Geometric Algebra

Currently Playing: Reflections | Geometric algebra episode 12

#geometricalgebra #projections #reflections #rotations #sandwichproduct Now that we know how to perform projections, it's very easy to create reflections. A crucial step in the calculations is when we put the reflection vector and its inverse around the other vector, creating a sandwich product. Every other important operation in Clifford algebra uses a sandwich product, which is very clean and easy. I also show you some pitfalls that you should avoid when working with reflections in geometric algebra. On Patreon, Rémy pointed out that I made a mistake in this video. When giving you the geometric interpretation of the sandwich product, I forgot to mention that this only works in projective geometric algebra, not in the plain "vector" variant that we talk about here. Sorry for the confusion, and thanks to Rémy for setting this straight. You can support us on Patreon, where you can already watch all the Geometric Algebra videos and get access to exclusive content. Unfortunately, we won't be publishing any new videos after Geometric Algebra, at least not in the near future. Your support is still more than welcome of course: https://www.patreon.com/user?u=86649007 [EIG 1] https://www.youtube.com/watch?v=AFwc0DPoFe8 Around 8:30, you can find a simple calculation that proves that a reflection doesn't change the length of the input vector. [MOM 1] https://www.youtube.com/watch?v=ERpcSJzX448 A video about projections and reflections in geometric algebra. The entire series by Mathoma is really good. 0:00 Creating a reflection from a projection 1:46 We get a sandwich product 3:10 Passive objects become active transformations 4:27 Properties of the reflection formula 5:40 Potential pitfalls, alternative formulas 8:54 Conclusion This video is published under a CC Attribution license ( https://creativecommons.org/licenses/by/4.0/ )