I'm real sorry, I still think I'm missing something here so I'll use moonlord's taper algorithm for the circle to line argument and someone tell me which step I've got wrong.
1. Take two bodies, A and B. Each of them is embed in the minimal dimensional hyperspace (the bounding space of each - alpha and beta). Place the two bodies in such a manner that these two hyperspaces do not intersect.
Body A: Circle
Body B: Line
2. Find a way to continuously transform A into B by changing it's parameters, like length, height, trenght, radius and so on. If it doesn't exist, tapering is not possible (not sure here).
If we draw 2 perpendicular axes on a 2 dimensional sheet of paper (embedding space) and draw a circle of radius 1, for instance, then we can shrink the "height" down to zero. What I literally mean is connect everything in the upper-half plane to a single corresponding point in the lower half plane with vertical lines parallel to the y-axis like in the diagram below
(clearly I got lazy with drawing lines so imagine a blue line connecting every point above the x-axis with a corresponding point below it) Now shrink each of these lines down to length zero (excluding the endpoints because they're already at "height" zero). Now we have a line, no?
3. Transform A into B (cont.) and in the same time translate the current body towards B so that when the transformation is complete (A has become B), the result is in the position of B.
So like I said in my previous example, each time the circle "shrinks" a little bit it gets a little bit "closer" to the line. So it's similar to a cone except that it has a line for a tip rather than a point.
4. Consider the whole "trail" one body, C.
So my line-cone is the taper?
(from above: except that it should be a circle, not an ellipse. Oops)
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Now, since there are infinitely many ways to taper a circle to a line depending on orientation of the line with respect to the shrinking axis and/or the angle at which the circle approaches the line in the time translation, which one do we consider the canonical tapering? (You mentioned that the embedding spaces are parallel, but you did not mention whether A and B are "on top" of eachother in a strictly informal use of the word.
Also, if a circle cannot be tapered to a line could you please explain to me via the same steps I just took how a disc can? I just don't see where the difference is, especially if a circle can be tapered to a point. See, you say that two objects are taperable if they are both special cases of one thing. A circle is a special case of an ellipse (you said). A line is a special case of an ellipse too, no? Ellipse has two axes of different lengths. Shrink the length of one of these axes down to zero and you have a line. (I'm certain this is where I'm getting my definition mixed up.) Thus circle and line are taperable.[/img]