De Casteljau's algorithm and splitting dCB curves
Insights into Mathematics
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In the Algebraic Calculus we are interested in a completely explicit and algebraic form of the subject, where computations can be run exactly with appeal to infinite processes running off the page (or off the universe!)
With such an approach, the basic objects become the polynomially parametrized curves or de Casteljau Bezier curves. These really ought to be the central objects of study in any calculus course!!
Here we look at an explicit example, show it visually with GeoGebra, and then illustrate Paul de Casteljau's algorithm to compute explicit coordinates of points on the curve, and how to use that to split a dCB curve into two. This is useful both practically in applications, and also theoretically, as it allows us to regard (connected) pieces of a dCB curve as dCB curves themselves.
There is certainly some magic here, and you are invited to roll up your sleaves and investigate it yourself. Try to verify what I am doing, and to extend and generalize it. In other words, first check what I am doing, then convince yourself that it is more generally true, ... and if you are ambitious, see if you can formulate a proof that demonstrates it.
The full Algebraic Calculus course, is available at https://www.openlearning.com/courses/... while the screenshot pdf (251 pages) of all the lectures is at https://www.wildegg.com/store/p123/Al...
This is part of a series of video lectures designed for motivated amateur, or professional, mathematicians wanting to delve into pure maths research without the abstraction and complexity that currently besets much of the subject. We will aim to always connect our abstract musings to computational reality by focusing on examples and explicit, concrete calculations - naturally without any pretense of being "able to do an infinite number of things"!
To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at https://www.youtube.com/channel/UCriF..., or become a Patron at https://www.patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One advantage: you get to access pdf summaries of some of the videos (such as this one) along with GeoGebra worksheets (for example the dCB curve worksheet I use in this video).
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Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
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Algebraic Calculus One: https://www.youtube.com/playlist?list...
Algebraic Calculus Two: https://www.youtube.com/playlist?list...
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