Complex Coordinates in Real Geometry

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Complex numbers usually appear solely as a way to solve polynomial equation. Here, I want to show you what that means for doing geometry.
This video complements my other one about perspective drawings: • Introduction to Projec...
While they can be understood completely independent from one another, they cover similar ideas from different perspective. It's definitely beneficial to watch both.
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Resources, references & further reading material:
Animations were made with CindyJS:
cindyjs.org/
Richter-Gebert, J. (2011) Perspectives in Projective Geometry:
link.springer.com/book/10.100...
Homogeneous Coordinates on Wikipedia:
en.wikipedia.org/wiki/Homogen...
Conic sections on Wikipedia:
en.wikipedia.org/wiki/Conic_s...
I & J on Wikipedia:
en.wikipedia.org/wiki/Circula...
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Chapters:
[00:00] Opening
[00:34] Homogeneous Coordinates
[05:03] Lines
[08:57] Conics
[13:28] Complex Numbers
[14:42] Circles
[17:42] Beyond Circles
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Music:
spacedust by airtone (c) copyright 2022 Licensed under a Creative Commons Attribution (3.0) license. dig.ccmixter.org/files/airtone...

Опубликовано:

6 авг 2024

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Комментарии : 7

@doctorbobstone Год назад

I've been spending a lot of time on conic sections lately, so definitely a very interesting topic. I haven't quite figured out if this will provide actual tools I need, but it's certainly interesting to learn. Thanks!

@wafikiri_ Год назад

When I was taught affine geometry (similar to the projective geometry in this video, but different), there was an extra, parametric element in vectors. You could use certain determinants to represent a conic, and analyse the determinant to learn what kind of conic it was, or use it in equations to compute problems like finding the family of circles in the plane tangent to a given straight line and passing through a given point.

@d.l.7416 11 месяцев назад

If you look at the two points where all circles intersect, if you scale them by a complex unit then they rotate. one of them turns clockwise, one goes anticlockwise. you can see this if you look at the real part.

@rustycobalt5072 Год назад

Quite a nice explanation and visually amazing video... (the infamous) but Isn't this a reformalization of grassman algebra?

@sumandproduct Год назад

Thank you! Yes, most of these things can be formulated via Grassmann algebras. For this simple 2D case it's a bit overkill, but it's crucial in higher dimensions.