It's actually possible to extend complex numbers to handle 3D rotations and translations. The 3D analog of the complex numbers are well known as the quaternions, but there also exist the dual-quaternions which are capable of describing any proper rigid transformation, ie rotation and translation. There's also an interesting way to extend these to higher dimensions as well as other types of transformations. While the components grow faster than matrices, doubling with each additional underlying dimension rather than going to the next square, they provide much smoother interpolation. I actually noticed a few times in this video where an object appeared to shrink as it was moving before ending up at the same size as it started.
Indeed, and interestingly, the complex numbers are the even sub-algebra of the 2D geometric algebra and the quaternions are the even sub-algebra of the 3D geometric algebra.
@@cstockman3461 Every geometric algebra is the even subalgebra of a higher geometric algebra. The dual-quaternions are the even subalgebra of 3D projective geometric algebra, and 3D vanilla geometric algebra is the even subalgebra of spacetime algebra. Geometrically, the PGA interpretation can apply to other algebras, in which case not only in VGA algebraically a subalgebra of PGA, but geometrically too, with a given VGA multivector representing the exact same subspace and transformation in PGA. Geometric algebra isn't really an algebra. It's more like a Matryoshka Doll of algebras. Even the basic Real numbers can be considered the even subalgebra of the complex numbers. Geometric algebras all the way down... And going all the way _up_ eventually brings you to the mythical Universal Geometric Algebra (UGA) aka Cl(∞,∞)
Very cool. Now, it is just a small step to quaternions😀. By the way, since there was a short blender clip inside the video, I just wanted to mention that I'm working on a library that realizes much of the manim tools inside blender. If you are interested, let me know.
so everyone knows, nowdays is common to hear that matrices do transformations, which is misleading what is actually happening is that in algebra, there is a concept called linear transformations that are just equations with some constraints this equations end up as a system of equations with each equation having a series of products between constants and variables, such as: a*x + b*y = k c*x + d*y = h and all linear transformations have a matrix representation, which, in this case, is: | a b | | x | | k | | c d | * | y | = | h | so the matrix abcd represents a specific linear transformation over some coordinates xy this transformation can be whatever you want, but if you want specific properties for this transformations, you can specify it in the original equations, figure them out and then the matrix comes in free
The scaling factor being a higher dimensional shear operation seems so obvious in retrospect.
Thank you so much for the animation!
Great video!
One off the best I've seen on the topic!
@@RickCharf Thank you!
Great graphics and explanation. I thought I was watching a 3 Blue 1Brown video at times. Well done!
It's actually possible to extend complex numbers to handle 3D rotations and translations. The 3D analog of the complex numbers are well known as the quaternions, but there also exist the dual-quaternions which are capable of describing any proper rigid transformation, ie rotation and translation. There's also an interesting way to extend these to higher dimensions as well as other types of transformations. While the components grow faster than matrices, doubling with each additional underlying dimension rather than going to the next square, they provide much smoother interpolation. I actually noticed a few times in this video where an object appeared to shrink as it was moving before ending up at the same size as it started.
Indeed, and interestingly, the complex numbers are the even sub-algebra of the 2D geometric algebra and the quaternions are the even sub-algebra of the 3D geometric algebra.
@@cstockman3461 Every geometric algebra is the even subalgebra of a higher geometric algebra. The dual-quaternions are the even subalgebra of 3D projective geometric algebra, and 3D vanilla geometric algebra is the even subalgebra of spacetime algebra.
Geometrically, the PGA interpretation can apply to other algebras, in which case not only in VGA algebraically a subalgebra of PGA, but geometrically too, with a given VGA multivector representing the exact same subspace and transformation in PGA.
Geometric algebra isn't really an algebra. It's more like a Matryoshka Doll of algebras. Even the basic Real numbers can be considered the even subalgebra of the complex numbers. Geometric algebras all the way down... And going all the way _up_ eventually brings you to the mythical Universal Geometric Algebra (UGA) aka Cl(∞,∞)
any translation in low dimensions can be represented as a transformation in higher dimensions (n+1). Great illustration !
Super cool video, really helpful to build intuition.
Fan of your channel!
when a student tries to become a teacher. thats you and this video. NICE
Very cool. Now, it is just a small step to quaternions😀. By the way, since there was a short blender clip inside the video, I just wanted to mention that I'm working on a library that realizes much of the manim tools inside blender. If you are interested, let me know.
So cool! It's definitely going to be helpful for me
Great video! I didn't knew homogeneous coordinates intuitively. ! Nice visuals
Thank you so much!
Now make a 4D game using 5D matrices (5x5 matrices)
thats EZ
CodeParade be like
awesome job!
this is exactly what i wanted!!
Linear algebra is still a very new concept for me but this video was very nifty! Awesome work :)
Wow😲. So helpful to me. Thanks a lot.
It is possible to visualize 4D geometry, and even to show it graphically and animate it.
Great explanation, thanks a lot!!!
so everyone knows, nowdays is common to hear that matrices do transformations, which is misleading
what is actually happening is that in algebra, there is a concept called linear transformations that are just equations with some constraints
this equations end up as a system of equations with each equation having a series of products between constants and variables, such as:
a*x + b*y = k
c*x + d*y = h
and all linear transformations have a matrix representation, which, in this case, is:
| a b | | x | | k |
| c d | * | y | = | h |
so the matrix abcd represents a specific linear transformation over some coordinates xy
this transformation can be whatever you want, but if you want specific properties for this transformations, you can specify it in the original equations, figure them out and then the matrix comes in free
excellent explanations
Thank you!
Great video
Really nice
excellent
great video! at 6:32 please use 'dots' instead of 'x' for matrix multiplication :)
why is that
@@bbrother92 X implies the cross product which is a different type of multiplication
@@matthewjames7513 im p sure cross product is only defined for vectors