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

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.

An “Animated Spatial Time Machine” in Co-Creation: Reconstructing History Using Gamification Integrated into 3D City Modelling, 4D Web and Transmedia Storytelling

“…Time is not always seen as the 4th dimension; Ohori et al promote the Level Of Detail (LOD) as the 4th dimension [34]. By seeing time and scale as higher dimensions, changes in 3D objects or various LOD could be integrated [35]. Time must ideally be part of the higher dimensionality and is not only an attribute of the Euclidean 3D space [36].…”

Interactive 4-D Visualization of Stereographic Images From the Double Orthogonal Projection

“…Commented videos with interactive environment discussing quaternions and stereographic projection are at [10]. A stereographic and double-stereographic projection of an arbitrary object in the 4-space was described in [4]. As an example of a recent application, a stereographic projection to a 3-sphere is used in [16] to analyze multiple disconnected anatomical structures mathematically represented as a composition of compact finite three-dimensional surfaces.…”