G-Code-Based Geometric Estimation of Minimum Cross-Sections and Their Locations in Material Extrusion: Point-Count Versus Union-Area Representations

Material extrusion (MEX) builds parts from discrete polymer beads, so the cross-section that carries a tensile load is the union of the deposited beads rather than the rectangle measured with a caliper, and it varies with the infill pattern and along the gauge length. Digital models that reconstruct this section from the G-code need an estimator of the load-bearing area that is fast and geometrically faithful. We compare four estimators on identical toolpaths: a line–plane intersection point-count index (RLPI), the additive sum of elliptical bead sections, the rasterized union of those sections and an obliquity-corrected union. The corpus comprises 357 ASTM D638 Type V toolpaths (17 infill patterns, seven densities, three layer heights) cut by 77 transverse planes, with the width of every bead read from the slicer output. Additive summation overestimates the union area by 3.1% to 10.6% because neighboring beads overlap; obliquity adds 0.9% on average. The point-count index and the minimum union area rank the patterns differently: six of the seventeen patterns move by seven positions or more between the two orderings, Hilbert curve from 14th by point count to 3rd by union area and Lightning from 6th to 16th, while the rank correlation between the two orderings is weak and does not reach significance (Spearman ρs = 0.36, p = 0.15, n = 17). The two estimators also locate the minimum section to be 1.83 mm apart on average, 24% of the gauge length. Sensitivity analyses on the sectioning interval, the obliquity cap, the tie-handling rule and the bead cross-section model leave these conclusions unchanged. Within this purely geometric scope, a self-normalized point count does not reproduce the solid area of the section across infill topologies; which estimator better predicts strength and fracture location remains to be tested experimentally.