Extract
A geodesic map projection for quadrilaterals.
Introduction
Buckminster Fuller's map projections are based upon intersections of geodesics (great circle arcs) that connect a point on one edge of a spherical figure to a point on another edge and pass through the point being projected. The best known, Fuller's Dymaxion[TM] (1) map projection, uses the triangular faces associated with an icosahedron (Gray 1995; Crider 2008). However, his patented map projection (Fuller 1946) is based on the cuboctahedron, which has eight triangular faces and six square faces. On a spherical square corresponding to a square face, Fuller describes a two-dimensional grid of intersecting geodesics equally spaced at the edges of the figure. He states that the grid can be subdivided as finely as needed in order to specify the position of a point within the figure. The word "geodesic" (derived from "geodesy") originally meant a great circle arc, or orthodrome. The concept has been generalized to include a curve on a surface that is the shortest path between two points on the surface, and, in graph theory, to mean the shortest distance between two nodes in a graph. Thus, it includes a straight line on a plane, and it is a useful concept for ellipsoids and other surfaces that may be used in cartography. However, in this paper, its use is limited to great circle arcs on spheres and to straight lines on planes. In this paper, Fuller's square map projection is expanded to spherical quadrilaterals that have opposite sides equal. The result is a new class of map projections based on great circle arcs (the geodesics of the sphere) mapping to lines (the geodesics of the plane). The map projections use a parameterized intermediate coordinate system that identifies the intersecting geodesics of the grid. The parameterized intermediate coordinates are then linearly transformed into map projection coordinates. The map projections have closed form inverses. The unit sphere is used here to represent the globe; points on the globe are represented as position vectors. The projections have not been extended to an ellipsoid. The map projections include rectangles and squares that map portions of the globe, as well as families of tiling sets of diamonds (spherical rhombi) that cover the globe. All the families of diamond tiling sets are described in detail. Projection distortions are examined. Finally, other applications besides map projections are discussed for the coordinate system and the diamond tiling sets; the applications are in geometry, astronomy, and computer graphics. Geodesic Coordinate System A new coordinate system is describe...See the full content of this document
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