In topology, when a shape is morphed into another shape, do the points inside of the shape move with the morphing of the shape, or does the boundary envelope or lose points as it changes? Visualization of the Question: either the shape's points move, or the boundary envelopes new points
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Welcome to math.SE! <> One snag in translating the informal idea of "morphing" into mathematics is the care required. If morphing is to represent a "continuous" operation, we normally work with mappings rather than sets (homotopy); if instead we want to work with sets of points, we generally imagine "morphing" to be a "discontinuous jump" from the initial to final shape (diffeomorphism). What works best for your situation depends on your aims. – Andrew D. Hwang Oct 24 '22 at 15:23
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@AndrewD.Hwang so it is a matter of preference? If so, then how might one choice be preferable in a given situation? What "aims" determine this choice? – Yamden Oct 26 '22 at 15:45
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For example, if you're animating a transition or modeling a physical process of deformation, homotopy is likely to be more appropriate. If instead you want to show two regions are "topologically equivalent" it's enough to find a diffeomorphism. <> The diagram (as I understand it) shows something more like a physical process of abrasion and deposition, which doesn't fit neatly into either framework. – Andrew D. Hwang Oct 26 '22 at 17:57