Point calculus as an ontological basis for representing geometric objects in design

In engineering design, the selection of an appropriate method for representing geometry largely determines the effectiveness of the system being developed. Traditional methods—boundary representations, structural solid geometry, and functional descriptions—remain the principal means of design automation; however, each of these methods captures only a specific aspect of an object, which requires the alignment of their descriptions. This leads to redundancy and increased complexity of data transformation procedures, limits the ability to model objects with complex internal structures, and reduces the versatility of digital models. This paper proposes to consider point calculus as an alternative concept for representing geometric objects. Within the point calculus framework, all geometric objects are considered as ordered sets of points defined through local and global simplices. In the context of solid modeling, this enables the simultaneous description of both the boundary shape and the internal volume of an object. Under this interpretation, boundary representation becomes a special case of the proposed approach, thereby ensuring compatibility with existing computer- aided design systems. The use of point calculus eliminates the separation between geometry and functional properties, making it possible to model anisotropic structures, internal channels, and variable materials within a unified parametric space. Applying point calculus to the representation of design objects supports the integrity of digital models and reduces cognitive load when working with their multiple representations. The proposed approach is particularly promising for applications in digital twins, additive manufacturing, and generative design, where the integration of flexibility, universality, and formal rigor is required.