Research on the Equal-Area Projection for the Construction of the Rhombic Triacontahedron Grid Systems

Objectives: Discrete global grid systems (DGGS) are new reference frameworks for the Digital Earth， and are suitable for organization, integration and analysis of multi-source big geospatial datasets. Among the existing construction methods of DGGS, using polyhedron projection to the sphere can generate DGGS with superior geometric properties and has been widely used at present. Basic polyhedrons and projection methods are main factors of DGGS design choices with respect to grid cell geometrics. Most existing DGGS schemes choose the platonic solids to approximate the Earth, where the icosahedron achieves the smallest distortion due to the most faces but still has difference with spherical surface. To break through the thinking of platonic solids and constructs more uniform and efficient grid systems, this paper chooses a new basic polyhedron, the rhombic triacontahedron, and researches its equal-area projection methods. Methods: The "Slice-and-Dice" approach provides the equal-area mapping between the spherical triangles and the planar triangles for the polyhedral projections, which has different implementations according to the partitioning strategies, and in this paper the vertex-oriented great circle projection is chosen to realize the equal-area mapping between the surface of the rhombic triacontahedron and the sphere because of more uniform angular distortion compared with the Snyder equal-area polyhedral projection. Each diamond face of the rhombic triacontahedron is divided into two triangles along the short diagonal as the basic planar triangles for projection. First, according to the equal surface area of the rhombic triacontahedron and the sphere, we calculate the edge and angle parameters of the rhombic triacontahedron and its corresponding spherical polyhedron. Then, the forward and inverse projection formulas are derived based on the equal-area conditions of the vertex-oriented great circle projection. All procedures are closed form without iterations like the inverse Snyder equal-area polyhedral projection. Results: Three experiments are realized and the results show that: (1) compared with the icosahedron using Snyder equal-area polyhedral projection, the angular distortion caused by the proposed method is reduced by 51%. The mean of angular distortion declines from 0.166 radians to 0.082 radians, and the standard deviation declines from 0.055 to 0.023; (2) the proposed scheme is used to generate three types of global grids with different cell shapes—triangles, quadrilaterals and the hexagons, which verifies the validity of the proposed scheme; (3) the efficiency of grid generation using the proposed inverse projection of the rhombic triacontahedron is about four times that of the icosahedron grids based on the inverse Snyder equal-area projection, which needs iterations. Conclusions: A new DGGS scheme is proposed by introducing the rhombic triacontahedron, which provides a new idea for the development and application of DGGS. The new scheme has more angular distortion, which helps generate more uniform grids and improve the accuracy of subsequent data processing and representation, and can be applied to different cell shapes. The next step is to research data modeling, spatial analysis algorithms and the combination of DGS and high-performance computing based on this scheme, so as to provide better solutions for the organization, processing and analysis of big data on the Earth.