A polyhedral map projection is a map projection based on a spherical polyhedron. Typically, the polyhedron is overlaid on the globe, and each face of the polyhedron is transformed to a polygon or other shape in the plane. The best-known polyhedral map projection is Buckminster Fuller's Dymaxion map. When the spherical polyhedron faces are transformed to the faces of an ordinary polyhedron instead of laid flat in a plane, the result is a polyhedral globe.[1]
Often the polyhedron used is a Platonic solid or Archimedean solid. However, other polyhedra can be used: the AuthaGraph projection makes use of a polyhedron with 96 faces, and the myriahedral projection allows for an arbitrary large number of faces.[2]
Although interruptions between faces are common, and more common with an increasing number of faces, some maps avoid them: the Lee conformal projection only has interruptions at its border, and the AuthaGraph projection scales its faces so that the map fills a rectangle without internal interruptions. Some projections can be tesselated to fill the plane, the Lee conformal projection among them.
To a degree, the polyhedron and the projection used to transform each face of the polyhedron can be considered separately, and some projections can be applied to differently shaped faces. The gnomonic projection transforms the edges of spherical polyhedra to straight lines, preserving all polyhedra contained within a hemisphere, so it is a common choice. The Snyder equal-area projection can be applied to any polyhedron with regular faces.[3] The projection used in later versions of the Dymaxion map can be generalized to other equilateral triangular faces,[4] and even to certain quadrilaterals.[5]
Polyhedral map projections are useful for creating discrete global grids, as with the quadrilateralized spherical cube and Icosahedral Snyder Equal Area (ISEA) grids.[6]
^Cite error: The named reference Pedzich was invoked but never defined (see the help page).
^van Wijk, Jarke J. (2008). "Unfolding the Earth: Myriahedral Projections". The Cartographic Journal. 45 (1): 32–42. doi:10.1179/000870408X276594. ISSN 0008-7041. S2CID 218692689.
^Snyder, John P (1992-03-01). "An Equal-Area Map Projection For Polyhedral Globes". Cartographica: The International Journal for Geographic Information and Geovisualization. 29 (1): 10–21. doi:10.3138/27H7-8K88-4882-1752. ISSN 0317-7173.
^Crider, John E. (2008-03-01). "Exact Equations for Fuller's Map Projection and Inverse". Cartographica: The International Journal for Geographic Information and Geovisualization. 43 (1): 67–72. doi:10.3138/carto.43.1.67. ISSN 0317-7173.
^Crider, John E. (2009-01-01). "A Geodesic Map Projection for Quadrilaterals". Cartography and Geographic Information Science. 36 (2): 131–147. doi:10.1559/152304009788188781. ISSN 1523-0406. S2CID 128390865.
^Sahr, Kevin; White, Denis; Kimerling, A.J. (2003). "Geodesic discrete global grid systems" (PDF). Cartography and Geographic Information Science. 30 (2): 121–134. doi:10.1559/152304003100011090. S2CID 16549179.
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