Visualising higher-dimensional space-time and space-scale objects as projections to ℝ3

9 Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object’s shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from R4 to R3. We present three projections that we believe are particularly intuitive for this purpose: (i) a simple ‘long axis’ projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (S3) followed by a stereographic projection to R3, which results in an inwards-outwards fourth axis. Our focus is in using these projections from R4 to R3, but they are formulated from R to Rn−1 so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API. 10 11 12 13 14 15 16 17 18 19 20 21 BACKGROUND 22 Projecting the 3D nature of the world down to two dimensions is one of the most common problems at 23 the juncture of geographic information and computer graphics, whether as the map projections in both 24 paper and digital maps (Snyder, 1987; Grafarend and You, 2014) or as part of an interactive visualisation 25 of a 3D city model on a computer screen (Foley and Nielson, 1992; Shreiner et al., 2013). However, 26 geographic information is not inherently limited to objects of three dimensions. Non-spatial characteristics 27 such as time (Hägerstrand, 1970; Güting et al., 2000; Hornsby and Egenhofer, 2002; Kraak, 2003) and 28 scale (Meijers, 2011a) are often conceived and modelled as additional dimensions, and objects of three or 29 more dimensions can be used to model objects in 2D or 3D space that also have changing geometries 30 along these non-spatial characteristics (van Oosterom and Stoter, 2010; Arroyo Ohori, 2016). For example, 31 a single 4D object can be used to represent the changes in a 3D object’s shape across time or all the best 32 representations of a 3D object at various levels of detail (van Oosterom and Meijers, 2014; Arroyo Ohori 33 et al., 2015a,c). 34 Objects of more than three dimensions can be however unintuitive (Noll, 1967; Frank, 2014), and 35 visualising them is a challenge. While some operations on a higher-dimensional object can be achieved by 36 running automated methods (e.g. certain validation tests or area/volume computations) or by visualising 37 only a chosen 2D or 3D subset (e.g. some of its bounding faces or a cross-section), sometimes there is 38 no substitute for being able to view a complete nD object—much like viewing floor or façade plans is 39 often no substitute for interactively viewing the complete 3D model of a building. By viewing a complete 40 model, one can see at once the 3D objects embedded in the model at every point in time or scale as well 41 as the equivalences and topological relationships between their constituting elements. More directly, it 42 also makes it possible to get an intuitive understanding of the complexity of a given 4D model. 43 For instance, in Fig. 1 we show an example of a 4D model representing a house at two different levels 44 of detail and all the equivalences its composing elements. It forms a valid manifold 4-cell (Arroyo Ohori 45 PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.2844v1 | CC BY 4.0 Open Access | rec: 2 Mar 2017, publ: 2 Mar 2017