We discuss natural and canonical deformations between complete, Riemannian metrics of constant curvature, and cone metrics with cone angles an integral multiple of $\pi$. We illustrate with Euclidean constructions which shed light on natural geometry of moduli spaces of hyperbolic surfaces of low genus. Other applications will also be discussed, in particular to CAT(0) piecewise Euclidean spines of hyperbolic link complements, which are naturally unions of Euclidean tori.