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.
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!
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.
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.
This is absolutely RP^2. Nice explanation bro. Keep it up🎉
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.
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!
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.
Quite a nice explanation and visually amazing video... (the infamous) but
Isn't this a reformalization of grassman algebra?
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.
Great stuff. Thank you.