Abstract
This paper introduces and formalizes a frame of reference for projective relations in 3D space that can be used to model human visual perception. While in 2D space visibility information can be derived from the concept of collinearity (thus, as ternary relations), in 3D space it can be derived from coplanarity, which calls for quaternary relations. Yet, we can retain ternary relations by anchoring our frame to an ubiquitous reference element: a general sense of vertical direction that, on Earth, can be the expression of gravity force or, in other cases, of the asymmetries of an autonomous agent, either human or robotic, that is, its vertical axis. Based on these observations, the presented frame of reference can be used to model projective and visibility information as ternary relations. Granularity and complexity of the models can be adjusted: we present two differently detailed realizations and discuss possible applications in Geographic Information Systems.
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Notes
- 1.
- 2.
If living in an imaginary 2D space, visibility would follow quite different rules as in the perilous life experience of Flatland characters (Abbott 1884).
- 3.
The acceptance area of an n-ary relation, can be defined as a parametric subset of the relation’s domain, having as a parameter a tuple of n − 1 domain objects.
- 4.
Pyramids, if considering polyhedral objects.
- 5.
Acceptance volumes are the generalization to 3D space of the notion of acceptance areas, discussed in Sect. 2.
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Fogliaroni, P., Clementini, E. (2015). Modeling Visibility in 3D Space: A Qualitative Frame of Reference. In: Breunig, M., Al-Doori, M., Butwilowski, E., Kuper, P., Benner, J., Haefele, K. (eds) 3D Geoinformation Science. Lecture Notes in Geoinformation and Cartography. Springer, Cham. https://doi.org/10.1007/978-3-319-12181-9_15
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