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

Higher-dimensional modelling of space, time and scale

There are a great number of models of geographic information, but most consider space, time and scale separately. For instance, space can be modelled using primitive instancing (Foley et al., 1995; Kada, 2007), constructive solid geometry (Requicha & Voelcker, 1977) or various boundary representation approaches (Muller & Preparata, 1978; Guibas & Stolfi, 1985; Lienhardt, 1994), among others. Time can be modelled on the basis of snapshots (Armstrong, 1988; Hamre, Mughal & Jacob, 1997), space–time composites (Peucker & Chrisman, 1975; Chrisman, 1983), events (Worboys, 1992; Peuquet, 1994; Peuquet & Duan, 1995), or a combination of all of these (Abiteboul & Hull, 1987; Worboys, Hearnshaw & Maguire, 1990; Worboys, 1994; Wachowicz & Healy, 1994). Scale is usually modelled based on independent datasets at each scale (Buttenfield & DeLotto, 1989; Friis-Christensen & Jensen, 2003; Meijers, 2011b), although approaches to combine them into single datasets (Gröger et al., 2012) or to create progressive and continuous representations also exist (Ballard, 1981; Jones & Abraham, 1986; Günther, 1988; Van Oosterom, 1990; Filho et al., 1995; Rigaux & Scholl, 1995; Plümer & Gröger, 1997; Van Oosterom, 2005).

As an alternative to the all these methods, it is possible to represent any number of parametrisable characteristics (e.g. two or three spatial dimensions, time and scale) as additional dimensions in a geometric sense, modelling them as orthogonal axes such that real-world 0D–3D entities are modelled as higher-dimensional objects embedded in higher-dimensional space. These objects can be consequently stored using higher-dimensionaldata structures and representation schemes Čomić & de Floriani (2012); Arroyo Ohori, Ledoux & Stoter (2015b). Possible approaches include incidence graphs (Rossignac & O’Connor, 1989; Masuda, 1993; Sohanpanah, 1989; Hansen & Christensen, 1993), Nef polyhedra Bieri & Nef (1988), and ordered topological models Brisson (1993); Lienhardt (1994). This is consistent with the basic tenets of n-dimensional geometry (Descartes, 1637; Riemann, 1868) and topology (Poincaré, 1895), which means that it is possible to apply a wide variety of computational geometry and topology methods to these objects.

In a practical sense, 4D topological relationships between 4D objects provide insights that 3D topological relationships cannot (Arroyo Ohori, Boguslawski & Ledoux, 2013). Also, McKenzie, Williamson & Hazelton (2001) contends that weather and groundwater phenomena cannot be adequately studied in less than four dimensions, and Van Oosterom & Stoter (2010) argue that the integration of space, time and scale into a 5D model for GIS can be used to ease data maintenance and improve consistency, as algorithms could detect if the 5D representation of an object is self-consistent and does not conflict with other objects.

Basic transformations and the cross-product in nD

The basic transformations (translation, scale and rotation) have a straightforward definition in n dimensions, which can be used to move and zoom around a scene composed of nD objects. In addition, the n-dimensional cross-product can be used to obtain a new vector that is orthogonal to a set of other n − 1 vectors in ℝⁿ. We use these operations as a base for nD visualisation and are thus described briefly below.

The translation of a set of points in ℝⁿ can be easily expressed as a sum with a vector $t=\left[t_0,\dots ,t_n\right]$, or alternatively as a multiplication with a matrix using homogeneous coordinates¹ in an (n + 1) × (n + 1) matrix, which is defined as: ${\displaystyle T=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill t_0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill t_1\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill t_n\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right].}$

Scaling is similarly simple. Given a vector $s=\left[s_0,s_1,\dots ,s_n\right]$ that defines a scale factor per axis (which in the simplest case can be the same for all axes), it is possible to define a matrix to scale an object as: ${\displaystyle S=\left[\begin{array}{cccc}\hfill s_0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill s_1\hfill & \hfill \cdots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill s_n\hfill \end{array}\right].}$

Rotation is somewhat more complex. Rotations in 3D are often conceptualised intuitively as rotations around the x, y and z axes. However, this view of the matter is only valid in 3D. In higher dimensions, it is necessary to consider instead rotations parallel to a given plane (Hollasch, 1991), such that a point that is continuously rotated (without changing the rotation direction) will form a circle that is parallel to that plane. This view is valid in 2D (where there is only one such plane), in 3D (where a plane is orthogonal to the usually defined axis of rotation) and in any higher dimension. Incidentally, this shows that the degree of rotational freedom in nD is given by the number of possible combinations of two axes (which define a plane) on that dimension (Hanson, 1994), i.e.  $\left(\begin{array}{c}n\end{array}2\right)$.

Thus, in a 4D coordinate system defined by the axes x, y, z and w, it is possible to define six 4D rotation matrices, which correspond to the six rotational degrees of freedom in 4D (Hanson, 1994). These respectively rotate points in ℝ⁴ parallel to the xy, xz, xw, yz, yw and zw planes: ${\displaystyle R_{xy}=\left[\begin{array}{cccc}\hfill cos\theta \hfill & \hfill -sin\theta \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill sin\theta \hfill & \hfill cos\theta \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]R_{xz}=\left[\begin{array}{cccc}\hfill cos\theta \hfill & \hfill 0\hfill & \hfill -sin\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill sin\theta \hfill & \hfill 0\hfill & \hfill cos\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]}$ ${\displaystyle R_{xw}=\left[\begin{array}{cccc}\hfill cos\theta \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -sin\theta \hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill sin\theta \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill cos\theta \hfill \end{array}\right]R_{yz}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill cos\theta \hfill & \hfill -sin\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill sin\theta \hfill & \hfill cos\theta \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]}$ ${\displaystyle R_{yw}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill cos\theta \hfill & \hfill 0\hfill & \hfill -sin\theta \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill sin\theta \hfill & \hfill 0\hfill & \hfill cos\theta \hfill \end{array}\right]R_{zw}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill cos\theta \hfill & \hfill -sin\theta \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill sin\theta \hfill & \hfill cos\theta \hfill \end{array}\right].}$

The n-dimensional cross-product is easy to understand by first considering the lower-dimensional cases. In 2D, it is possible to obtain a normal vector to a 1D line as defined by two (different) points p⁰ and p¹, or equivalently a normal vector to a vector from p⁰ to p¹. In 3D, it is possible to obtain a normal vector to a 2D plane as defined by three (non-collinear) points p⁰, p¹ and p², or equivalently a normal vector to a pair of vectors from p⁰ to p¹ and from p⁰ to p². Similarly, in nD it is possible to obtain a normal vector to a (n − 1)D subspace—probably easier to picture as an (n − 1)-simplex—as defined by n linearly independent points p⁰, p¹, …, pⁿ⁻¹, or equivalently a normal vector to a set of n − 1 vectors from p⁰ to every other point (i.e.,  p¹, p², …, pⁿ⁻¹) (Massey, 1983; Elduque, 2004).

Hanson (1994) follows the latter explanation using a set of n − 1 vectors all starting from the first point to give an intuitive definition of the n-dimensional cross-product. Assuming that a point pⁱ in ℝⁿ is defined by a tuple of coordinates denoted as $\left(p_0^i,p_1^i,\dots ,p_{n-1}^i\right)$ and a unit vector along the ith dimension is denoted as ${\hat{x}}_i$, the n-dimensional cross-product $\overrightarrow{N}$ of a set of points p⁰, p¹, …, pⁿ⁻¹ can be expressed compactly as the cofactors of the last column in the following determinant: ${\displaystyle \overrightarrow{N}=\left|\begin{array}{ccccc}\hfill \left(p_0^1-p_0^0\right)\hfill & \hfill \left(p_0^2-p_0^0\right)\hfill & \hfill \cdots \hfill & \hfill \left(p_0^{n-1}\right)\hfill & \hfill {\hat{x}}_0\hfill \\ \hfill \left(p_1^1-p_1^0\right)\hfill & \hfill \left(p_1^2-p_1^0\right)\hfill & \hfill \cdots \hfill & \hfill \left(p_1^{n-1}\right)\hfill & \hfill {\hat{x}}_1\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill \left(p_{n-1}^1-p_{n-1}^0\right)\hfill & \hfill \left(p_{n-1}^2-p_{n-1}^0\right)\hfill & \hfill \cdots \hfill & \hfill \left(p_{n-1}^{n-1}\right)\hfill & \hfill {\hat{x}}_{n-1}.\hfill \end{array}\right|}$

The components of the normal vector $\overrightarrow{N}$ are thus given by the minors of the unit vectors ${\hat{x}}_0,{\hat{x}}_1,\dots ,{\hat{x}}_{n-1}$. This vector $\overrightarrow{N}$–like all other vectors—can be normalised into a unit vector by dividing it by its norm $\left(\begin{array}{c}\hfill \overrightarrow{N}\hfill \end{array}\right)$.

Previous work on the visualisation of higher-dimensional objects

There is a reasonably extensive body of work on the visualisation of 4D and nD objects, although it is still more often used for its creative possibilities (e.g., making nice-looking graphics) than for practical applications. In literature, visual metaphors of 4D space were already described in the 1880 sin Flatland: A Romance of Many Dimensions (Abbott, 1884) and A New Era of Thought (Hinton, 1888). Other books that treat the topic intuitively include Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions (Banchoff, 1996) and The Visual Guide To Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces (McMullen, 2008).

In a more concrete computer graphics context, already in the 1960s, Noll (1967) described a computer implementations of the 4D to 3D perspective projection and its application in art (Noll, 1968).

Beshers & Feiner (1988) describe a system that displays animating (i.e. continuously transformed) 4D objects that are rendered in real-time and use colour intensity to provide a visual cue for the 4D depth. It is extended to n dimensions by Feiner & Beshers (1990).

Banks (1992) describes a system that manipulates surfaces in 4D space. It describes interaction techniques and methods to deal with intersections, transparency and the silhouettes of every surface.

Hanson & Cross (1993) describes a high-speed method to render surfaces in 4D space with shading using a 4D light and occlusion, while Hanson (1994) describes much of the mathematics that are necessary for nD visualisation. A more practical implementation is described in Hanson, Ishkov & Ma (1999).

Chu et al. (2009) describe a system to visualise 2-manifolds and 3-manifolds embedded in 4D space and illuminated by 4D light sources. Notably, it uses a custom rendering pipeline that projects tetrahedra in 4D to volumetric images in 3D—analogous to how triangles in 3D that are usually projected to 2D images.

A different possible approach lies in using meaningful 3D cross-sections of a 4D dataset. For instance, Kageyama (2016) describes how to visualise 4D objects as a set of hyperplane slices. Bhaniramka, Wenger & Crawfis (2000) describe how to compute isosurfaces in dimensions higher than three using an algorithm similar to marching cubes. D’Zmura, Colantoni & Seyranian (2000) describe a system that displays 3D cross-sections of a 4D virtual world one at a time.

Similar to the methods described above, Hollasch (1991) gives a simple formulation to describe the 4D to 3D projections, which is itself based on the 3D to 2D orthographic and perspective projection methods described by Foley & Nielson (1992). This is the method that we extend to define n-dimensional versions of these projections and is thus explained in greater detail below. The mathematical notation is however changed slightly so as to have a cleaner extension to higher dimensions.

In order to apply the required transformations, Hollasch (1991) first defines a point from ∈ ℝ⁴ where the viewer (or camera) is located, a point to ∈ ℝ⁴ that the viewer directly points towards, and a set of two vectors $\overrightarrow{up}$ and $\overrightarrow{over}$. Based on these variables, he defines a set of four unit vectors $\hat{a}$, $\hat{b}$, $\hat{c}$ and $\hat{d}$ that define the axes of a 4D coordinate system centred at the from point. These are ensured to be orthogonal by using the 4D cross-product to compute them, such that: ${\displaystyle \hat{d}=\frac{to-from}{\parallel to-from\parallel }}$ ${\displaystyle \hat{a}=\frac{up\times over\times \hat{d}}{\parallel up\times over\times \hat{d}\parallel }}$ ${\displaystyle \hat{b}=\frac{over\times \hat{d}\times \hat{a}}{\parallel over\times \hat{d}\times \hat{a}\parallel }}$ ${\displaystyle \hat{c}=\hat{d}\times \hat{a}\times \hat{b}.}$

Note two aspects in the equations above: (i) that the input vectors $\overrightarrow{up}$ and $\overrightarrow{over}$ are left unchanged (i.e.,  $\hat{b}=\overrightarrow{up}$ and $\hat{c}=\overrightarrow{over}$) if they are already orthogonal to each other and orthogonal to the vector from from to to (i.e.,  to − from), and (ii) that the last vector $\hat{c}$ does not need to be normalised since the cross-product already returns a unit vector. These new unit vectors can then be used to define a transformation matrix to transform the 4D coordinates into a new set of points E (as in eye coordinates) with a coordinate system with the viewer at its centre and oriented according to the unit vectors. The points are given by: ${\displaystyle E=\left[\begin{array}{c}\hfill P-from\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill \hat{a}\hfill & \hfill \hat{b}\hfill & \hfill \hat{c}\hfill & \hfill \hat{d}\hfill \end{array}\right].}$

For an orthographic projection given $E=\left[\begin{array}{ccc}\hfill e_0\hfill & \hfill e_1{e}_2\hfill & \hfill e_3\hfill \end{array}\right]$, the firstthree columns e₀, e₁ and e₂ can be used as-is, while the fourth column e₃ defines the orthogonal distance to the viewer (i.e., the depth). Finally, in order to obtain a perspective projection, he scales the points inwards in direct proportion to their depth. Starting from E, he computes $E^{\prime }=\left[\begin{array}{ccc}\hfill e_0^{\prime }\hfill & \hfill e_1^{\prime }{e}_2^{\prime }\hfill & \hfill e_3^{\prime }\hfill \end{array}\right]$ as:

${\displaystyle e_0^{\prime }=\frac{e_0}{e_{3}tan\vartheta ∕2}}$ ${\displaystyle e_1^{\prime }=\frac{e_1}{e_{3}tan\vartheta ∕2}}$ ${\displaystyle e_2^{\prime }=\frac{e_2}{e_{3}tan\vartheta ∕2}}$ ${\displaystyle e_3^{\prime }=e_3.}$

Where ϑ is the viewing angle between x and the line between the from point and every point as shown in Fig. 2. A similar computation is done for y and z. In E′, the first three columns (i.e.,  $e_0^{\prime }$, $e_1^{\prime }$ and $e_2^{\prime }$) similarly give the 3D coordinates for a perspective projection of the 4D points while the fourth column is also the depth of the point.

‘Long axis’ projection

First we aim to replicate the idea behind the example previously shown in Fig. 1—a series of 3D objects that are shown next to each other, seemingly projected separately with the correspondences across scale or time shown as long edges (as in Fig. 1) or faces connecting the 3D objects. Edges would join correspondences between vertices across the models, while faces would join correspondences between elements of dimension up to one (e.g. a pair of edges, or an edge and a vertex). Since every 3D object is apparently projected separately using a perspective projection to 2D, it is thus shown in the same intuitive way in which a single 3D object is projected down to 2D. The result of this projection is shown in Fig. 3 for the model previously shown in Figs. 1 and in 4 for a 4D model using 3D space with time.

Although to the best of our knowledge this projection does not have a well-known name, it is widely used in explanations of 4D and nD geometry—especially when drawn by hand or when the intent is to focus on the connectivity between different elements. For instance, it is usually used in the typical explanation for how to construct a tesseract, i.e., a 4-cube or the 4D analogue of a 2D square or 3D cube, which is based on drawing two cubes and connecting the corresponding vertices between the two (Fig. 5). Among other examples in the scientific literature, this kind of projection can be seen in Fig. 2 in Yau & Srihari (1983), Fig. 3.4 in Hollasch (1991), Fig. 3 in Banchoff & Cervone (1992), Figs. 1–4 in Arenas & Pérez-Aguila (2006), Fig. 6 in Grasset-Simon, Damiand & Lienhardt (2006), Fig. 1 in Paul (2012) and Fig. 16 in Van Oosterom & Meijers (2014).

Conceptually, describing this projection from n to n − 1 dimensions, which we hereafter refer to as a ‘long axis’ projection, is very simple. Considering a set of points P in ℝⁿ, the projected set of points P′ in ℝⁿ⁻¹ is given by taking the coordinates of P for the first n − 1 axes and adding to them the last coordinate of P which is spread over all coordinates according to weights specified in a customisable vector ${\hat{x}}_n$. For instance, Fig. 3 uses ${\hat{x}}_n=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$, resulting in 3D models that are 2 units displaced for every unit in which they are apart along the n-th axis. In matrix form, this kind of projection can then be applied as $P^{\prime }=P\left[\begin{array}{cc}\hfill I\hfill & \hfill {\hat{x}}_n\hfill \end{array}\right]$.

Orthographic and perspective projections

Another reasonably intuitive pair of projections are the orthographic and perspective projections from nD to (n − 1)D. These treat all axes similarly and thus make it more difficult to see the different (n − 1)-dimensional models along the n-th axis, but they result in models that are much less deformed. Also, as shown in the 4D example in Fig. 6, it is easy to rotate models in such a way that the corresponding features are easily seen.

Based on the description of 4D-to-3D orthographic and perspective projection described from Hollasch (1991), we here extend the method in order to describe the n-dimensional to ( n − 1)-dimensional case, changing some aspects to give a clearer geometric meaning for each vector.

Similarly, we start with a point from ∈ ℝⁿ where the viewer is located, a point to ∈ ℝⁿ that the viewer directly points towards (which can be easily set to the centre or centroid of the dataset), and a set of n − 2 initial vectors ${\vec{v}}_1,\dots ,{\vec{v}}_{n-2}$ in ℝⁿ that are not all necessarily orthogonal but nevertheless are linearly independent from each other and from the vector to − from. In this setup, the ${\vec{v}}_i$ vectors serve as a base to define the orientation of the system, much like the traditional $\overrightarrow{up}$ vector that is used in 3D to 2D projections and the $\overrightarrow{over}$ vector described previously. From the above mentioned variables and using the nD cross-product, it is possible to define a new set of orthogonal unit vectors ${\hat{x}}_0,\dots ,{\hat{x}}_{n-1}$ that define the axes x₀, …, x_n−1 of a coordinate system in ℝⁿ as: ${\displaystyle {\hat{x}}_{n-1}=\frac{to-from}{\parallel to-from\parallel }}$ ${\displaystyle {\hat{x}}_0=\frac{{\vec{v}}_1\times \cdots \times {\vec{v}}_{n-2}\times {\hat{x}}_{n-1}}{\parallel {\vec{v}}_1\times \cdots \times {\vec{v}}_{n-2}\times {\hat{x}}_{n-1}\parallel }}$ ${\displaystyle {\hat{x}}_i=\frac{{\vec{v}}_{i+1}\times \cdots \times {\vec{v}}_{n-2}\times {\hat{x}}_{n-1}\times {\hat{x}}_0\times \cdots \times {\hat{x}}_{i-1}}{\parallel {\vec{v}}_{i+1}\times \cdots \times {\vec{v}}_{n-2}\times {\hat{x}}_{n-1}\times {\hat{x}}_0\times \cdots \times {\hat{x}}_{i-1}\parallel }}$ ${\displaystyle {\hat{x}}_{n-2}={\hat{x}}_{n-1}\times {\hat{x}}_0\times \cdots \times {\hat{x}}_{n-2}.}$

The vector ${\hat{x}}_{n-1}$ is the first that needs to be computed and is oriented along the line from the viewer (from) to the point that it is oriented towards (to). Afterwards, the vectors are computed in order from ${\hat{x}}_0$ to ${\hat{x}}_{n-2}$ as normalised n-dimensional cross products of n − 1 vectors. These contain a mixture of the input vectors ${\vec{v}}_1,\dots ,{\vec{v}}_{n-2}$ and the computed unit vectors ${\hat{x}}_0,\dots ,{\hat{x}}_{n-1}$, starting from n − 2 input vectors and one unit vector for ${\hat{x}}_0$, and removing one input vector and adding the previously computed unit vector for the next ${\hat{x}}_i$ vector. Note that if ${\vec{v}}_1,\dots ,{\vec{v}}_{n-2}$ and ${\hat{x}}_{n-1}$ are all orthogonal to each other, ∀0 < i < n − 1, ${\hat{x}}_i$ is simply a normalised ${\vec{v}}_i$.

Like in the previous case, the vectors ${\hat{x}}_0,\dots ,{\hat{x}}_{n-1}$ can then be used to transform an m × n matrix of mnD points in world coordinates P into an m × n matrix of mnD points in eye coordinates E by applying the following transformation: ${\displaystyle E=\left[\begin{array}{c}\hfill P-from\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill {\hat{x}}_0\hfill & \hfill \cdots \hfill & \hfill {\hat{x}}_{n-1}\hfill \end{array}\right].}$

As before, if E has rows of the form $\left[\begin{array}{ccc}\hfill e_0\hfill & \hfill \cdots \hfill & \hfill e_{n-1}\hfill \end{array}\right]$ representing points, e₀, …, e_n−2 are directly usable as the coordinates in ℝⁿ⁻¹ of the projected point in an n-dimensional to (n − 1)-dimensional orthographic projection, while e_n−1 represents the depth, i.e. the distance between the point and the projection (n − 1)-dimensional subspace, which can be used for visual cues² . The coordinates along e₀, …, e_n−2 could be made to fit within a certain bounding box by computing their extent along each axis, then scaling appropriately using the extent that is largest in proportion to the extent of the bounding box’s corresponding axis.

For an n-dimensional to (n − 1)-dimensional perspective projection, it is only necessary to compute the distance between a point and the viewer along every axis by taking into account the viewing angle ϑ between ${\hat{x}}_{n-1}$ and the line between the to point and every point. Intuitively, this means that if an object is n times farther than another identical object, it is depicted n times smaller, or $\frac{1}{n}$ of its size. This situation is shown in Fig. 7 and results in new $e_0^{\prime },\dots ,e_{n-2}^{\prime }$ coordinates that are shifted inwards. The coordinates are computed as: ${\displaystyle e_i^{\prime }=\frac{e_i}{e_{n-1}tan\vartheta ∕2},\mathrm{for }0\le i\le n-2.}$

The (n − 1)-dimensional coordinates generated by this process can then be recursively projected down to progressively lower dimensions using this method. The objects represented by these coordinates can also be discretised into images of any dimension. For instance, Hanson (1994) describes how to perform many of the operations that would be required, such as dimension-independent clipping tests and ray-tracing methods.

Stereographic projection

A final projection possibility is to apply a stereographic projection from ℝⁿ to ℝⁿ⁻¹, which for us was partly inspired by Jenn 3D (http://www.math.cmu.edu/ fho/jenn/) (Fig. 8). This program visualises polyhedra and polychora embedded in ℝ⁴ by first projecting them inwards/outwards to the volume of a 3-sphere³ and then projecting them stereographically to ℝ³, resulting in curved edges, faces and volumes.

In a dimension-independent form, this type of projection can be easily done by considering the angles ϑ₀, …, ϑ_n−2 in an n-dimensional spherical coordinate system. Steeb (2011, §12.2) formulates such a system as: ${\displaystyle r=\sqrt{x_0^2+\cdots +x_{n-1}^2}}$ ${\displaystyle {\vartheta }_i={cos}^{-1}\left(\frac{x_i}{\sqrt{r^2-\sum _{j=0}^{i-1}{x}_j^2}}\right),\mathrm{for}0\le i<n-2}$ ${\displaystyle {\vartheta }_{n-2}={tan}^{-1}\left(\frac{x_{n-1}}{x_{n-2}}\right).}$

It is worth to note that the radius r of such a coordinate system is a measure of the depth with respect to the projection (n − 1)-sphere Sⁿ⁻¹ and can be used similarly to the previous projection examples. The points can then be converted back into points on the surface of an (n − 1)-sphere of radius 1 by making r = 1 and applying the inverse transformation. Steeb (2011, §12.2) formulates it as: ${\displaystyle x_i=rcos{\vartheta }_i\prod _{j=0}^{i-1}sin{\vartheta }_j,\mathrm{for}0\le i<n-2}$ ${\displaystyle x_{n-1}=r\prod _{j=0}^{n-2}sin{\vartheta }_j.}$

The next step, a stereographic projection, is also easy to apply in higher dimensions, mapping an (n + 1)-dimensional point x = (x₀, …, x_n) on an n-sphere Sⁿ to an n-dimensional point x′ = (x₀, …, x_n−1) in the n-dimensional Euclidean space ℝⁿ. Chisholm (2000) formulates this projection as: ${\displaystyle x_i^{\prime }=\frac{x_i}{x_n-1},\mathrm{for }0\le i<n.}$

The stereographic projection from nD to (n − 1)D is particularly intuitive because it results in the n-th axis being converted into an inwards-outwards axis. As shown in Fig. 9, when it is applied to scale, this results in models that decrease or increase in detail as one moves inwards or outwards. The case with time is similar: as one moves inwards/outwards, it is easy to see the state of a model at a time before/after.