Using azimuthal orthographic projection in QGIS?

Newer versions of QGIS have 'simplified geometries' enabled by default. As a result you might get an image like this:

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To disable simplified geometries, untick the "Simplify geometries" option in the layer rendering options.

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I'll update the blog post when I can.

I recommend reprojecting the shapefile first.

The On-the-fly-reprojection of QGIS is sometimes not able to cope with strange projections.

And make sure you are always working on a sphere. Mixing ellipsoid and sphere might also produce artefacts.

The method by Hamish works for me. I added a Vector -> Geometry -> Densify on the grid before reprojecting to Ortho:

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If you reproject the clipped part of the world to aeqd, it does not really look the same:

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The degree grid does not give a perspective view as the ortho projection provides. The aeqd definition promises equdistance, that's opposite to the perspective ortho view.