Bidirectional mapping between rhombic triacontahedron and icosahedral hexagonal discrete global grid systems

ABSTRACT

The icosahedron is currently the mainstream polygon in research and application of discrete global grid systems (DGGS). However, compared to the rhombic triacontahedron (RT), the icosahedron has disadvantages, such as lower sphere-fitting accuracy, greater projection distortion, and difficulty in incorporating the matrix structure for geospatial data storage. More importantly, the special positional relationship between the rhombic triacontahedron and the Earth enables it to effectively support event simulations related to geographical locations. To this end, bidirectional mapping of the hexagonal grid between the RT and icosahedron was proposed, which can efficiently integrate the existing datasets and algorithms of icosahedral DGGS into RT DGGS, thereby achieving seamless conversion between heterogeneous grid systems. We established geometric and topological correlations between the RT and icosahedron, abstracted the spatial algebraic structures of hexagonal grids on the two different polygons, and constructed mapping relationships between them. Finally, conversion between heterogeneous grid indices was achieved using dual quaternions. Experiments revealed that the proposed method was 3.9150 and 2.8151 times more efficient at grid conversion from RT to icosahedron and from icosahedron to RT, respectively, than was a method using latitude/longitude coordinates as a medium.

KEYWORDS:

• Discrete global grid systems
• grid conversion
• hexagonal grid
• polygon

1. Introduction

The rapid development of Earth observation technology and systems as well as satellite, airborne, and ground remote sensing systems provides data with high spatial, temporal, and radiometric resolutions (Guo, Zhang, and Zhu 2015). The exponential growth of data has led to the availability of ‘big data’, which exhibits multi-source, multi-resolution, multi-dimension, and spatial–temporal characteristics (Lee and Kang 2015; Li 2016). Discrete global grid systems (DGGS), considered the framework for the next-generation Digital Earth, utilize regular geometries to recursively divide the surface of Earth and form multiresolution reference systems. These systems also support large-scale massive data storage, processing, and analysis (Huang et al. 2024; Peterson et al. 2015; Purss et al. 2016; Thompson et al. 2022).

The mainstream cell geometries used in previous studies on polyhedron-based DGGS were the (1) triangle, (2) rhombus, and (3) hexagon (Uher 2019). As the shape closest to a circle, the hexagon has higher coverage efficiency, higher angular resolution, more directions, and less quantization error than do the triangle and rhombus, which are widely used in DGGS research at present (Sahr 2011; Stough et al. 2020).

Platonic solids have long been the preferred choice for polyhedron-based DGGS, owing to their highly symmetrical geometric structures and excellent geometric properties (Figure 1). Based on the cube, Gibb (2016) extended HEALPix to an ellipsoid of revolution and developed rHEALPix. Dutton (1999) proposed a quaternary triangular mesh based on the octahedron that is widely used in global data organization and management. Ben et al. (2015) proposed a construction algorithm for hexagonal grid systems based on the octahedron. Among all Platonic solids, the icosahedron is the most studied and used polygon at present, owing to its superior spherical approximation efficiency and regular triangular faces. Sahr, White, and Kimerling (2003) constructed the icosahedral Snyder equal area aperture 3 hexagonal grid (ISEA3H) and proposed central pace indexing (Sahr 2019); Ben et al. (2018) proposed an algebraic encoding scheme for aperture 3 icosahedral hexagonal DGGS; Peterson (2013) designed the hierarchical coding method PYXIS for ISEA3H; Mahdavi-Amiri, Harrison, and Samavati (2015) combined triangles on an icosahedron to form ‘virtual rhombuses’ and realized efficient visualization of a hexagonal grid; and Zhou et al. (2023) developed the hexagon hierarchy on uniform tiles and designed efficient cell navigation methods.

Figure 1. Five Platonic solids. Except for dodecahedron, the remaining Platonic solids are widely applied in the construction of polyhedral DGGSs due to their highly symmetrical geometric structures and regular surfaces.

Constrained by the number of faces and inherent geometric properties of the icosahedron, most existing grid systems struggle to achieve high spherical approximation efficiency. To this end, Hall et al. (2020) proposed the Disdyakis triacontahedron DGGS. Bernardin et al. (2011) proposed Crusta for real-time visualization of sub-meter digital topography based on the rhombic triacontahedron (RT), and Liang et al. (2022) proposed a construction method for an RT DGGS in which the spherical cells have less distortion and higher compactness than those of the icosahedron. Overall, the RT has a higher spherical approximation efficiency than the icosahedron, and its rhombic surfaces can be effectively integrated with quadtree-based structures for data storage and processing purposes (Wang et al. 2023). Moreover, the Open Geospatial Consortium has attempted to establish specifications for the interoperability of DGGS (Alderson et al. 2020; Huang et al. 2024; Purss et al. 2016). However, various grid systems have been designed based on specific requirements and interoperation between them must be achieved by latitude/longitude as the medium, which is efficient. Bidirectional mapping can integrate the existing datasets and algorithms of icosahedral DGGS into RT DGGS to make use of the geometric advantages of the RT for spatial analysis. Furthermore, mapping provides a new approach to the interoperability of heterogeneous grids.

To this end, we (a) introduce the concept of homogeneous coordinates in computer vision which represent points in projective geometry utilizing an additional coordinate, allowing for the seamless handling of transformations and representing points at infinity. Furthermore, we also expand two-dimensional coordinates on a polyhedral surface into quaternion form; (b) establish geometric topological correlations between the icosahedron and RT and utilize dual quaternions which are simply the unification of dual-number theory with hypercomplex numbers (Kenwright 2012), to express the mapping relationship; (c) abstract the algebraic structure of hexagonal cells onto an RT and icosahedron, and rigorously prove the existence of a bijection between the two types of grids.

2. Basic concepts

The aperture refers to the area ratio between cells at successively coarser grid resolutions, where a hexagonal grid can be constructed for any aperture $i^2+\mathrm{ij}+j^{2}i,j\in {\mathbb{Z}}^{+}$ (Purss et al. 2019). Simultaneously, hexagonal cells can be classified into two types (Sahr 2008): cells parallel to edges (Class I) and cells perpendicular to edges (Class II), as shown in Figure 2.

Figure 2. Two different types of hexagonal cell.

In this study, the RT and icosahedron Snyder equal area projections (Snyder 1992) are defined as RTEA (Liang et al. 2022) and ISEA, respectively, and a hexagonal grid with aperture X is denoted by XH. This study focused on ISEA -4H -CI, ISEA -4H -CII, RTEA -4H -CI, and RTEA -4H -CII.

3. Bidirectional mapping of hexagonal grids

The fundamental goal of conversion between different polyhedral grids is to establish a one-to-one correspondence relationship between discrete points in the polyhedral spaces. Utilizing the intrinsic geometric connections between different polyhedrons, a global coordinate system (GCS) was established to connect different local coordinate system (LCS), ultimately completing the construction of the mapping function.

3.1. Geometric topological correlations between the icosahedron and RT

Let the edge length of the RT be denoted by $L_1$, the obtuse angle of the rhombic surface by $\alpha$, and the surface area of the RT by $S_{\mathrm{RT}}=30\sin \alpha L_1^2$. Let the icosahedron, denoted by ICO, with a surface area equal to that of the RT, have an edge length $L_2$, an angle of triangular faces $\text{β}$, and a surface area $S_{\mathrm{ICO}}=5\sqrt{3}{L}_2^2$. Based on the condition of equal surface area $S_{\mathrm{ICO}}=S_{\mathrm{RT}}$, the following can be easily obtained as. (1) $L_2=\sqrt{2\sqrt{3}\sin \alpha }{L}_1,$(1) where $\alpha ={116.57}^{\circ }$.

Connecting the long diagonals of the rhombuses on the $\mathrm{RT}$ forms its dual icosahedron, denoted as ICO’, with edge length $L{{}^{\mathrm{\prime }}}_2$, as illustrated in Figure 3(a) and (b). The centers of ICO and ICO’ are located at the same position in space, and the ratio $k$ between $L_2$ and $L{{}^{\mathrm{\prime }}}_2$ is expressed as. (2) $k={\displaystyle \frac{L{{}^{\mathrm{\prime }}}_2}{L_2}={\displaystyle \frac{2\sin {\displaystyle \frac{\alpha }{2}}}{\sqrt{2\sqrt{3}\sin \alpha }},}}$(2)

Figure 3. RT and its dual icosahedron. (a) Triangular surface formed by the long diagonals of rhombuses on the RT (b) The dual icosahedron of the RT.

3.2. Mapping relationship between icosahedral and RT coordinate systems

Polyhedron-based DGGS divides a polyhedron into several sub-domains using basic surfaces (triangles, rhombuses) and establishes a local planar coordinate system in each sub-domain to describe the position of cells (Robertson et al. 2020).

A LCS $O-{\mathbf{i}}^{\mathrm{\prime }}{\mathbf{j}}^{\mathrm{\prime }}{\mathbf{k}}^{\mathrm{\prime }}$ was constructed by taking the center $O$ of the two-dimensional basic surface as the origin and building orthogonal imaginary axes $\mathbf{i}{}^{\mathrm{\prime }},\mathbf{j}{}^{\mathrm{\prime }}$as well as an imaginary axis $\mathbf{k}{}^{\mathrm{\prime }}$ along the normal direction of the surface. A GCS $O_{\mathrm{Global}}-\mathbf{ijk}$ was also established by considering the center $O_G$ of the polyhedron as the origin, and any point $P$ on the polyhedral surface could be represented as a pure quaternion $P=\left[0,\mathbf{v}\right],\mathbf{v}=[x,y,z{]}^T$.

Let $f_{\mathrm{Global}-\mathrm{RT}}$, $f_{\mathrm{RT}-\mathrm{Global}}$ denote the mapping between the LCS and GCS of the RT, and $f_{\mathrm{Global}-\mathrm{ICO}{}^{\mathrm{\prime }}}$, $f_{\mathrm{ICO}{}^{\mathrm{\prime }}-\mathrm{Global}}$ denote the mapping between the LCS and GCS of the ICO’. Let $f_{\mathrm{RT}-\mathrm{ICO}{}^{\mathrm{\prime }}}$, $f_{\mathrm{ICO}{}^{\mathrm{\prime }}-\mathrm{RT}}$ denote the mapping between the LCS of the RT and ICO’, as shown in Figure 4. The above mapping has a Jacobian of 1, indicating that the transformation satisfies the property of equal areas.

Figure 4. Mapping relationship between the LCS and GCS of the RT and ICO’.

3.2.1. Mapping between the LCS and GCS

We denoted the rhombic surface of the RT ${\mathrm{\Pi }}_s,s\in \left\{1,2\dots 30\right\}$, the vertices composing each rhombus $v_m=\left(x_m,y_m,z_m\right),m\in \left\{1,2,3,4\right\}$, the center coordinate $v_{\mathrm{center}}={\displaystyle \frac{\sum v_m}{4}}$, and the normal vector $\overrightarrow{F}=\left(A,B,C\right)$. We denoted the triangular surface of the ICO $\mathrm{\Pi }{{}^{\mathrm{\prime }}}_s,s\in \left\{1,2\dots 20\right\}$, the vertices of each triangle $v{{}^{\mathrm{\prime }}}_m=\left(x{{}^{\mathrm{\prime }}}_m,y{{}^{\mathrm{\prime }}}_m,z{{}^{\mathrm{\prime }}}_m\right),m\in \left\{1,2,3\right\}$, the center $v{{}^{\mathrm{\prime }}}_{\mathrm{center}}={\displaystyle \frac{\sum v{{}^{\mathrm{\prime }}}_m}{3}}$, and the normal vector $\overrightarrow{F}{}^{\mathrm{\prime }}=\left(A{}^{\mathrm{\prime }},B{}^{\mathrm{\prime }},C{}^{\mathrm{\prime }}\right)$. The normal vector of the GCS was denoted by ${\overrightarrow{F}}_G=\left(A_G,B_G,C_G\right)=\left(0,0,1\right)$.

Using linear algebra, the analytical expressions for ${\mathrm{\Pi }}_i$ and $\mathrm{\Pi }{{}^{\mathrm{\prime }}}_i$ can be obtained as. (3) $\mathrm{\Pi }=\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1\end{array}\right|=\mathrm{Ax}+\mathrm{By}+\mathrm{Cz}+D=0,$(3) (4) ${\mathrm{\Pi }}^{\mathrm{\prime }}=\left|\begin{array}{ccc}x-x{{}^{\mathrm{\prime }}}_1& y-y{{}^{\mathrm{\prime }}}_1& z-z{{}^{\mathrm{\prime }}}_1\\ x{{}^{\mathrm{\prime }}}_2-x{{}^{\mathrm{\prime }}}_1& y{{}^{\mathrm{\prime }}}_2-y{{}^{\mathrm{\prime }}}_1& z{{}^{\mathrm{\prime }}}_2-z{{}^{\mathrm{\prime }}}_1\\ x{{}^{\mathrm{\prime }}}_3-x{{}^{\mathrm{\prime }}}_1& y{{}^{\mathrm{\prime }}}_3-y{{}^{\mathrm{\prime }}}_1& z{{}^{\mathrm{\prime }}}_3-z{{}^{\mathrm{\prime }}}_1\end{array}\right|=A{}^{\mathrm{\prime }}x+B{}^{\mathrm{\prime }}y+C{}^{\mathrm{\prime }}z+D{}^{\mathrm{\prime }}=0.$(4)

Dual quaternions have been utilized to achieve a unified implementation of rotation and translation, providing a compact, singularity-free, concise, and rigid transformation (Kenwright 2012). Therefore, they were introduced to simplify mapping operations and improve conversion efficiency and were expressed as. (5) $\mathbf{q}={\mathbf{q}}_r+{\mathbf{q}}_dϵ=\left[{\mathbf{q}}_r,{\mathbf{q}}_d\right]\mathrm{with}ϵ\ne 0,{ϵ}^2=0,$(5) where ${\mathbf{q}}_r,{\mathbf{q}}_d$ are quaternions representing the real and dual parts, respectively, and $ϵ$ is the dual operator.

Typically, a unit dual quaternions is utilized to represent displacement, where the real part ${\mathbf{q}}_r$ denotes the pure rotational process and the dual part ${\mathbf{q}}_d$ the pure translational process and is formulated as. (6) ${\mathbf{q}}_r=\left[\mathbf{r},0\right],{\mathbf{q}}_d=\left[1,{\displaystyle \frac{1}{2}\mathbf{tr}}\right],$(6) where $\mathbf{r}$ and $\mathbf{t}$ are pure quaternions.

By combining unit dual quaternions for translation and rotation into $\mathbf{q}$, any point $\mathbf{p}$ that undergoes rigid body motion (Figure 5) can be represented as $\mathbf{p}{}^{\mathrm{\prime }}$, shown as. (7) $\mathbf{q}={\mathbf{q}}_d\times {\mathbf{q}}_r,$(7) (8) $\mathbf{p}{}^{\mathrm{\prime }}=\mathbf{qp}{\mathbf{q}}^{\ast }.$(8)

Taking mapping from the LCS to GCS as an example, we decomposed the transformation into two processes: rotation ($u_r$) and translation ($u_t$). The offset of the two different coordinate origins was defined as $\overrightarrow{t}=\left(x_{\mathrm{offset}},y_{\mathrm{offset}},z_{\mathrm{offset}}\right)$, and $u_t$ was represented by the unit dual quaternion. (9) $u_t=\left[1,{\mathbf{q}}_t\right],{\mathbf{q}}_t=\left[0,{\displaystyle \frac{1}{2}\overrightarrow{t}}\right].$(9)

$u_r$ was further decomposed into rotation $u_{q_1}=\left[{\mathbf{q}}_{r_1},0\right]$ around the imaginary axis $\mathbf{k}$ with angle $\theta$ and rotation $u_{q_2}=\left[{\mathbf{q}}_{r_2},0\right]$ around the intersection plane of $O-\mathbf{ij}$ and $O_G-\mathbf{IJ}$ with angle $\phi$. Let the projection of the axis $\mathbf{k}{}^{\mathrm{\prime }}$ of the LCS into the GCS be denoted as ${\overrightarrow{v}}_{\mathrm{proj}}$, and let the vector of $\mathbf{k}$ be denoted as $\overrightarrow{v}$. $\theta$ and $\phi$ were both obtained as (10) ${\overrightarrow{v}}_{\mathrm{proj}}=\left(x_j-A_G{\displaystyle \frac{w}{l},y_j-B_G{\displaystyle \frac{w}{l},z_j-C_G{\displaystyle \frac{w}{l}}}}\right),$(10) (11) $\theta =\arccos {\displaystyle \frac{{\overrightarrow{v}}_{\mathrm{proj}}\cdot \overrightarrow{v}}{\left|{\overrightarrow{v}}_{\mathrm{proj}}\right|\left|\overrightarrow{v}\right|},}$(11) (12) $\phi =\arccos {\displaystyle \frac{\left|r\right|}{\sqrt{l\cdot l{}^{\mathrm{\prime }}}},}$(12) where $w=A_G\cdot x_j+B_G\cdot y_j+C_G\cdot z_j+D_G$, $l=A_G^2+B_G^2+C_G^2$, $l{}^{\mathrm{\prime }}=A^2+B^2+C^2$, $r=A_G\cdot A+B_G\cdot B+C_G\cdot C$.

Figure 5. Rigid body motion in different coordinate systems.

Then, it is easy to obtain $u_r$, which is presented by. (13) ${\mathbf{q}}_{r_1}=\left[\cos {\displaystyle \frac{\theta }{2},\sin {\displaystyle \frac{\theta }{2}{\overrightarrow{n}}_1}}\right],{\overrightarrow{n}}_1=\left(0,0,1\right),$(13) (14) ${\mathbf{q}}_{r_2}=\left[\cos {\displaystyle \frac{\phi }{2},\sin {\displaystyle \frac{\phi }{2}{\overrightarrow{n}}_2}}\right],{\overrightarrow{n}}_2=\overrightarrow{F}\times {\overrightarrow{F}}_G,$(14) (15) $u_r=u_{q_2}{u}_{q_1}.$(15)

Finally, any point ${\mathbf{q}}_{\mathrm{Local}}=\left[0,x_l,y_l,0\right]$ in the LCS can be converted to a global coordinate ${\mathbf{q}}_{\mathrm{Global}}=\left[0,x_g,y_g,z_g\right]$ through the unit dual quaternion $\mathbf{u}$, which is formulated by. (16) ${\mathbf{q}}_{\mathrm{Global}}=\mathbf{u}{\mathbf{q}}_{\mathrm{Local}}{\mathbf{u}}^{\ast },$(16) where $\mathbf{u}=u_t\times u_r$.

According to the properties of the unit dual quaternion, inverse transformation can also be easily implemented.

3.2.2. Mapping between the LCS of the RT and ICO’

Based on Section 3.2.1, we further investigated the correspondence relationship between the LCS of the RT and ICO’. The triangular surface ${\mathrm{\Pi }}_{T_n},n\in \left\{1,2\dots 20\right\}$ of the ICO’ was divided into three regions: $T_{n_1}$, $T_{n_2}$, and $T_{n_3}$ (Figure 6). The correspondence relationship is presented as. (17) $\{\begin{array}{c}{T}_{n_1}=\left\{x⩾0\cap y⩾0\cap y<{\displaystyle \frac{1}{\sqrt{3}}x}\right\}\\ T_{n_2}=\left\{x<0\cap y<0\cap y>{\displaystyle \frac{-1}{\sqrt{3}}x}\right\}\\ T_{n_3}=\left\{y⩾{\displaystyle \frac{1}{\sqrt{3}}x\cup y⩽{\displaystyle \frac{-1}{\sqrt{3}}x}}\right\}\end{array},$(17) where ${\mathrm{\Pi }}_{T_n}=T_{n_1}\cup T_{n_2}\cup T_{n_3}$ and $T_{n_1}\cap T_{n_2}\cap T_{n_3}=\mathrm{\varnothing }$.

Figure 6. Correspondence relationship between regions on the icosahedron and RT.

Similarly, the rhombic surface ${\mathrm{\Pi }}_{D_m},m\in \left\{1,2\dots 30\right\}$ of the RT was divided into two regions, denoted by $D_{m_1}$ and $D_{m_2}$ (Figure 6). The correspondence relationship is. (18) $\{\begin{array}{c}{D}_{m_1}=\left\{x⩾0\right\}\\ D_{m_2}=\left\{x<0\right\}\end{array},$(18) where ${\mathrm{\Pi }}_{D_m}=D_{m_1}\cup D_{m_2}$ and $D_{m_1}\cap D_{m_2}=\mathrm{\varnothing }$.

Because direct conversion between the two LCSs is not possible, we chose the GCS as the medium to realize the mapping. $f_{\mathrm{RT}-\mathrm{ICO}}$ and $f_{\mathrm{ICO}-\mathrm{RT}}$ can be achieved in two steps: first by converting from the initial LCS to the GCS using the unit dual quaternion ${\mathbf{u}}_1$, and second from the GCS to the target LCS using the unit dual quaternion ${\mathbf{u}}_2$, demonstrated as. (19) $\{\begin{array}{c}{f}_{\mathrm{ICO}-\mathrm{RT}}=f_{\mathrm{ICO}-\mathrm{Global}}\cdot f_{\mathrm{Global}-\mathrm{RT}}\\ f_{\mathrm{RT}-\mathrm{ICO}}=f_{\mathrm{RT}-\mathrm{Global}}\cdot f_{\mathrm{Global}-\mathrm{ICO}}\end{array}.$(19)

Thus, any point ${\mathbf{q}}_{\mathrm{Local}}=\left[0,x_l,y_l,0\right]$ in the initial LCS can be converted to the local coordinates ${\mathbf{q}}_{\mathrm{Res}}=\left[0,x{{}^{\mathrm{\prime }}}_l,y{{}^{\mathrm{\prime }}}_l,0\right]$ in the target LCS, which is described by (20) ${\mathbf{q}}_{\mathrm{Res}}=\left({\mathbf{u}}_1{\mathbf{u}}_2\right){\mathbf{q}}_{\mathrm{Local}}({\mathbf{u}}_1{\mathbf{u}}_2{)}^{\ast }.$(20)

3.3. Mapping relationship between icosahedral and RT grids

Sections 3.1 and 3.2 revealed the geometric correlation between the RT and icosahedron and established the bidirectional mapping of different coordinate systems. As the grid is essentially a discretization of the continuous polyhedral space, conversion between different grid systems requires the establishment of a mapping function $f{}^{\mathrm{\prime }}$ for polyhedral discretized spaces and $f{}^{\mathrm{\prime }}$ must be either injective or bijective.

Theorem 3.1:

The mapping ($f{{}^{\mathrm{\prime }}}_{\mathrm{RT}-\mathrm{ICO}}$ and $f{{}^{\mathrm{\prime }}}_{\mathrm{ICO}-\mathrm{RT}}$) between RTEAH4 and ISEAH4 is bijective.

Proof 3.1:

According to Section 3.2.2, the regions ${\pi }_{T_n}$ and ${\pi }_{D_m}$ are projections of each other on another polyhedron, that is, ${\pi }_{T_n}=\mathrm{Proj}\left({\pi }_{D_m}\right)$ and ${\pi }_{D_m}=\mathrm{Pro}{j}^{-1}\left({\pi }_{T_n}\right)$. Thus, $\mathrm{Proj}$ is a bijection.

The set of infinite and grid points on $\pi$(${\pi }_{T_n}$ or ${\pi }_{D_m}$) were denoted by $S$ and $S_{\mathrm{Hex}}$, respectively. Since $\mathrm{Proj}$ is a bijection and $S_{\mathrm{Hex}}\subseteq S$, we can always find the image $S_{\mathrm{Image}}$ of $S_{\mathrm{Hex}}$ in $\mathrm{Proj}\left(\pi \right)$. At this point, we only need to prove that $S_{\mathrm{Image}}$ is equal to the set of grid points $S{{}^{\mathrm{\prime }}}_{\mathrm{Hex}}$ in $\mathrm{Proj}\left(\pi \right)$, which will prove the existence of a bijection between the hexagonal grids.

The number of cells in the different hexagonal grid systems is listed in Table 1. The numbers of cells in ISEA -4H -CI and RTEA -4H -CII, as well as in ISEA -4H -CII and RTEA -4H -CI, are clearly the same, with a difference of one level. We can easily deduce that $f{{}^{\mathrm{\prime }}}_{\mathrm{RT}-\mathrm{ICO}}$ and $f{{}^{\mathrm{\prime }}}_{\mathrm{ICO}-\mathrm{RT}}$ are surjective.

Table 1. Numbers of cells in different grid systems.

Level	ISEA -4H -CI	RTEA -4H -CII	ISEA -4H -CII	RTEA -4H -CI
1	32	122	12	42
2	122	482	42	162
3	482	1922	162	642
4	1922	7682	642	2562
…	…	…	…	…
L	$30\ast 4^{L-1}+2$	$30\ast 4^L+2$	$10\ast 4^{L-1}+2$	$10\ast 4^L+2$

We selected icosahedral and RT grids with the same numbers of cells at levels L and L-1 ($L>1$), respectively, and classified them for discussion.

(1)	ISEA -4H -CII and RTEA -4H -CI

Firstly, we combined triangles on ICO’ to form 10 ‘virtual rhombuses’ as shown in Figure 7(a). Let $\omega ={\displaystyle \frac{1}{2}+{\displaystyle \frac{\sqrt{3}}{2}\mathbf{i}}}$, where $\mathbf{i}$ represents an imaginary quantity. The set $E=\left\{0,\omega ,{\omega }^5,{\omega }^6\right\}$ was defined to describe the positions of cells, and $e$ represents any element in $E$. Excluding the two special hexagons at the north and south poles, the set of hexagonal cells ${\mathbb{T}}_L$ in ISEA -4H -CII can be represented as. (21) ${\mathbb{T}}_L={\cup }_{n=1}^{10}{\cup }_{l=1}^L{\displaystyle \frac{1}{2^l}e.}$(21)

Based on the projection $\mathrm{Proj}$ mentioned above, we mapped the rhombic surfaces of the RT onto the triangular faces of ICO’, and the resulting new rhombus had an $\alpha$ of 120° and a long diagonal equal to the length of the triangular faces. ‘Vertex tiles’ and ‘face tiles’ of the RT were defined as demonstrated in Figure 8. The set of hexagonal cells ${\mathbb{D}}_{L-1}$ in RTEA -4H -CI is expressed as (22) ${\mathbb{D}}_{L-1}=\left({\cup }_{n=1}^{30}{F}_n^{L-1}\right)\cup \left({\cup }_{m=1}^{10}{V}_m^{L-1}\right),L⩾1.$(22)

Each ‘virtual rhombus’ can be equivalently represented by three $F_{\mathrm{RT}}$ and one $V_{\mathrm{RT}}$, as shown in Figure 8. (23) ${\cup }_{l=1}^L{\displaystyle \frac{1}{2^l}e={\cup }_{l=1}^{L-1}{\displaystyle \frac{1}{2^l}e\cup \left({\displaystyle \frac{1}{2}\omega +{\cup }_{l=1}^{L=1}{\displaystyle \frac{1}{2^l}e}}\right)\cup \left({\displaystyle \frac{1}{2}{\omega }^5+{\cup }_{l=1}^{L=1}{\displaystyle \frac{1}{2^l}e}}\right)\cup \left({\displaystyle \frac{1}{2}{\omega }^6+{\cup }_{l=1}^{L=1}{\displaystyle \frac{1}{2^l}e}}\right),}}$(23) (24) ${\mathbb{D}}_{L-1}={\cup }_1^{10}\left({\cup }_1^3{F}^{L-1}\cup V^{L-1}\right)={\cup }_{n=1}^{10}{\cup }_{l=0}^L{\displaystyle \frac{1}{2^l}e={\mathbb{T}}_L.}$(24)

(2)	ISEA -4H -CI and RTEA -4H -CII

Figure 7. (a) ISEA -4H -CII and RTEA -4H -CI (b) ISEA -4H -CI and RTEA −4H -CII.

Figure 8. Spatial relationship between the ISEA -4H -CII and RTEA -4H -CI grid cells.

Twelve ‘vertex tiles’ $V_{\mathrm{ICO}}$ (two of which were located at the north and south poles) and twenty ‘face tiles’ $F_{\mathrm{ICO}}$ of ICO’ were defined. Excluding the tiles at the poles, the set of hexagonal cells $\mathbb{T}{{}^{\mathrm{\prime }}}_L$ in ISEA -4H -CI is defined as. (25) $\mathbb{T}{{}^{\mathrm{\prime }}}_L=\left({\cup }_{n=1}^{20}F{{}^{\mathrm{\prime }}}_n^L\right)\cup \left({\cup }_{m=1}^{10}V{{}^{\mathrm{\prime }}}_m^L\right).$(25)

Each ‘virtual rhombus’ was equal to three rhombic surfaces of the RT and can also be equivalently represented by two $F_{\mathrm{ICO}}$ and one $V_{\mathrm{ICO}}$, as depicted in Figure 9. The set $E{}^{\mathrm{\prime }}=\left\{0,{\displaystyle \frac{\omega +{\omega }^2}{2},{\displaystyle \frac{\omega +{\omega }^6}{2},{\displaystyle \frac{{\omega }^5+{\omega }^6}{2}}}}\right\}$ was defined, and $e{}^{\mathrm{\prime }}$ represents any element in $E{}^{\mathrm{\prime }}$. The set of hexagonal cells $\mathbb{D}{{}^{\mathrm{\prime }}}_{L-1}$ in RTEA -4H -CII is expressed as (26) $\mathbb{D}{{}^{\mathrm{\prime }}}_{L-1}={\cup }_{n=1}^{30}{\cup }_{l=1}^{L-1}{\displaystyle \frac{1}{2^l}e{}^{\mathrm{\prime }}.}$(26)

Similarly, based on the equivalence relation, the following can be obtained. (27) $\mathbb{D}{{}^{\mathrm{\prime }}}_{L-1}={\cup }_1^{10}\left({\cup }_{n=1}^3{\displaystyle \frac{1}{2^l}e{}^{\mathrm{\prime }}}\right)={\cup }_{n=1}^{10}\left({\cup }_1^{2}F{}^{\mathrm{\prime }}\cup V{}^{\mathrm{\prime }}\right)=\left({\cup }_{n=1}^{20}F{{}^{\mathrm{\prime }}}_n^L\right)\cup \left({\cup }_{m=1}^{10}V{{}^{\mathrm{\prime }}}_m^L\right)=\mathbb{T}{{}^{\mathrm{\prime }}}_L.$(27)

According to the discussion above, it can be deduced that all elements in the set of hexagonal grids in the RT and ICO’, denoted as $S_{\mathrm{RT}}^{L-1}$ and $S_{\mathrm{ICO}}^L$, respectively, can be mapped to each other through $f_{\mathrm{RT}-\mathrm{ICO}}$ and $f_{\mathrm{RT}-\mathrm{ICO}}$ with unique images.

Figure 9. Neighborhood search of multi-structural elements.

Q.E.D.

According to Theorem 3.1 and Section 3.2, seamless conversion between ISEA-4H and RTEA -4H can be achieved using dual quaternions, which enables the integration and extension of the existing indexing schemes and algorithms of icosahedral DGGS into RT DGGS.

Moreover, except for the conversion between ICO and RT DGGS, the proposed method has potential application with heterogeneous DGGS with a dual relationship between polyhedrons such as octahedron and rhombic dodecahedron.

4. Results and discussion

First, the dual quaternion class was constructed to implement the coordinate transformation relationships established in this study. Second, the conversion between the heterogeneous grids was achieved by conducting mapping. Third, the proposed method was compared with the traditional conversion method utilizing latitude/longitude coordinates as the medium, and the efficiency ratio was calculated. Finally, we evaluated the compactness of the converted cells and selected specific areas in which to test the correctness. Each program was compiled on the same laptop (Intel Core [email protected] GHz,16GB RAM, Windows 10 Pro For Workstations; MSVC++ 2022 Current (Release X64) version 17.0.0).

Using the open-source matrix library Eigen (https://eigen.tuxfamily.org/) (Guennebaud, Jacob, and others 2010), we constructed a dual quaternion class DualQuaternion, whose basic structure is shown in Figure 10. The members of the class were the real and imaginary parts, and the main implemented functions included dot product, cross product, and scalar multiplication of dual quaternions.

Figure 10. Class diagram of DualQuaternion.

The accuracy of the conversion between ISEA- 4H -CII and RTEA −4H -CI as well as between ISEA- 4H -CI and RTEA- 4H- CII was tested. The basic approach was as follows: the four types of grids were initialized and used to generate the corresponding grid code. The grid code was converted into local Cartesian coordinates and extended to quaternion form. Using the mapping relationship established in this study, the current coordinates were converted into the target coordinate system and finally into the target grid code, as depicted in Figure 11. The conversion results are plotted in Figure 12.

Figure 11. Conversion process between ISEA -4H and RTEA -4H.

Figure 12. Results of conversion between (a) RTEA -4H -CII and ISEA −4H -CI and between (b) RTEA -4H -CI and ISEA -4H -CII.

The experiments showed that the proposed mapping method correctly reflects the spatial relationships between the heterogeneous grids and that ISEA −4H and RTEA −4H cells correspond one-to-one after transformation, indicating generality and robustness.

Furthermore, the proposed method was compared with a traditional method using latitude/longitude coordinates as the medium, and the grid level was set at 5–12. The results are shown in Figure 13 and Figure 14 and Tables 2–5.

Figure 13. Efficiency ratio of grid conversion between RTEA -4H -CII and ISEA −4H -CI.

Figure 14. Efficiency ratio of grid conversion between RTEA-CI-H4 and ISEA-CII-H4.

Table 2. Conversion from RTEA -4H -CII to ISEA -4H -CI.

Level	Time taken by the latitude/longitude method (ms)	Time taken by the proposed method (ms)	Efficiency ratio
5	0.0082	0.0021	3.8393
6	0.0334	0.0085	3.9345
7	0.1331	0.0339	3.9288
8	0.6375	0.1534	4.1546
9	2.1172	0.6338	3.3405
10	8.4343	2.1683	3.8898
11	33.6679	8.6556	3.8897
12	139.0001	37.5791	3.6989

Table 3. Conversion from ISEA -4H -CI to RTEA -4H -CII.

Level	Time taken by the latitude/longitude method (ms)	Time taken by the proposed method (ms)	Efficiency ratio
5	0.0045	0.0016	2.8182
6	0.019	0.0070	2.75642
7	0.0773	0.0285	2.7166
8	0.354	0.495	3.1630
9	1.4168	0.6338	2.8640
10	5.0187	1.8029	2.7837
11	20.7136	7.6743	2.6991
12	80.6029	29.0698	2.7727

Table 4. Conversion from RTEA -4H -CI to ISEA -4H -CII.

Level	Time taken by the latitude/longitude method (ms)	Time taken by the proposed method (ms)	Efficiency ratio
5	0.0085	0.0021	4.0922
6	0.0334	0.0084	3.9549
7	0.1319	0.0344	3.8347
8	0.5566	0.1463	3.8056
9	2.1991	0.5833	3.7699
10	8.4047	2.1703	3.8725
11	33.5425	8.6698	3.8689
12	138.4990	41.4339	3.3426

Table 5. Conversion from ISEA -4H -CII to RTEA -4H -CI.

Level	Time taken by the latitude/longitude method (ms)	Time taken by the proposed method (ms)	Efficiency ratio
5	0.0045	0.0016	2.8003
6	0.0191	0.0070	2.7421
7	0.0774	0.0277	2.7968
8	0.3386	0.1119	3.0251
9	1.3833	0.4723	2.9288
10	5.0124	1.8835	2.6612
11	20.1806	7.3107	2.7604
12	80.5240	29.2597	2.7520

The experiments indicated that the average conversion from RTEA- 4H -CII to ISEA −4H -CI using the proposed method was 3.8345 times more efficient than that using the latitude/longitude method, from RT4HCI to IS4HCII 3.9954 times more efficient, from ISEA −4H -CI to RTEA −4H -CII 2.8217 times more efficient, and from ISEA −4H -CII to RTEA −4H -CI 2.8084 times more efficient. The efficiency ratio remained stable as the grid level increased. Overall, the efficiency improvement when converting from the RT to icosahedron was more significant, primarily because the RT had more surfaces than the icosahedron, resulting in a longer processing time during the Snyder equal-area projection process (Snyder 1992).

Although there was a one-to-one correspondence between the grid cells, owing to the different projection methods between the different polygons and the sphere, the cells projected onto the surface of the sphere inevitably experience distortion, even if their areas are equal. This can be regarded as an uneven result generated by different discretization methods of the spherical space. The degree of cell deformation was evaluated by introducing zonal standard compactness (ZSC) (Kimerling et al. 1999; Wang et al. 2021), and the results are shown in Table 6 and Figure 15. (28) $\mathrm{ZSC}={\displaystyle \frac{\sqrt{4\pi s^2-s^2/r^2}}{p}}$(28) where $s$ represents the area of the cells, $r$ represents Earth's radius, and $p$ represents the perimeter of the cells.

Figure 15. The compactness of cells on RTEA-4H and ISEA-4H.

Table 6. The compactness of cells on RTEA -4H and ISEA -4H.

Level	Ratio (RT/ICO)
	Maximum	Average	Median
3	0.8852	0.9569	1.0002
4	0.8800	0.9787	1.0001
5	0.8769	0.9898	1.0005
6	0.8753	0.9954	1.0005
7	0.8746	0.9982	1.0006
8	0.8745	0.9997	1.0006
9	0.8752	1.0004	1.0007

The results indicate that RT exhibits superior compactness in grid cells, and compactness tends to converge with the increasing hierarchy of the grid. This observation underscores the practicality of the proposed method: achieving seamless transformation of grid centers between two heterogeneous DGGSs while enhancing the precision of data modeling.

Finally, RTEA -4H -CII and the converted ISEA -4H -CI were tested for discretization of the same area, with the grid level set at 5–8, as shown in Figure 16.

Figure 16. Discretization of the mainland United States using RTEA-4H -CII and the converted ISEA −4H -CI.

Overall, the cells converted using the proposed method maintained similar degrees of compactness to those of the original cells, and there were no topological errors, such as holes or missing areas. This method supported the extension and integration of the current icosahedral grid dataset into the RT DGGS. Moreover, the existing data model and simulation algorithms can achieve seamless migration due to the one-to-one corresponding relationship between grid centers.

5. Conclusions

Icosahedral DGGS are widely used in Earth information science and spatial analysis. Therefore, it is necessary to establish a fast and seamless model for conversion between icosahedral and RT DGGS to achieve efficient integration of existing datasets and algorithms.

First, based on the inherent geometric properties of the RT and icosahedron, spatial topological relationships were established. Secondly, the ‘dimensionality enhancement’ concept was adopted using quaternions, and the two-dimensional coordinates were extended to three dimensions. A GCS was then established along with the conversion relationships between different LCSs. Moreover, the algebraic structures of RTEA -4H and ISEA -4H were studied, and a rigorous proof of the bijective relationship between RTEA -4H and ISEA -4H was provided. Finally, dual quaternions were used for efficient mapping between heterogeneous grid cells. Experiments showed that compared with the latitude/longitude method, the proposed method significantly improved efficiency while also supporting the integration of existing grid datasets, models, algorithms, and interoperability between different grid systems.

This study refers to the idea of homogeneous coordinates in computer vision and uses quaternions to describe grid cells in a hyper-spherical space. The results could also be applied to the modeling of three-dimensional DGGS and four-dimensional spatio-temporal DGGS. Furthermore, using dual quaternions for rotation in the three-dimensional GCS may solve the problem of crossing faces in the DGGS, significantly improving computational efficiency.

In this study, we focused on the mapping relationship of the aperture 4 hexagonal grid between the RT and icosahedron because studies are lacking on the algebraic structure of the aperture 3 and aperture 7 hexagonal grids on the RT. We plan to constructed bidirectional mapping between RT and icosahedral DGGS with different aperture in our future work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Defense Science Innovation Special Zone Program of China: [Grant Number 20-163-14-LZ-001-003-01]; the special science fund for innovation ecosystem construction of National Supercomputing Center in Zhengzhou: [Grant Number 201400210100].