# Disdyakis Triacontahedron DGGS

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## Abstract

**:**

## 1. Introduction

- Initial Polyhedron: The coarsest discretization of the Earth into cells
- Projection: A mapping between each point on the Earth and a corresponding point on the polyhedron (see Figure 1)
- Indexing: Assigning a unique identifier to each cell
- Refinement: A way to hierarchically subdivide coarse cells into finer ones creating multi-resolution cells

## 2. Related Work

#### 2.1. Initial Polyhedron

#### 2.2. Projection

#### 2.3. Refinement and Cell Types

#### 2.4. Indexing

#### 2.5. DGGS

## 3. Creating DT DGGS

#### 3.1. Initial Polyhedron

#### 3.2. Equal Area Projection

#### 3.3. Slice and Dice Area Preserving Projection

#### 3.4. Inverse Projection for the DT

#### 3.5. Refinement and Indexing

## 4. Connectivity and Location

#### 4.1. Neighborhood Queries

#### 4.2. Hierarchical Queries

#### 4.3. Point-to-Cell Query

## 5. Results

## 6. Conclusions

#### Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DE | Digital Earth |

DGGS | Discrete Global Grid System |

DT | Disdyakis Triacontahedron |

RT | Rhombic Triacontahedron |

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**Figure 1.**Projection between the polyhedron and the Earth. Mirror symmetry is present along the edges of all adjacent triangles in the Disdyakis Triacontahedron (DT).

**Figure 3.**From left to right, Hilbert, Peano, Sierpinski and Morton space filling curves. The order that the curve visits the cells determines the indexing. Images taken from [36], licensed under Creative Commons BY 4.0.

**Figure 4.**A Bipyramid with 64 faces. All faces are identical and mirror images of each other along the adjoining edge but as we increase the number of faces the compactness of the triangles reduces creating longer and skinnier triangles.

**Figure 5.**Point P cuts spherical triangle $ABC$ into two parts with a great circle arc emanating from vertex B. Point ${P}^{\prime}$ within a given planar triangle ${A}^{\prime}{B}^{\prime}{C}^{\prime}$ is solved for such that the relationship in equation 1 holds.

**Figure 6.**Notice that regions $BDC$ and $BPE$ are both portions of a spherical cap. The area of a spherical cap is proportionate to the angle that it subtends so the area of $BDC$ is proportionate to $1-cos(x+y)$ and the area of $BPE$ is proportionate to $1-cos\left(x\right)$. Also notice that regions ${B}^{\prime}{D}^{\prime}{C}^{\prime}$ and ${B}^{\prime}{P}^{\prime}{E}^{\prime}$ are similar triangles. The area of similar planar triangles is proportionate to the square of their side length so the area of the triangle ${B}^{\prime}{D}^{\prime}{C}^{\prime}$ is proportionate to ${({x}^{\prime}+{y}^{\prime})}^{2}$ and the area of ${B}^{\prime}{P}^{\prime}{E}^{\prime}$ is proportionate to ${x}^{\prime 2}$.

**Figure 7.**Common refinement of triangles includes (

**a**) 1:2 (

**b**) 1:3 and (

**c**) 1:4. Shown in (

**d**), a 1:4 composite refinement scheme can also be created by two applications of 1:2 refinement in (

**a**).

**Figure 8.**The rhombic triacontahedron (RT) is a subset of the vertices and edges of the DT. Two of the rhombic faces and the rhombus indexing are shown in red. The black indexing corresponds to rhombuses in Figure 10.

**Figure 9.**(

**a**) Connectivity maps can be assigned to two adjacent triangles to form a quadrilateral domain parameterized by u and v ranging from 0 to 1. (

**b**) Applying traditional 1:4 refinement twice on a triangular connectivity map produces small quad domains within the connectivity map that can be indexed by parameters u and v.

**Figure 10.**An example of the indexing scheme at resolutions 1, 2 and 3 from left to right. Note that this is duplicated for every rhombus in RT. These Z shaped patterns within small rhombuses are repeated along rows and columns of small rhombuses in row major order. We index small rhombuses row by row, most evident in resolution level 3 on the right.

**Figure 11.**Cell indices of two adjacent base rhombuses. Notice that for any cell c in the small rhombus indicated in red, two neighbours are at indices $c\oplus 1$ and $c\oplus 2$. This is true of any index in any small rhombus. The final neighbour depends on if we are on a boundary between rhombuses or not.

**Figure 12.**The rhombus highlighted in red will become a parent rhombus when we decrease subdivision levels. Rows are divided with a blue line—columns with a red line. The green indices are the start of two consecutive blocks of indices within the red rhombus. Specific indices become part of specific parents as seen in Figure 13.

**Figure 13.**Parent cells are highlighted in blue and red. Note that 16 child cells make up the four parents in the rhombus.

**Figure 14.**A DT with great circle arcs (longitude) between the North and South pole highlighted in red. The region between these arcs contains 12 base triangles. A point P on the Earth can be quickly checked to see if it lies within the partition created by the red arcs by simply checking its longitude. The dashed lines represent the next subdivision level. The barycentric coordinates of the point within the base triangle determines which child cell P lies in.

**Figure 15.**Circles on the spherical face of a DT shown in blue are projected to ellipses on the planar face of a DT shown in green. We can see that the angular distortion is minor because the major and minor axis of the ellipses are close to the same.

**Figure 16.**(

**a**) Angular distortion mapped to a greyscale value across a face of an icosahedron. (

**b**) Angular distortion mapped to a greyscale value across a face of a DT. Black is low distortion and white is high distortion. We have used the same greyscale mapping for both for direct comparison.

**Figure 18.**Temperature data interpolated across the DT Discrete Global Grid System (DGGS). Cold areas are blue and red areas are hot. This data can be queried and visualized across the Earth in real time.

**Table 1.**Mean and standard deviation of angular distortion (in radians) of the projection to the icosahedron and the DT. The mean for the DT is approximately four times less and the standard deviation is lower showing that projection to the DT has significantly reduced angular distortion.

Mean Angular Distortion (Radians) | Standard Deviation | |
---|---|---|

Icosahedron | 0.144 | 0.027 |

Disdyakis Triacontahedron | 0.039 | 0.016 |

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**MDPI and ACS Style**

Hall, J.; Wecker, L.; Ulmer, B.; Samavati, F.
Disdyakis Triacontahedron DGGS. *ISPRS Int. J. Geo-Inf.* **2020**, *9*, 315.
https://doi.org/10.3390/ijgi9050315

**AMA Style**

Hall J, Wecker L, Ulmer B, Samavati F.
Disdyakis Triacontahedron DGGS. *ISPRS International Journal of Geo-Information*. 2020; 9(5):315.
https://doi.org/10.3390/ijgi9050315

**Chicago/Turabian Style**

Hall, John, Lakin Wecker, Benjamin Ulmer, and Faramarz Samavati.
2020. "Disdyakis Triacontahedron DGGS" *ISPRS International Journal of Geo-Information* 9, no. 5: 315.
https://doi.org/10.3390/ijgi9050315