Blending equal-area map projections

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Qaanol
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Blending equal-area map projections

Postby Qaanol » Thu Feb 05, 2015 6:36 am UTC

I want to create an equal-area map projection that is a blend of the Hammer and Mollweide projections, but I’m not sure how to do it or if it’s even possible.

Near a certain location (0°N, 150°E) I want the map to be as the Mollweide projection. Near the antipodal point (0°N, 30°W) I want it to be as the Hammer projection. And I want them to smoothly meld throughout the rest of the map, while maintaining the equal-area property.

My first thought is, for each point (θ,φ) on the globe, find the angular distance from (0°N, 150°E), call it α, and take the weighted average of the projections of that point under each mapping, as (α/π)Hammer(θ,φ)+(1-α/π)Mollweide(θ,φ). However, I haven’t been able to show whether or not this preserves areas.

Any ideas?
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PM 2Ring
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Location: Mid north coast, NSW, Australia

Re: Blending equal-area map projections

Postby PM 2Ring » Thu Feb 05, 2015 10:17 am UTC

I have no idea if your weighted mean will preserve areas, but it sounds reasonable. I guess you could brute-force test it by writing a program to project small circles & spherical triangles of known area.

It's a pity that Daniel "daan" Strebe doesn't post here any more. You could contact him via the mapthematics site. However, he might not be interested in doing such consultation for free.

Nicias
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Re: Blending equal-area map projections

Postby Nicias » Thu Feb 05, 2015 12:16 pm UTC

You can just math this.

Call you new map N(\theta, \phi). You just need to find the determinant of the Jacobian of your map. If it isn't cos(theta), you projection isn't equal area.

[edit]
After messing around, the math appears to be a PITA. I can't seem to find a formula for | J(N) | that doesn't require an explicit formula for the Mollweide Projection, which does not exist.

Nicias
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Re: Blending equal-area map projections

Postby Nicias » Thu Feb 05, 2015 9:02 pm UTC

Ok, doesn't work. Here is the Mathematica code I used:

Code: Select all

ClearAll[Evaluate[Context[]<>"*"]]
$Assumptions=  -Pi<=p<=Pi && -Pi/2<=l<=Pi/2 ;
sol=NDSolve[{2T'[p]+2Cos[2T[p]]T'[p]==Pi Cos[p] ,T[0]==0},T,{p,-1.5,1.5}];
Theta=sol[[1]][[1]][[2]];
p0=0;l0=0;
M[l_,p_]:=Sqrt[2]*{2 l/Pi*Cos[T[p]],Sin[T[p]]}
H[l_,p_]:= Sqrt[2]*{2 Cos[p]Sin[l/2],Sin[p]}/Sqrt[1+Cos[p]Cos[l/2]]
Alpha[l_,p_]:=ArcCos[Sin[p]Sin[p0]+Cos[p]Cos[p0]Cos[l-l0]]
PM[l_,p_]:=(Alpha[l,p]* H[l,p]+(Pi-Alpha[l,p])*M[l,p])/Pi
Jac[l_,p_]:=D[PM[x,y],{{x,y}}]/. {T->Theta, x->l, y->p}
Error[l_, p_]:=(Det[Jac[l,p]]-Cos[p])Sec[p];
Plot3D[Error[l,p], {l,-3,3},{p,-1.5,1.5}]

The Error ranges from -0.1 to 0.05. So the area is off by up to -10% in some areas and +5% in other.
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Strebe
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Re: Blending equal-area map projections

Postby Strebe » Sun Feb 04, 2018 8:32 pm UTC

In general, weighted averages do not preserve areas. Here is a link to my paper that describes how to do this for •any• two equal-area projections:
Regards
— daan Strebe


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