Longitude Mix

[Piotr]

If this request becomes true, then in future daan will announce a fun feature, officially called "Longitude Mix". You multiply the longitudes, by a factor.
Q: "Piotr, isn't that already possible by shrinking or growing the database horizontally around the center while preserving image size?"
A: That's the method I used for the preview, but you didn't read the entire feature. Try to blend a normal projection with a projection done with your method.

If it's possible in the projection (always possible in equatorial cylindrical and pseudocylindrical projections), you can show the entire world while the longitude is grown. There is also an option to preserve the graticule, or multiply the longitude graticule lines together with the map longitudes.

This can be used to generalize Aitoff and Hammer. Hemispheric Stereographic is shown below, with a factor of 0.5 and widened by 2x:


Conformal Longitude Mix: shrink or grow the Mercator projection in both x and y to multiply longitudes while preserving angles. With a factor of 0.5, the world is projected into a hemisphere into a manner similar (identical?) to Gilbert's two-world globe.

How should this work if the map isn't equatorial?

[daan]

I think it is longitudes, not latitudes, that you have multiplied by a factor in your graphic?

This “meridian duplication”, as I call it, is the basis of the Aitoff and Hammer projections, as well as of Geocart’s “heart” projection. It’s part of Wagner’s Umbeziffern process, which also shrinks latitudes. Hammer’s innovation was to realize that Aitoff’s method preserves areas. Because the heart projection uses the same technique on Bonne, it, too, is equal-area. Das Umbeziffern uses meridian duplication, and also provides an equal-area transform for latitudes for those cases when you wish to preserve area measure.

I do aim to add more general ways to modify projections to Geocart. Mathematically, it’s easy. The problem is the boundary description.

Regards,
— daan

[Piotr]

Oops, sorry for the mistake. Corrected everywhere, except in the title of your post.

[Piotr]

You should do both Longitude Mix, Conformal Longitude Mix and Latitude Mix, with an option for these to be relative to graticule, or relative to the projection being done with this option at a certain center and recreated on a custom one. For example, Aitoff would be the latter option with Longitude mix 0.5, and the longitude multiplying relative center to be at 0, 0, then stretched 2x. The Latitude Mix comes from Miller projection, and should have a latitude centering, for example if latitude center is 90° and the factor is 2, then 89° becomes 88°, 45° becomes 0°, 0° becomes -90° and so on.

[Piotr]

Let's analyze, what would happen in case of Longitude Mix with a factor of 0.5:

Forwards formula: Multiply longitude by 0.5, then apply the formula.
Backwards formula: Apply the formula, then divide longitude by 0.5.
Boundary description: As this Longitude Mix projects the world onto a hemisphere, the boundary would be the same as that of a hemisphere of the projection.
Derivative (necessary in Tissot Indicatrix): It would be very complicated.
Distortion Visualization: It would be very complicated.