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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Amazing, Life Changing Tutor
Before I met Jonathan, I was struggling through most of my STEM classes because I was simply not studying properly. He taught me all kinds of new study habits that would help me to save time, raise my grades, and lower my stress. Jonathan was specifically tutoring me in vector calculus and is an amazing tutor on the subject. You will walk away from a lesson with a game plan knowing exactly what you need to do before your next session with him or before your next exam in order to do well in the course. I highly recommend him as a tutor.