Gilbert two-world perspective doesn't make sense

[daan]

Where are these regions of overlap? I’m only aware of the reverse problem: The projection does not show the entire globe.

— daan

[Piotr]

Where are these regions of overlap? I’m only aware of the reverse problem: The projection does not show the entire globe.

— daan

Overlap is two or more points on the globe mapping to the same point on the map. Now imagine instead two or more points on the map globing to the same point on the globe. It's a non-bijectivity as well.

[daan]

Where does that happen?

[Piotr]

Where does that happen?

Due to the perspective, the North pole is a point **in** the map, not just the boundary. As a result, regions very much near the North pole are twice on the map.

[daan]

Due to the perspective, the North pole is a point **in** the map, not just the boundary. As a result, regions very much near the North pole are twice on the map.

I don’t think this is true. It is true that Geocart renders this way. Geocart renders this way because DeLucia and Snyder’s paper state that the projection is bounded by a circle of radius R. However, the paper’s statement seems to be erroneous. The true boundary is slightly smaller and possibly not a circle.

— daan

[quadibloc]

The North Pole is indeed "in" the Gilbert two-world perspective projection, and regions very close to it do appear twice on the map.
That's because the projection, in its normal aspect, shows a globe tilted slightly forwards, so that its North Pole is visible.
This isn't just due to the aspect of the projection, and is instead part of the projection itself.
In the case of an Orthographic projection of the globe, you can tilt the globe forwards simply by a choice of the orientation of the globe with respect to the globe that is being projected. This wouldn't work for the Gilbert two-world projection.
What instead happens there is that the two worlds to one world transformation (think of mapping two half-scale Mercator projections to one full-scale one) happens in Earth coordinates, and then the coordinates of the transformed globe are altered relative to an orthographic (or perspective?) projection to tilt the double-world globe forwards.
If one tilted the Earth forwards instead, instead of looking like a Mercator-on-a-globe, it would look like an oblique Mercator-on-a-globe.
I'd explain this with a picture, but my web site domain renewal (which I have now successfully purchased) is now percolating through the Internet. I hope none of the maps in any of my posts have been temporarily replaced by porn.

[daan]

The North Pole is indeed "in" the Gilbert two-world perspective projection, and regions very close to it do appear twice on the map.
That's because the projection, in its normal aspect, shows a globe tilted slightly forwards, so that its North Pole is visible.
This isn't just due to the aspect of the projection, and is instead part of the projection itself.

Yes. I don’t dispute that you can do this, just as you can do this to most projections whose generating formulæ don’t treat even multiples of the longitude as the same longitude. On the other hand, there is no reason it must be that way. And Delucia and Snyder’s bounding circle is simply wrong. It’s slightly too large, even given the one-to-many mapping near the north pole.

I will probably eliminate the duplication and provide a correct boundary in Geocart eventually.

— daan

[daan]

And Delucia and Snyder’s bounding circle is simply wrong. It’s slightly too large, even given the one-to-many mapping near the north pole.

I got a chance to look more deeply into this. I had misunderstood something in their paper: They give a value of 2.502° for a constant they call φ₁′. I thought that that value was how they defined φ₁′ but it turns out that it was merely a low-precision approximation for a transformation of 5°. Using the transformation at full precision, the bounding circle does have radius R.

I am not willing to use a mapping that isn’t bijective, though, so I have changed the boundary definition in Geocart to eliminate the excess in the region very close to the pole. This occurs north of latitude 89.9453537969439358390° on the outer meridian.

— daan