## 2 TTRS and Mathematical Modeling  
### Technologically and Topologically Related Surfaces (TTRS)  
As mentioned above, we adopted the TTRS model [25] as the main reference to support tolerance analysis models. This paper proposes an interpretation of the theory and a definition of the TTRS concept that is compatible with the definitions provided by other authors, but that presents a clear orientation for SysML modeling.  
The TTRS model is based on the geometric invariance class concept, which classifies surfaces into seven types, among which, the three translation and three rotation degrees of freedom (DoF) can be variant (in which a change modifies the position of the surface in the space) or invariant (in which a change keeps the surface invariant). The seven invariance classes are spherical, cylindrical, planar, helical, revolute, prismatic, and complex. Each invariance class has a minimum set of reference geometric elements, named Minimum Geometric Reference Elements (MGREs). Each MGRE is composed of one, two, or three Reduced Geometric Elements (RGEs), which can be a point, line, or plane, that fulfill some geometrical constraints between them, depending on the invariance class.  
From a technological and topological perspective, surfaces are usually associated. A TTRS is an assembly (set) formed by two surfaces (or a surface and a TTRS, or two TTRSs) belonging to the same solid (topological aspect) and being located in the same kinematic loop in a given mechanism (technological aspect). Moreover, many authors have also transferred this definition to the relationships between surfaces of different solids, introducing the concept of Pseudo TTRS [18].  
A TTRS associates two surfaces through the definition of constraints between their MGREs (or between the RGEs owned by each MGRE). These associations are characterized, giving rise to 44 cases of combinations of surfaces according to the invariance class of each of them and the possible constraints imposed between them. Each of these combinations gives rise to a whole characterization through a new invariance class (reclassification mechanism).  
These 44 cases are shown in Figure 2, where a color code is used to differentiate three particular sets of MGRE associations: association between simple MGRE, composed by one RGE (green); associations with at least one complex MGRE, composed by more than one RGE (yellow); and associations where a helical MGRE participates (gray), which are identified separately, because they are not considered in this work (hence, only the remaining 35 cases were modeled). Differentiation between green and yellow associations is important, because cases with complex MGREs are supported by a combination of different constraints defined in the 13 simple cases.  
From the perspective of our proposal, the analysis of pairwise surfaces and the constraints between them is very interesting for the representation of geometric specifications and tolerances, as has been explored in [27].  
TTRS is suitable for representing tolerance zones (TZ) according to geometric dimensioning and tolerancing (GD&T) standards [28], especially basic cases such as planar, cylindrical, or spherical TZs. These TZs can be defined by their corresponding MGRE (according to their invariance class), and their size (basically width or diameter). According to GD&T standards, more complex TZs can be constructed as a result of the intersection or union of simple TZs.