displaying panoramas on fulldome

I had explored displaying panoramic photos on our mirrordome setup in an earlier post, just using Filters -> Distorts -> Polar Co-ordinates in Gimp on a square image. Here are some more details. The filter dialog box was as below,

Here, the map backwards prevents the resulting polar image from being left-right inverted. If Map from top is removed, the result is like these mini planets. The offset angle is entered so that Hanuman statue is centred on the resulting fulldome image.
Merely adding more 'sky' and making the pan square is not enough. The resultant fulldome image will appear squished down vertically. Example below.

This image, which is square,

becomes this image on doing polar co-ords distort

So, the image has to be vertically stretched 2.5x for it to look reasonably good. The image below is stretched 3x, it looks a bit vertically elongated on the dome:

3x stretched pan made into a square,

and then polar:

Now, if the image is not made into a square, still the polar distortion can be done. But the resulting image is as if the original had been stretched to make it into a square and then the polar distortion was done. For example, with this original image,

we get this,

which is the same as we get if we distort this image got by stretching the pan into a square,

Now, these are not 360 degree pans - more like 180 degrees or even less. Assuming 180 degrees, we would get an image like this if we were to add white space for the missing 180 more degrees - I have put in a text background colour to highlight the white space.

But this also yields a vertically squished image,

 

As our audience is front-facing, the region which is well visible to our audience is highlighted below,

So I can try and make pans with some clouds or a fade-to-grey or something for the non-visible part, and push the relevant pixels into the visible part. The non-visible part is around 25%, so we can try adding 25% to the width of the image and then making it polar.