Context. High-resolution imaging in scattered light has revealed complex morphologies in protoplanetary disks. Measuring their vertical height is key to understanding disk structure, evolution, and the properties of embedded dust. Aims. This work develops a robust method for fitting elliptical shapes to scattered-light images of protoplanetary disks in order to extract vertical height profiles of the dust-scattering surface (τ = 1) across a large and morphologically diverse disk sample. The dataset comprised 92 unique near-infrared polarimetric images of individual systems obtained with VLT/SPHERE. The goal was to identify trends in vertical structure across morphologies and test for correlations with stellar mass, age, and disk dust mass. Using the height profiles, we also investigated the implications of the constrained height for the masses of potential embedded planets. Methods. We implemented a structure extraction and ellipse fitting (SEEF) algorithm using edge detection and Gaussian fitting to locate the structure within protoplanetary disks. Fitting ellipses to the structure revealed spatial offsets from the centre of the ellipse and the star, which we interpreted as the vertical height assuming a circular ring geometry. We also derived the disk inclination, position angle, and aspect ratio (hτ=1/r). Results. The SEEF algorithm obtained successful vertical height measurements for 92 unique disks, revealing variations in height profiles consistent with flared disk geometries. Analysis of the full sample showed that the vertical height profile cannot be confidently described by a single power-law relation. Subdivision of the sample by disk morphology revealed no strong correlations within most categories, with the exception of extended disks (router ≥ 150 au), which exhibited a strong correlation with a single power-law trend. Investigation into underlying disk properties revealed no correlation for its effect to the vertical height structure. Conclusions. We present a consistent method for measuring the vertical structure of circumstellar disks using ellipse fitting on scattered-light images. While trends in height structure remain weakly correlated for the full sample and many disk morphologies, extended disks (router ≥ 150 au) stand out as the only subgroup showing a clear power-law flaring trend. The lack of a strong correlation across other morphologies and with system properties such as stellar or dust mass suggests that either differing disk morphologies exhibit different vertical height profiles or that another, unidentified factor affects the disk flaring.
Byrne et al. (Thu,) studied this question.