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Lecture 7: Intro to Geometry, Splines, and Bezier Curves (32)

Why a cubic polynomial specifically?


I think it's because we have 4 constraints on our interpolation function: 2 function values and 2 derivatives, which determine a cubic polynomial.


What does the "no good approximation" mean here? I understand that if the u samples are close, the estimation could be less reliable, but I'm a little confused on why the "no good approximation" is equivalent to linearly independent.


No good approximation means you can't substitute/approximate x^k with the linear combination of all other x^i. That's why the elements are linearly independent.


There is no good approximation because if i unequal to k, then u^i is either a higher-order infinitesimal quantity of u^k or a higher-order infinitesimal quantity of it,,which means lim(u^i/u^k) = 0 or C(C unequal to 1).

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