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Why can't you get all three properties? Is that because there aren't enough degrees of freedom? Would quadric splines help?


If achieving all three properties is impossible, is there a generally preferred two properties we would want to get?


Do the three properties have some connection such that it's proved that all three cannot coexist?


Can we get all three properties if we use non-cubic splines?


What's the importance ranking of these three properties?


Is the reason for not using other types of splines because they are complex? Or that there isn't a type that satisfies all 3?


I'm having a hard time understanding why locality fails for natural splines. It seems like if you change one piece of the spline, it should only affect the one piece and maybe its immediate neighbors, since they may need to adjust their endpoint positions or derivatives to match. Is this effect on the immediate neighbors enough to say that natural splines do not have locality, or is there some effect on the entire spline that I'm failing to see here?


When is it better to favor one property over another?


When would you ever want to give up exact interpolation? Animation seems to require (or at least strongly desire) all of the keyframes to be hit exactly.


Is there a theoretical proof that we cannot get all of the three properties?


Will these properties be considered in rendering? How important are them?


Is there any particular reason we stick with cubic splines if we can't get all three properties? Are these still the best overall?


Is it a matter of mathematical fact (proven) that you can't get all three? Or is it just an intuition given what we've discovered so far?


Does not affecting the whole curve mean that only its neighboring control points are affected or something like only a few coefficients can change from the originally determined curve? Or is there a range of change that is acceptable?