Table of Contents
Unusual Map Projections
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Subjects To Be CoveredPartial List
The Mapping ProcessCommon Surfaces Used in cartography
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The Surface of the Earth Is TwoDimensional
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Mercator’s Projection Is Not Perspective
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The Loximuthal Projection
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The Entire Earth On Half A Globe
Conventional Way of Tracking Satellites
Instead of straight meridians and parallels with curved satellite tracks, as on the previous map, let us bend the meridians so that the satellite track becomes a straight line. This is convenient for automatic plotting of the satellite tracks.
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Bend The Meridians Instead
This Map ShowsMagnetic ‘Parallels’ And ‘Meridians’
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Map Which Straightens The Magnetic CoordinatesStudent Drawing
Therefore the interstate highway system can be drawn as orthogonal equidistant lines on a map and the meridians and parallels bent to fit this. Try it!
US Highway CoordinatesStudent Drawing
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The World On A Moebius StripPrint upside down on back and give it a twistthen glue the ends together
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Two Point Equidistant MapLondon to Wellington
For three points Wellman Chamberlin of the National Geographic Society invented the “trimetric” projection.
Constructing A Trimetric Projection
In order to best preserve distances from more than three points one can use advanced techniques. The next map demonstrates this.
Optimal Distance Preserving Projection Of The United States
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Mediterranean Sea Preserving Loxodromic Distances
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Laskowski’s Trioptimal Projection
Computing coordinates from distances is known in cartography as trilateration. If one takes road distances from a Rand McNally road atlas and uses these distances to compute the location of the places one can then interpolate the latitude/longitude graticule, and from this draw a map with state boundaries and coastlines. The resulting map projection illustrates the distortion introduced by the road system.
Road Distance Map Of The United StatesStudent drawing
The next map is an azimuthal map projection with all places shown at their correct direction from Seattle. But the map scale is in parcel postage cents.
Parcel Post View From Seattle
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World Ocean Distances MapBased on Shipping Distances Between 42 Ports
The distinctive cordiform StabWerner projection is equal area and shows distances correctly, to scale, from the map center only. On the rendition shown the center is just south of the North Pole.
Werner’s (1515) Equidistant Equal Area Projection
Oblique versions of Werner’s projection are rare. The one shown here has its center at New York city, with the central axis directed towards Seattle. Note the position of the North Pole.
An Oblique Version Of Werner’s ProjectionCentered at New York With Direction to Seattle
It is often asserted that transportation costs increase at a decreasing rate with geographic distance. In other words, they have a concave down shape. On the map that follows the transport cost idea is represented by the square root of the spherical distance from the map center, but the map has also been made to preserve spherical areas.
Equal Area Projection With Square Root Distances From CenterPolar Case
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Oblique Version Of The Previous ProjectionCentered on New York and Directed Toward Seattle
Map Projections Come In Many ShapesTobler’s hyperelliptical projection series
A Series of Equal Area Projections by Thomas
DistanceDirection DiagramDistance from Santa Barbara read down. Direction from Santa Barbara read across.The line across the top represents Santa Barbara.
An Unusual Azimuthal ProjectionDirections are correct from the intersection of Greenwich and the Equator. Based on an idea by J. Craig, Cairo, 1910.
Lee’s Conformal Map Within A TriangleL.P. Lee, 1973, The Conformal Tetrahedric Projection with Some Practical Applications, Cartographic J., 10(1):2228.
A NewThree Sided Equal Area Mappolar case
It is relatively easy to fit equal area maps into regular N sided polygons. One computer program can do them all, starting, as in the previous illustration, with N=3. Beyond about N=20 it is not very interesting because the maps all converge to Lambert’s (1772) azimuthal equal area projection with a circular boundary.W. Tobler 1972, “Lambert’s Notes on Maps”, Introduction and Translation of his “Beitraege, 1772”, Ann Arbor, 125 pages.
A New Equal Area Map In A Five Sided Polygonpolar case
Maps on the platonic solids have been known for a long time. They can be equal area or conformal. The gnomonic projection is particularly easy to do on the surface of these solids.
Equal Area Projection on A Pyramid
Here Is A Little FootstoolEqual area of courseFrom Tobler’s hyperelliptical series
A Map Projection for Quick Computer Display
The equations used areX = R { cos(?o) *??  sin(?o) * ?? *?? }Y = R { ?? + 0.5 * sin(?o) * cos(?o) * ?? * ?? },where R is in kilometers per degree on the mean radius sphere (computed by the program), ?? is the latitude minus the average latitude ?o, and ?? is the longitude minus the average longitude ?o The X and Y coordinates are given in kilometers.
The next two maps illustrate what happens when the equations are, inappropriately, applied to the entire sphere.
The Bow Tie ProjectionBy Tobler
The Floppy Bow Tie ProjectionBy Tobler
Here Is an Equal Area Projection for the Pacific Ocean
This One is Equal Area and Makes the Earth Look Elliptical
Polyconic and polycylindric (a.k.a. pseudocylindric) projections are often used. Shown next is the development from three cylinders to the limiting case of an infinite number of cylinders resulting in the equal area sinusoidal projection.
Many Projection Surfaces Are UsedHere Is a Polycylindrical Development.From three cylinders to infinitly many, resulting in a continuous map.
Mercator’s projection is the most famous anamorphose
The equations show that equal area projections are a special case of area cartograms
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US Population By One Degree Quadrilaterals
Now use the TransformSolveInvert paradigm
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World Population by Latitude
World Population by Longitude
On the next map the tick marks indicate the latitude  longitude grid. The map is based on world population given by five degree quadrilaterals, but uses only the marginal distribution of the population (population by latitude and population by longitude) hence it is a “pseudo” cartogram. Computer rendition using a program with an anamorphizing map projection.
Pseudocartogram Of The World According To Population
Composite Equal Area Projections
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Lambert Above MollweideMight make a nice bowl
Lambert Above Sinusoidal
Mollweide Above Lambert A loaf of bread?
Sinusoidal Above LambertA cap, maybe?
Sinusoidal Above Mollweide
Mollweide Above SinusoidalA spinning top?
Mollweide Above LambertSquashed With An Equal Area Affine Transformation. A church window?
Lambert Above Mollweide Squashed With An Equal Area Affine Transformation
Sinusoidal Above LambertSquashed With An Equal Area Affine Transformation.Another church window?
Lambert Above SinusoidalSquashed With An Equal Area Affine Transformation
Sinusoidal Above MollweideSquashed With An Equal Area Affine TransformationObviously a teardrop.
Mollweide Above SinusoidalSquashed With An Equal Area Affine Transformation.A very unstable world.
Sinusoidal ProjectionSquashed With An Equal Area Affine Transformation.A nice lens.
Mollweide ProjectionSquashed With An Equal Areal Affine Transformation.The world on a diet.
Mollweide ProjectionSquashed Into An Equal Area CircleA new, attractive, world in a circle
Lambert Cylindrical ProjectionSquashed into an equal area square for use as a global quadtree
Another (New) Equal Area Map in a Square
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The Azimuthal Projections
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The Radial Function Display Of Azimuthal Projections
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Snyder’s Magnifying Glass Projection In the radial function display, with two scales and a discontinuity.
In studying migration about the Swedish city of Asby, Hägerstrand used the logarithm of the actual distance as the radial scale. This enlarges the scale in the center of Asby, near which most of the migration takes place. Actually, but not shown, there is a small hole in the middle of the map since the logarithm of zero is minus infinity.
Hägerstrand’s Logarithmic Map
Draw Your OwnI’ve drawn a quarter circle, but you can invent your own azimuthal map projection.This one encourages myopia.
Or reverse the effect to combat myopia.
The Santa Barbarian ViewA cube root distance azimuthal projection
A New Yorker’s ViewSquare root azimuthal projection, with obvious distortion
The View From Michiganwith less obvious distortion
A Case Using Double ProjectionThe graph for ? = 15o shrinks the middle of the map
There Are Many More Unusual Projections
Retroazimuthal projections show reverse directions to a center. This property can be combined with correct distances to the center.
A radio station was established at Rugby to broadcast a time signal to British colonies overseas. The equidistant retroazimuthal projection was used to let the colonials know in which direction to point their radio antennas.
Hinks’ Retro Azimuthal ProjectionCentered at Rugby, UK
Here is a New RetroAzimuthal ProjectionCentered at 20N, 40E, Close to Mecca Mecca along top. Down is distance, left to right is direction.The map contains a hole and overlaps itself.
In 1935 Brown Published A Mathematical Paper That Included
From Brown’s Paper1 of 4
From Brown2 of 4
From Brown3 of 4
From Brown4 of 4
Here Is Another Novel ProjectionSuggested by C. ArdenClose. Implemented by P.B. Anderson.
Thank You For Your Attention
