Formulas for Rounded-Behrmann

[RogerOwens]

(I decided that "Rounded-Behrmann" might be a better name for the grafted projection that I've been proposing.)

Formulas for Rounded-Behrmann:
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All distances in these formulas are in terms of the radius of the Earth or globe on which a Lambert CEA map is based. …where Behrmann is gotten by multiplying that Lambert’s equator-length (2*pi) by .75, to get Behrmann’s equator length of 3*pi/2.
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(But of course all that’s necessary is that the unit for distances is the same throughout the formulas).
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Map co-ordinates are in capitals, X and Y.
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In the above-graft part, the lower-case “y” stands for Y minus the Y value in Behrman for lat 42.
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r = radius of quarter circles at corners of the map = .341490755
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P = length of pole-line = 4.0294075
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Y1 = Y value at lat 42 = .669130606
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y = Y - Y1
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Above lat 42:
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sin(lat) =
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-(2/(3pi))([{90-arcsin(y/r)}/180]*pi*(r^2) - y*sqrt((r^2)-(y ^2))) + ((r^2)/3) + y(1-(4/(3pi))) + sin(42)
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lon = X (180/[{P/2}+ sqrt{(r^2) – (y^2)}])


I should add that, for the formula relating sin(lat) and Y, lat and Y are only defined for positive values of Y and sin(lat).

For -Y, sin(lat) is the negative of what it is for Y.

In other words: if, for positive Y, sin(lat) is F(Y), then F(-Y) = -F(Y).
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(Most likely, formula format can be written by using the code option at the top of the edit-screen, but I don’t know the code.)
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Below lat 42:
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Y = sin(lat)
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X = (3pi/4) * (lon/180)
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Michael Ossipoff

[RogerOwens]

I should add that, for the formula relating sin(lat) and Y, lat and Y are only defined for positive values of Y and sin(lat).

For negative Y, sin(lat) is the negative of what it is for Y.

In other words: if, for positive Y, sin(lat) is F(Y), then F(-Y) = -F(Y).

Michael Ossipoff

[RogerOwens]

I left out a factor in a term.

In the 2nd-to-last term, 4/3pi should be multiplied by (r - sqrt(r^2 - y^2))

So here is the formula with that substitution:

Above lat 42:
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sin(lat) =
-(2/(3pi))([{90-arcsin(y/r)}/180]*pi*(r^2) - y*sqrt((r^2)-(y ^2))) + ((r^2)/3) + y(1-[4/(3pi)][r - sqrt(r^2 - y^2)]) + sin(42)
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lon = X (180/[{P/2}+ sqrt{(r^2) – (y^2)}])

[RogerOwens]

No, the 2nd-to-last term in that long expression for sin(lat) should just be: P*y(2/(3*pi)).

So here are the correct formulas for Rounded-Behrmann:

Above lat 42:

sin(lat) = - (2/(3pi))([{90-arcsin(y/r)}/180]*pi*(r^2) - y*sqrt((r^2)-(y ^2))) + ((r^2)/3) + P*y*(2/(3*pi)) + sin(42)
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lon = X (180/[{P/2}+ sqrt{(r^2) – (y^2)}])

Below lat 42:
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Y = sin(lat)
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X = (3pi/4) * (lon/180)

Michael Ossipoff

[daan]

Y = sin(lat)

That’s not Behrmann’s y; that’s Lambert’s.

— daan

[RogerOwens]

Y = sin(lat)

That’s not Behrmann’s y; that’s Lambert’s.

— daan

But I shortened the equator-length, and all the X values, by a factor of .75

Michael Ossipoff

[daan]

But I shortened the equator-length, and all the X values, by a factor of .75

Why wouldn’t you just use the standard Behrmann’s formulation below 41°24′35″ and enlarge your formulation above it? Your entire map is going to be 3/4 the size it should be given any usual notion of “scale”.

— daan

[RogerOwens]

But I shortened the equator-length, and all the X values, by a factor of .75

Why wouldn’t you just use the standard Behrmann’s formulation below 41°24′35″ and enlarge your formulation above it? Your entire map is going to be 3/4 the size it should be given any usual notion of “scale”.

— daan

I didn't know the usual way of saying it, but it just seemed easier that way, and, either way, I'm specifying the same map-shape(s), so it didn't seem to matter.

I rounded the graft-latitude up to 42 for simplicity-of-definition.

I like Rounded-Behrmann as a way of getting the advantages of CEA, as far up as Behrmann has good shapes and no absolute compression, and then, above that latitude, reducing the distortion that CEA would otherwise have.

Admittedly, both above and below the graft, a map like Rounded Behrmann isn't going to have as good shapes a purely pesuedocylindrical map could. So it gives up a little shape-accuracy in return for the cylindrical advantages up to lat 42.

Those cylindrical advantages include:

1. Conformality along two parallels instead of just at two points.

2. A latitude-ruler can have a scale for measuring longitude, at any latitude.

3. A latitude rule can have a scale that tells the north-south scale at any Y value.

4. Because the conformal-latitude is in the cylindrical section, then, for a given equator-length, the map is as large as it would be it were Behrmann all the way up.

Another advantage of cylindnicals is their stark simplicy...but that doesn't apply to Rounded Behrmann, because of the larger formula expressions for its above-graft part.

The version that you posted, with a shorter pole-line, has, in the inward-curved-meridians section, better shapes at the central meridian, than does Rounded Behrmann. So it's just a matter of whether one is more interested in shape-accuracy at a favored middle longitude, or trying to reduce the distortion at more peripheral longitudes instead, at latitudes not too far above the graft.

I'm not entirely sure which of those ways I'd prefer it.

I guess that's the decisive issue with Eckert IV vs Eckert VI too: Central meridian region vs more peripheral longitudes, so either projection could be considered better,depending on whether one is more interested in central longitudes or all longitudes.

Michael Ossipoff