The main innovation of this article is the determination of the impact of curvature of a shape cut out in a brittle material using an abrasive water jet (AWJ) process as an important factor of the machined surfaces. The curvature of a shape, resulting from the size of the radius of the cutting head trajectory, is one of the key requirements necessary for ensuring the required surface quality of materials shaped by the abrasive water jet process, but very few studies have been carried out in this regard. An important goal of the experimental studies carried out here and presented in this work was to determine its influence on the quality of the inner and outer surfaces of the cutting kerf. This goal was accomplished by cutting the shape of a spiral in soda–lime glass. For such a shape, the effect of radius of the trajectory of the cutting head on selected parameters of the surface texture of the inner surface of the cutting kerf (IS) and the outer surface of the cutting kerf (OS) was studied. The obtained results of the experimental studies confirmed that the effect of the curvature of the cut shape is important from the point of view of the efficiency of the glass-based brittle material-cutting process using AWJ. Analyses of the surface textures of the areas located in the upper part of the inner and outer surfaces separated by the use of AWJ machining showed that the OS surfaces are characterized by worse technological quality compared with IS surfaces. Differences in the total height of surface irregularities (given by St amplitude parameter), determined on the basis of the obtained results of the measurements of both surfaces of the cutting kerf, were as follows: ΔStr = 50 = 0.6 μm; ΔStr = 35 = 1 μm; ΔStr = 15 = 1.3 μm. The analysis of values measured in areas located in the more sensitive zone of influence of the AWJ outflow proved that the total height of irregularities (St) of the OS was higher. Differences in the total heights of irregularities for inner and outer surfaces of the cutting kerf were as follows: ΔStr = 50 = 2.1 μm; ΔStr = 35 = 3 μm; ΔStr = 15 = 14.1 μm, respectively. The maximum difference in the total heights of irregularities (St), existing between the surfaces considered in a special case (radius 15 mm), was almost 20%, which should be a sufficient condition for planning cutting operations, so as to ensure the workpiece is shaped mainly by internal surfaces.