Topologically mirages can be represented by imagining that what we actually see by eye is pasted onto a transparent sphere. This is called the 'image sphere'. The eye or camera is at the sphere's centre.
The object creating the mirage can be imagined pasted onto a more distant 'object sphere'.
The refractions and curved rays occur between the two spheres. However, the beauty of this topology approach is that we do not have to worry about how or why the ray paths get curved. All we have to do is look at the relation between the images pasted on the two spheres.
The mirage from the inner sphere can be transformed into the object and vica-versa.
To do so make a rubbery stretchable copy of the inner sphere, Call it the transfer surface. Then inflate it so that it expands towards the outer object sphere. The aim is to make the mirage images on the transfer sphere exactly match the single object sphere image. In topology, stretching and folding is allowed - but in this case any cutting, snipping and discontinuities are forbidden.
Below left: The three mirage images are made to overlay each other and be congruent to the object by folding the transfer sphere twice. If there had been five images then another two folds would have done the trick. Had there been four images it would have been impossible.
The point of all this rubbery playing is that it generates deep insights into the underlying rules and structure of mirages. For example, allowed mirages are erect, inverted, erect,.. Mirages like erect, erect, inverted are not because they cannot be made by smooth folding of the inner sphere.
Another result is that in a complete mirage there are always an odd number of images (including the 'real' one). Try folding the sphere to make an even number! Many mirages are incomplete because the ground or an air layer cuts them off - we do not then see all the images.
For a more complete description see this page.
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