This study analyzes the directional properties of geological faults using triangulations to model displaced horizons. We investigate two scenarios: one without elevation uncertainties and one with such uncertainties. Through formal mathematical proofs and computational experiments, we explore how triangular surface data can reveal geometric characteristics of faults. Our formal analysis introduces four propositions of increasing generality, demonstrating that in the absence of elevation errors, duplicate elevation values lead to identical dip directions. For the scenario with elevation uncertainties, we find that the expected dip direction remains consistent with the error-free case. These findings are further supported by computational experiments using a combinatorial algorithm that generates all possible three-element subsets from a given set of points. The results offer insights into predicting fault geometry in data-sparse environments and provide a framework for analyzing directional data in topographic grids with imprecise elevation data. This work has significant implications for improving fault modeling in geological studies, particularly when dealing with limited or uncertain data.