The search for admissible stress states in masonry structures has motivated several researchers to model internal forces in masonry structures as thrust lines (e.g., [9; 10; 11; 12; 13; 14]), or as thrust surfaces (e.g., [15; 16; 17; 18]) or thrust networks (e.g., [19; 20; 21]) to account for the full 3D mechanical responses of masonry vaults. This work focuses on the last method, modelling the internal force equilibrium as compressive thrust networks, contained within the vault's envelope, defined by its intrados and extrados. With such a method, a connected network is considered, in which the compressive forces are lumped along the edges, and the loads are applied to the vertices of the network. Recent work has framed the search for thrust networks in different optimisation processes [21; 22; 23; 24]. In [25] the main objective was to calculate the geometric safety factor (GSF) values of vaults by directly computing their minimum thicknesses. The GSF is defined as the ratio between the actual thickness of the structure and its limit thickness. The limit, or minimum thickness, is defined as the smallest thickness, as an offset of the initial geometry, for which the structure is still stable; i.e., an admissible stress state can still be found [8]. By calculating the GSF, one can quantify the level of stability of a given masonry structure [5]. Many contributions are available in the literature that deal with the problem of determining the minimum thicknesses of arches, which yields the well-known Couplet problem [14; 26]. In [27; 28; 29], pointed arches were considered, and in [30], the Couplet problem was solved for parametric pointed arches, where the author found that for specific radius-over-length values, the thickness of the arch attains a global minimum when a 7-hinge mechanism in the arch is encountered. Although the solution of the Couplet problem is well-known for arches, its extension to 3D geometries and particularly for pointed groin vaults remains open in the literature. The extensive literature review on the analysis of groin vaults has shown that the problem in 3D is intricate, since multiple force flows are possible due to the high indeterminacy of the problem [21]. In [31], the minimum thrust of rounded groin vaults was calculated using the kinematic theorem of limit analysis. In [32], a funicular model was used to explore different stress states on cross vaults. In [33], discrete-element modelling (DEM) was used to compute the collapse state of masonry vaults subjected to self weight, and support displacement. Additionally, several works discussed the role of ribs in cross vaults [34; 35]. Given the lack of literature about the minimum thickness problem in groin vaults, this paper aims to fill this gap using the thrust network analysis (TNA)-based optimisation framework described in [25]. A parametric investigation is developed describing the vault's geometry with respect to different radius-over-length ratios and springing angles. This approach allows investigating which the relevant geometric parameters are that influence the stability of groin vaults. Moreover, special attention is paid to connecting the outcome of the proposed research to the 2D parametric analyses available in the literature, particularly [30]. The GSF evaluation is not sufficient by itself to fulfil all requirements for a proper structural assessment [2]. Therefore, to give additional information about the robustness of the structure, in the present paper stability domains are also defined. This domain measures the size of the space of admissible stress states by computing the vault's minimum and maximum thrusts and observing how these evolve towards the limit state. The stability domain is computed for some relevant specific geometries, extracted from the proposed parametric study. The implementation of this paper considers the flow of forces, i.e., the topology, and geometry of the thrust network, fixed in its horizontal projection, as in [21]. The equilibrium is computed by making use of independent force densities [36].