## 1 Introduction  
Astronomy, geoscience, computer graphics, and many other disciplines rely on research that uses spherical data. A fundamental problem lies in determining the optimal means of organizing, representing, or approximating these data. The most natural representation of spherical data is the spherical function \(f:\mathbb{S}^{2}\to X\) or samples of the function \(f\). Because of the intrinsic geometric properties of the sphere, the choice of an appropriate coordinate system is crucial for formulating an efficient and accurate numerical method for solving problems in spherical geometry. Spherical polar coordinates, although arguably the most natural coordinates for representing a point on a sphere \(\mathbf{p}\in\mathbb{S}^{2}\), present several disadvantages (collectively known as pole problems) when used for numerical computations over the entire spherical surface. The other approach to representing data is to map the spherical domain onto a plane \(M:\mathbb{S}^{2}\to\mathbb{R}^{2}\), such as the method used in traditional cartography. Then, points on a plane can be employed to process spherical signals. Unfortunately, in mathematics, the spherical surface is non-developable. That is, it is impossible to map a portion of the sphere onto the plane without introducing distortion (i.e.,'stretching' or 'compressing'). For these reasons, a new class of spherical coordinates, spherical area coordinates (SACs), is presented in this paper. The SACsystem is free of any singularity and naturally defined on the sphere surface, thus avoiding planar mapping distortion. Research in science and engineering often needs a method for partitioning the sphere or some subset of it into sampling or analysis units (sphere tessellation or discretization). In geographical information systems (GISs), researchers have presented different kinds of Discrete Global Grid Systems (DGGSs) for organizing global data [1; 2; 3]. The most commonly used regular DGGSs are those based on the geographic (latitude-longitude) coordinate system. Raster global data sets often employ cell regions with the edges defined by arcs of equal-angle increments of latitude and longitude (for example, the \(2.5^{\circ}\times 2.5^{\circ}\) and \(10^{\circ}\times 10^{\circ}\) NASA Earth Radiation Budget Experiment (ERBE) grids [4] and National Geophysical Data Center (NGDC) 5-minute Gridded Earth Topography Data [5]). Grid systems based on the geographic coordinate system have numerous practical advantages. The geographic coordinate system is in line with people's habits, and a wide array of existing data sets are based on it, so square grid regions are by far the most familiar to users. However, such grids have limitations: the cells of the grid become increasingly distorted in area, shape, and interpoint spacing as one moves north and south from the equator (Figure 1). The north and south poles, both points on the surface of the globe, map to lines on the longitude \(\times\) latitude plane. The top and bottom row of grid cells are, in fact, triangles, not squares as they appear in the other row. These polar singularities have forced applications, such as global climate modeling to make use of specific grids for polar regions. To solve these problems, researchers have proposed another kind of discrete global grid system called the Geodesic Global Grid System (Geodesic DGGS) [6], which is based on the geodesic dome invented by [7]. A geodesic dome is a spherical or partially spherical shell structure or lattice shell network on the surface of a sphere (Figure 2). These geodesic DGGSs are not a coordinate isoline grid but rather constructed by some specialized subdivision schemes [8]. Figure 1: The distortions of the geographical coordinate grid. The top and bottom row of grid cells are, in fact, triangles, not squares as they appear in the other row. Furthermore, some of these grid systems have useful features, such as an equal-area hierarchical structure and regularity in shape. Rather than using great circle arcs for subdivision, [PERSON] et al.