We propose a novel algorithm to decompose a 3D object into an atlas of disk-like charts. Decomposition into charts with controlled shape and topology is relevant in many engineering areas, such as spline fitting, compression and re-meshing. We produce our chartifications by jointly exploiting the Reeb graph of a guiding function and its gradient aligned flow paths. The key advancements of our method with respect to similar approaches are: (i) a novel strategy to provably remove all T-junctions; (ii) a stable system to trace flow paths starting far from critical points; (iii) the exploitation of the regularity of certain functions under isometries (e.g., harmonic ones) to produce structurally equivalent chartifications for families of objects posed differently. The charts produced by our system can be of two types: topological quads and topological octagons. Both of them can be easily gridded to produce full quadrilateral meshes, as we demonstrate in the second part of the article.