CATpie: A Python tool for quantifying membrane curvature, area, and geometrically faithful normal-line thickness from...

Membrane shape is commonly described by curvature, area, and thickness, which underpin bilayer mechanics and protein-membrane interactions. In molecular dynamics (MD) simulations, these quantities must be inferred from discrete coordinates, and common grid-based smoothing approaches can introduce resolution-dependent artefacts and ambiguities, especially under periodic boundary conditions. Here we present CATpie, a Python framework for a unified geometric analysis of simulated membranes represented as smooth leaflet height fields over a periodic domain (from user-defined leaflet reference atoms or beads). CATpie fits real Fourier series to the upper and lower leaflets, constructs the mid-surface, and evaluates mean curvature, Gaussian curvature, and real area from analytic expressions of the fitted surfaces. It further defines a normal-line thickness as the distance between the intersections of each mid-surface normal with the two leaflets, providing a geometrically controlled measure of leaflet separation that complements the commonly used vertical separation D_zz. Validation on analytic benchmark surfaces, including a sinusoidal surface with zero Gaussian curvature and a separable cosine surface with non-zero Gaussian curvature, shows high-accuracy recovery of curvature fields, real areas, and leaflet separation. Application to a mixed POPS/POPC bilayer demonstrates that mid-surface curvature and normal-line thickness quantify local deformations and thickness modulations, while the ratio T_norm/D_zz highlights regions where vertical separation overestimates normal-line spacing due to membrane tilt. CATpie supports atomistic and coarse-grained simulations of planar and undulated bilayers that remain representable as single-valued leaflet height fields z=Z(x,y) over a periodic domain (without overhangs or topological changes), enabling systematic geometric characterisation of membrane simulations.