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The Gnomonic Projection

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The Gnomonic Projection Gnomonic Projections onto Various Polyhedra By Brian Stonelake Table of Contents Abstract ................................................................................................................................................... 3 The Gnomonic Projection .................................................................................................................. 4 The Polyhedra ....................................................................................................................................... 5 The Icosahedron ............................................................................................................................................. 5 The Dodecahedron ......................................................................................................................................... 5 Pentakis Dodecahedron ............................................................................................................................... 5 Modified Pentakis Dodecahedron ............................................................................................................. 6 Equilateral Pentakis Dodecahedron ........................................................................................................ 6 Triakis Icosahedron ....................................................................................................................................... 6 Modified Triakis Icosahedron .................................................................................................................... 7 Equilateral Triakis Icosahedron ................................................................................................................ 7 Mapping a Polyhedron ....................................................................................................................... 8 Icosahedral Great Circles ............................................................................................................................. 8 Icosahedral Great Circle Type A ............................................................................................................................... 8 Icosahedral Great Circle Type B ................................................................................................................................ 9 Icosahedral Great Circle Type C ................................................................................................................................ 9 LCD Triangles ................................................................................................................................................. 11 LCD Classification ......................................................................................................................................................... 11 Values that Define the Projections ............................................................................................... 13 Exact Values for Relevant Angles ............................................................................................................ 13 Exact Values for LCD Vertices ................................................................................................................... 15 Exact Values for Centers of Projection .................................................................................................. 18 Icosahedron and Dodecahderon ............................................................................................................................ 18 Center of Projection formulas ................................................................................................................................. 18 Modifications of Dodecahedron and Icosahedron ......................................................................................... 21 Distortion .............................................................................................................................................. 25 Entire LCD triangle ....................................................................................................................................... 28 Modifications of the Dodecahedron and Icosahedron ................................................................................. 31 LCD sampling .................................................................................................................................................. 32 Continuous measure .................................................................................................................................... 33 Density Plots ................................................................................................................................................................... 35 Summary of distortion ................................................................................................................................ 39 Remaining Questions ........................................................................................................................ 41 Calculations .......................................................................................................................................... 42 Exact Coordinates for M 2.8 Calculations ............................................................................................... 43 Exact Coordinates for C2.7.8 Calculations .............................................................................................. 45 Pentakis Dodecahedron Center of Projection Calculations ........................................................... 47 Modified Pentakis Dodecahedron Center of Projection Calculations ......................................... 48 Equilateral Pentakis Dodecahedron Center of Projection Calculations .................................... 49 Triakis Icosahedron Center of Projection Calculations ................................................................... 50 Modified Triakis Icosahedron Center of Projection Calculations ................................................ 51 Equilateral Triakis Icosahedron Center of Projection Calculations ............................................ 52 Height of Pyramid Caps for Optimal Triakis Icosahedron .............................................................. 53 References: ........................................................................................................................................... 54 Image Credits: ................................................................................................................................................ 54 Abstract In this paper, I’ll attempt to build upon work done by RW Gray exploring gnomonic projections onto various polyhedra. Much of his work appears to be motivated by Buckminster Fuller, though many pioneers are recognized in the field. While this work was motivated by the practical application of geographic mapping, this paper will focus on the mathematics involved, rather than its applications. Finding closed form representations for relevant Quantities, for example, will be far more interesting to me than their decimal approximations. After a brief introduction on the method of mapping that I will use and the polyhedra onto which I will project, I’ll discuss the base unit with which I will tile the sphere. These units, termed “LCD triangles” are referenced in Fuller’s Synergetics and expanded upon beautifully by Gray. My contributions will be to develop a logical coding convention for these units and to derive closed form notation for the coordinates of the vertices and their images, relative to a specific orientation. I’ll also derive closed-form notation for other points necessary to explicitly define the projection, as well as functions allowing others to find relevant points for modifications of my polyhedra. Finally, I’ll offer several Quantifications of the distortion associated with these mappings. The goal here is more an exploration of the distortion that a search for an ideal polyhedron. The Gnomonic Projection A gnomonic projection (GP) is a bijection from the surface of a hemisphere (open) to a plane. It has the useful property that it maps great circle arcs to straight lines. This map can be visualized by letting a sphere sit atop an image plane. If we imagine the sphere as our globe and associate the point of tangency on the sphere with the South Pole, any point in the southern hemisphere is in the domain of this map. With this picture in mind, the image of a point in the southern hemisphere is just the intersection of the image plane and a ray from the sphere’s center through the point in Question. Figure 1: The Gnomonic Projection To see that great circle arcs are mapped to straight lines, one only needs to know that a great circle is, by definition, the intersection of a sphere and a plane defined by any two points on the sphere and the sphere’s center. Thus the ray that defines the GP for any point on the great circle arc lies on this plane, so the image is the intersection of two planes; a straight line (see figure 1). Figure 1 not only shows that the GP takes all great circle arcs to straight lines, but can be used to show the converse; that any straight line is the images of a great circle arc. To prove the latter, consider an arbitrary line in the image plane and two distinct points on that line. We can define a plane using these two points and the sphere’s center, and note that the intersection of this plane and the
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