The AuthaGraph projection (Narukawa 2017) has less area distortion. It also evenly distributes the distortion. And this map projection has a rectangular outline. AuthaGraph projection fits a sphere into a rectangle by projecting it onto a regular tetrahedron. It uses a projection plane so called “curved tetrahedron” named by the author. 96 faces form a “curved regular tetrahedron”. The method by using the projection plane reduces distortion. Most of the world maps so far have concentrated distortions in high latitude areas. Therefore, these maps have had difficulty in viewing the high latitude area. AuthaGraph projection improves this challenge. From different viewpoint, most of the world maps so far have the earth's axis as the axis of projection. That is, these maps follow latitude and longitude. Cylindrical projection and conic projection are typical examples of these.
On the other hand, the AuthaGraph projection uses a regular tetrahedron. The regular tetrahedron has a closed surface. By utilizing the characteristics of regular tetrahedron, the projection improves the visibility of polar regions. It was created with the aim of creating a world map that does not use this latitude and longitude line as a guideline. And it is made with the aim of giving an equal overview of the whole world. However, it pays a trade-off for this purpose. AuthaGraph projection has not been able to link latitude and longitude to map coordinates x and y. That has been a challenge to make the formula of it.
As a result, the world map by AuthaGraph projection has been created by making full use of computational power. In addition, the author operated a 3D modeling software to create a map of the world by AuthaGraph projection. Without the above manual work, the AuthaGraph projection could not be practiced. From another academic point of view, it is necessary to formulate the AuthaGraph projection. The formula allows map projection researchers to quantitatively verify the distortion of the projection. Based on the above viewpoints, this paper practices the following three ideas and comparative verification.
1. Formulate the AuthaGraph projection.
2. The formula quantitatively evaluates the distortion.
3. The evaluation of the distortion compares the existing rectangular area-equal world map with the AuthaGraph projection.
The earth shapes a spheroid. Its flattening is approximately equal to 1/300. However, in discussing the projection, the shape of the earth may be simplified as a sphere. This paper as well, set the earth as a sphere for the map projection’s formula.
