Formulas for 8 equal-area Power-Function maps

[RogerOwens]

Edit note, 1/6/16:

I've added the backwards formulas. They explicitly give lat and lon, in terms of x and y.

In the unsolved equation, I've greatly simplified and clarified the equation relating y to lat, by only having "abs(sin lat)" instead of "abs(y)".

That's simpler, and is possible because y is positive when sin(lat) is positive.

[end of edit note]

Edit note, 1/7/16:

Fixing formulas by adding absolute values where needed.

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First some general formulas for equal-area power-function maps:

lat is latitude in degrees

lon is longitude in degrees

The center of the map is the coordinate origin.

k is an adjustment constant to give y as a fraction of y's maximum magnitude.

The distance from the center of the map to the north pole is one unit.

f is an x-adjustment constant for the purpose of achieving conformality along the equator

(...for these equal-area maps, in this posting. For the linear maps in the previous formulas-posting, f was for bringing conformality to the point (lon 0, lat 45). For the linear maps in the previous formulas-post, the linear maps are conformal along the equator when f = 1.)

Longitude west (left) of the central meridian is negative.

y(lat) probably doesn't have a solution in closed-form, but this equation describes the relation between y and lat:

y- (y^(p+1))/(p+1) = k abs(sin lat)

If lat < 0, then y = - abs(y)
Else y = abs(y)

x = 2f(1-(abs(y))^p)(lon/180)

Backward Formulas:

(In the following formula, substitute "abs(y)" for "y")

abs(lat) = arcsin((1/k)(y-(y^(p+1))/(p+1)))

if y<0, then lat = -abs(lat)
Else lat = abs(lat)

lon = 90x/(f(1-(abs(y))^p))


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The 8 equal-area map's:

Sv/Sh = vertical scale divided by horizontal scale.

w/h = aspect ratio
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p=2, k = 2/3, f = 1.0:

Sv/Sh along equator: 1.0472

w/h: 2.0
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p = 2, k = 2/3, f = 1.0472

Sv/Sh along equator: 1.0

Aspect-Ratio: 2.0944
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p=3, k = .75, f = 1.0

Sv/Sh along equator1.178

Aspect ratio: 2.0
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p=3, k = .75, f = 1.178

Sv/Sh along equator: 1.0

Aspect-Ratio: 2.356
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p = 4, k = .8, f = 1.0

Sv/Sh along equator: 1.2566

Aspect-Ratio: 2.0
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p=4, k = .8, f = 1.2566

Sv/Sh along equator: 1.0

Aspect-Ratio: 2.5132

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p = 8.32, k = .8927, f = 1.0

Sv/Sh along equator: 1.4

Aspect-Ratio: 2.0
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p = 8.32, k = .8927, f = 1.4

Sv/Sh along equator: 1.0

Aspect-Ratio, 1.4

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Michael Ossipoff