In our last video, we talked about de Casteljau Bezier curves, mostly cubics, for design work. In this lecture we discuss another application of cubic splines—to the interpolation problem: finding a smooth curve passing through a finite number of points in the (x,y) plane.
Our approach to this question is somewhat novel, and focusses on the use of what we call Taylor coefficient vectors. A given cubic polynomial in our space P^3 has a 4-vector of Taylor coefficients at any point, and the relations between two such Taylor vectors is given by a linear transformation, essentially a Pascal matrix (see WLA23).
So our strategy is to create the cubic spline one segment at a time, transferring the knowledge of the Taylor coefficient vector at one endpoint to the other.
Although we are using calculus ideas, we develop them independently, so the viewer is not required to have had prior knowledge of calculus.
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Screenshot PDFs help you navigate through most of Norman’s video series: they can be found at http://www.wildegg.com/wildegg-store….
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/…. I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?… .