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To avoid the possibility of opinions, let me rephrase the question. Which software packages are able to produce an Oblique Mercator map between any two arbitrary points, similar to what is offered in Matlab (see http://www.mathworks.com/examples/mapping/mw/map-ex21842716-create-long-narrow-oblique-mercator-projection) but without Matlab's price tag of thousands of dollars?

From what I've learned, Google Maps does not seem to work well at the poles. I'm willing to pay a few hundred dollars for the package if necessary, or I can put in the time to learn to use Geotools, GMT or whatever tool that can produce these types of maps.

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No problem for QGIS:

Write the coordinates of both points in decimal degrees into a text file:

Name East North
Tokyo 139.75 35.6667
NewYork -73.96667 40.78333

Load the points file as delimited text

Create a aeqd projection on one point using a sphere:

+proj=aeqd +lat_0=40.78333 +lon_0=-73.96667 +x_0=0 +y_0=0 +a=6371000 +b=6371000 +units=m +no_defs

Set the project CRS to the aeqd projection

Measure the angle between a point North of NewYork (x=0) to NewYork, then to Tokyo (about -27°)

Create a line from NewYork to Tokyo, and densify the Geometry with 99 points

Create an omerc projection with these parameters:

+proj=omerc +lat_0=40.78333 +lonc=-73.96667 +alpha=-27 +k=1 +x_0=0 +y_0=0 +gamma=-90 +R=6371000 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs

Add a raster basemap, like Blue Marble

Set the project CRS to the omerc projection

You will get this picture:

enter image description here

For vector data, you have to clip the data to the visible extent to avoid artefacts. If it does not look as expected, check that QGIS uses your custom CRS on the files, and not a Generated CRS which might have different parameters.

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