panorama mapping formula

As mentioned in my previous post, the field of view is what decides the "stretching" if needed. As an empirical test, the original stadium pan in this post seems to be around 120 degrees horizontally and around 60 degrees vertically. So, vertically it would need to be multiplied by 3 to cover the full 180 from front to back, and horizontally it would need to be multiplied - again by 3 - to cover the 360 degrees all around the observer. And indeed, using white space instead of cloning to save time, using an image where the pan is the ower middle tile in a 3x3 set of tiles of the same size as the pan,

we get this polar image, which seems to be quite undistorted.

If we keep the same aspect ratio as the orig of this pic and reduce the white space, we get more coverage of the dome with the image. The extreme case of this would be to use the pan itself without any padding, since the padding is equal in horiz and vert directions in this case. Then we get this,

where we can see that the Hanuman statue which is further away from the horizon seems to be vertically stretched. On the dome also, it is seen to be vertically stretched. So, in general, we should shoot for images which are near the horizon, and we can clone the sky or mirror the image for filler.