distance. Its exponent is sharp even for smooth positive densities near the uniform measure. The identity $W_1(\nu_N,\sigma)=\pi/(2N)$ yields a general $O(N^{-1})$ bound for the regular grid. The sharper $O(N^{-2})$ rate follows from the sparse discrepancy spectrum and the inverse multiplier $(1-k^2)^{-1}$. Finally, a perturbation estimate gives sufficient conditions for preserving the quadratic rate and its leading constant.