The volume-based scale distribution is the 3D extension of 2D surface-based scale distribution originally defined by Dumouchel et al. (2008). It allows characterizing the multi-scale features of shapes with complex morphology and thus appears as an attractive metric for characterizing the primary atomization process where liquid structures are generally not spherical. On the other hand, curved surfaces such as liquid-gas interfaces can also be well represented by differential geometry and the use of intrinsic observables such as the two principal curvatures. In this study, we use results from differential geometry to build analytical bridges between the volume-based scale distribution, the geometry of the liquid-gas interface (namely the surface area, the mean and Gaussian curvatures) and probability density functions of ‘equivalent' systems constituted of either sheets, cylinder of droplets. This framework is then used, on the basis of data from Direct Numerical Simulations to assess the effect of Reynolds and Weber numbers on the structure of liquid jet primary atomization.