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Lecture 7: Intro to Geometry, Splines, and Bezier Curves (50)

Would there be any situation where it would benefit to use higher-order polynomials in interpolation or graphics in general?


I think this is more of a rule of thumb as opposed to an absolute must-follow, since the optimal order of the polynomial should be the order of the system you want to model. If you have this information and you points can fit perfectly, then you could choose to use a higher-order polynomial for a more accurate model. However, as mentioned here, the issue with going higher than cubic for interpolation is overfitting, which can cause one small error to be magnified all throughout the polynomial. In fact, it would better to underfit, since that effectively acts as a smoothing filter on the system, which we know from examples such as antialiasing, can be a good thing


I think another reason why cubics are used is because they are the smallest degree function, and therefore simplest function that can interpolate between 2 points, since if you take a quadratic, you have 3 like degrees of where you place the function, and as such 2 points set, leave you with only 1, wheras in a cubic, you have 2 and therefore it is more accurate than a quadratic, and you can just peicewise interpolate.Another point is that a quadratic function also can only really point in one direction at their endpoints, up or down, so the peicewise function would be more noticeably weird, and with larger degree functions its also easier to get weird fluctuations and oscilations in the graph.


My thought is that the cubic curves can provide enough information to generate a smooth signal. While the order gets higher, it will be much more computational costly, especially in real-world applications. Meanwhile, high-order polynomial may not generalize well for every 2D/3D signal we have. So cubics are more widely used.


Would it be a good idea to use piecewise function to define a curve, i.e. each piece is a controllable lower-order polynomial? I think it would resolve the stability problem.


Is there some rule (maybe based on the nature of the points) to determine if another degree polynomial could work better than a third-degree polynomial?


Intuitively it makes a lot of sense to me to use cubic functions instead of higher or lower-degree polynomials, as it provides a balance between complexity and simplicity. However, is there any mathematical proof that shows actually cubic polynomials is the best we could use (minimize some cost function), or is this just empirical choice?

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