Abstract
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 ℝ4to 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 ℝ4to ℝ3, but they are formulated from ℝnto ℝn−1so 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.
Highlights
Projecting the 3D nature of the world down to two dimensions is one of the most common problems at the juncture of geographic information and computer graphics, whether as the map projections in both paper and digital maps (Snyder, 1987; Grafarend and You, 2014) or as part of an interactive visualisation of a 3D city model on a computer screen (Foley and Nielson, 1992; Shreiner et al, 2013)
By viewing a complete model, one can see at once the 3D objects embedded in the model at every point in time or scale as well as the equivalences and topological relationships between their constituting elements
We show that with the help of low-level graphics APIs, all the required operations can be applied at interactive framerates for the 4D to 3D case
Summary
Projecting the 3D nature of the world down to two dimensions is one of the most common problems at the juncture of geographic information and computer graphics, whether as the map projections in both paper and digital maps (Snyder, 1987; Grafarend and You, 2014) or as part of an interactive visualisation of a 3D city model on a computer screen (Foley and Nielson, 1992; Shreiner et al, 2013). Non-spatial characteristics such as time (Hägerstrand, 1970; Güting et al, 2000; Hornsby and Egenhofer, 2002; Kraak, 2003) and scale (Meijers, 2011a) are often conceived and modelled as additional dimensions, and objects of three or more dimensions can be used to model objects in 2D or 3D space that have changing geometries along these non-spatial characteristics (van Oosterom and Stoter, 2010; Arroyo Ohori, 2016). It makes it possible to get an intuitive understanding of the complexity of a given 4D model
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