PRH | Educ | 6.6 • Blur Geometry (UG Version): Between Euclid and Lobachevsky

We introduce Blurrichevsky geometry, an observer-aware interpolation between Euclidean and hyperbolic geometry. A scale kernel first produces a blend parameter $\beta \in[0,1]$. Model A then uses the rotational metric

$$
g_\beta=d r^2+w_\beta(r)^2 d \theta^2, \quad w_\beta(r)=(1-\beta) r+\beta \frac{\sinh (\kappa r)}{\kappa},
$$

whose curvature is proved to satisfy $-\kappa^2 \leq K_\beta \leq 0$. Model B gives an equivalent transportfirst description: the Euclidean and hyperbolic Levi-Civita connection one-forms are blended in an orthonormal frame, and the resulting connection reconstructs exactly the same $w_\beta$. Model C is a separate projective realization based on correctly normalized Klein distances on a common disk; averaging these distances gives a genuine metric with straight chord geodesics, generally of Finsler rather than Riemannian type. Its selected observables can be calibrated to those of Model A, but no full metric equivalence is claimed. We distinguish the classical angle of parallelism from the opening of the non-intersection sector, formulate a useful local constant-curvature proxy, and describe observer-dependent landing through band-limited kernels. An appendix explains only where a blur parameter may sit inside a metric ansatz for the Einstein equation: general relativity is not derived from blur, but its established variational structure shows that blur can serve as a legitimate reduced coordinate on a family of geometries.