An example of where something like this is likely to occur in practice is if you animate a figure to flip over by scaling it with values that vary from 1.0 (normal rendering) down to -1.0 (flipped rendering). As you iterate through the scale values from 1.0 to -1.0 you can at some point in the middle have a scale of 0.0 along the axis you are flipping it over. That is the point at which, visually, the figure looks like it is temporarily "side on" to your point of view and hard to see because it is so thin. And at this point in time you will have a transform that is mapping all coordinates in the figure to that thin vertical or horizontal line. At that point, the transform also happens to be invertible...
As someone pointed out to me - that last word is wrong, the last line should read:
transform also happens to be *non*invertible...
...jim
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