Complex Coordinates in Real Geometry

Complex numbers usually appear solely as a way to solve polynomial equation. Here, I want to show you what that means for doing geometry.

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:

Richter-Gebert, J. (2011) Perspectives in Projective Geometry:

Homogeneous Coordinates on Wikipedia:

Conic sections on Wikipedia:

I & J on Wikipedia:

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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:

[doctorbobstone]

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!

[d.l.]

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.

[rustycobalt]

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