# proj: Oblique Mercator specifying two points on central line

I'm trying to use Oblique Mercator in QGIS 3.4, which uses `proj` to calculate coordinates conversion. I specify two points in the central line, rather than specifying the center and alpha.

Oblique Mercator: Usage explains

In the second case, the azimuth is given indirectly by specifying two points on the central line, using the options +lat_1, +lon_1, +lat_2, and +lon_2.

Here are the two locations I'm using:

• Tokyo: latitude: 35.652832, longitude: 139.839478
• New York: latitude: 40.730610, longitude: -73.935242

Calling `proj` command (Rel. 6.1.0) specifying the two locations for the central line of Oblique Mercator and converting the two points to x-y coordinates gives:

``````\$ echo -73.935242 40.730610 |proj +proj=omerc +lat_1=35.652832 +lon_1=139.839478 +lat_2=40.730610 +lon_2=-73.935242 +ellps=GRS80
5162164.32  14191389.18
\$ echo 139.839478 35.652832 |proj +proj=omerc +lat_1=35.652832 +lon_1=139.839478 +lat_2=40.730610 +lon_2=-73.935242 +ellps=GRS80
1450567.63  3987778.87
``````

In the output, I expect the "y" parts of the two coordinates are the same, because these two points are on the central line of projected coordinates. However, they are not (14191389.18 != 3987778.87).

Can somebody explain how omerc with two points in the central line works and provide correct parameter for `proj`?

• Possibly try two-point equidistant instead. it will align the two cities on a horizontal line or try adding +no_off to your omerc command. – mkennedy May 22 at 17:38
• @mkennedy Unfortunately `+no_off` did not change the output and I want to know how "omerc with two points in the central line" works rather than calculating their equidistant by myself. – suztomo May 23 at 14:10
• @mkennedy BTW, I feel it's a bug in proj. If you also think so and post it as an answer, I will be happy to accept that. – suztomo May 23 at 22:28

In the output, I expect the "y" parts of the two coordinates are the same, because these two points are on the central line of projected coordinates. However, they are not (14191389.18 != 3987778.87).

Your expected result is only true in the "unrectified" uv plane. Some academicians view the Hotine Oblique Mercator to be a triple-projector, i.e., (1) from ellipsoid/sphere to "aposhere", (2) and then from "aposphere" to uv plane ("unrectified"), (3) and finally from uv plane to grid plane ("rectified"). The picture below shows Tokyo and New York in the uv plane. (Note that the result in uv plane is flipped.) In PROJ 6's implementation, when you choose the Two-Point Method/Formula, it performs rectification at the Natural Origin. The result is in the picture below. Having said the above, please note the following :-

• The Hotine Oblique Mercator Two-Point Method/Formula first appeared in Snyder's Map Projections: A Working Manual. The IOGP Publication 373-7-2 acknowledged the Method/Formula, but (sort of) rejected it.
• Synder himself wrote that "Normally, the Oblique Mercator is used only to show the region near the central line and for a relatively short portion of the central line." This means distortion increases as we move further away from the Central Line and the Centre of Projection. For the Two-Point Method/Formula, I have not yet come across any publication as to where is the Centre of Projection or if there would be one in the first place.
• Thank you for great explanation with the images. Would you share a reference (or concrete command/tool) how you created the first image through the 3 steps? – suztomo Sep 28 at 11:52