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Comment by dvt | Hacker News Reader

I've seen this a while ago, and while it's pretty instructive, it's actually also pretty confusing. The magic happens in a seemingly-innocuous throwaway sentence at around 4:20 (after being introduced to the 2D plane):

> ... This can now include rotating along with some stretching and shrinking ...

It's entirely non-obvious WHY we should be okay with rotating all of a sudden. The real answer is not super complicated, but it deals with a couple of amazing relationships between exponential functions and trigonometric identities[1]. So really, we don't have to accept "rotating" as some weird new action we can do when moving into the complex plane, we just have to accept that trigonometry is weirdly related to analysis due to some very cool properties.

[1] https://www.phy.duke.edu/~rgb/Class/phy51/phy51/node15.html

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>It's entirely non-obvious WHY we should be okay with rotating all of a sudden.

Why shouldn't we? We have a set of object called multipliers that we want to generalize to the 2 dimensional plane. The presented generalization (eg, the multiplier identified by x maps the point at 1 to the point at x) seems natural; and it defiantly seems to be well defined. Additionally, it appears obvious that the presented generalization is equivalent to the original definition if considered only along the x axis.

From the perspective of the video, the fact that e^x is written as an exponential is "a vestige of its relationship with repeated multiplication". In the construction presented in this video, we should think of e^x as the "natural" homomorphism from adders to multipliers.

The fact that this function has anything to do with exponential, trigonometry, or analysis is neat, but not important in this context.

> Why shouldn't we? We have a set of object called multipliers that we want to generalize to the 2 dimensional plane. The presented generalization (eg, the multiplier identified by x maps the point at 1 to the point at x) seems natural

Rotating all of a sudden does not feel natural to me at all. Why not start cutting up the 2D plane? Why not fold it? In analytic terms, we would cause discontinuities, whereas rotating is "smooth". But this is definitely nontrivial. Rotating is just so arbitrary. I think that skewing or flipping, for example is just as "natural" as rotating.

> The fact that this function has anything to do with exponential, trigonometry, or analysis is neat, but not important in this context.

It's actually at the heart of why rotation (and not some other geometric operation) is key to e^iπ.

The better idea is to start with vectors. Adding vectors or scaling them is straight-forward. It’s trickier to figure out what it means to multiply or divide two vectors.

The quotient of two vectors v/u should be some kind of operator which transforms one into the other (that is, when you multiply it by one, you get the other, (v/u)u = v(u\u) = v, because we want multiplication to be associative). If those two vectors are the same length but different directions, the natural transformation to use is a rotation. The reason to use a rotation is that we want the transformation to make sense irrespective of any arbitrary coordinate system we decide to impose. If we used some kind of skew, it would break down under change of coordinates. As for reflections: if the quotient of two vectors was some kind of reflection, then we could square any quotient of vectors to get an identity transformation, which would not result in a very useful or consistent arithmetic.

The nicest and most useful formalism for defining multiplication of vectors is called geometric algebra, a.k.a. Clifford algebra. Start with http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Or see the recent blog post http://www.shapeoperator.com/2016/12/12/sunset-geometry/

Or see more links at https://news.ycombinator.com/item?id=12938727#12941658

We want some operation that maps the point 1 to the point x, while holding the origin constant. I agree that scale and rotate is not the only such function, but when I personally visualize taking a grid and moving one point while keeping another constant, that is what I visualize. Additionally, this has the added property of maintaining the grid structure.

It is not immediately obvious to me how skewing could define such an operation in the general case; or how flipping could define such an operation in most cases.

In any case, the system of adders and multipliers in 2-dimension he describes, even if not the only reasonable 2D generalization, is certainly a reasonable generalizaion, and one that has proved useful.

>It's actually at the heart of why rotation (and not some other geometric operation) is key to e^iπ.

In this video, e^x is defined as a mapping between multipliers and adders. Multipliers are defined to be a combination of rotation and scaling. Any relationship between exponential, calculus, infinite sums, etc is purely a result of these definitions (except, perhaps, the motivation for choicing i * pi to be the principle multiplier mapping that gets mapped to -1 under e.)

> Multipliers are defined to be a combination of rotation and scaling.

True, my point was only that the definition obscures the fact that rotation (and specifically the stunning relationship between trigonometry and exponential functions) is the key to the answer of "why". Without that, I think the video is just an exercise in indirection.

as a masters in mathematics, this is exactly what i came here to write. thanks!

To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula. I honestly think the best way to get a visual sense for why multiplication by rexp(itheta) is to look at the first few terms of the taylor series added together and see that the adders combine into a spiral that converges on rcos(theta) + isin(theta). that is still plenty visual for me.

Sure thing!

> .. see that the adders combine into a spiral that converges on rcos(theta) + isin(theta)

Incidentally, that's how Mathologer explains e^iπ and I like that explanation a lot more.

>To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula.

No you don't. You might notice this by simply working with imaginary numbers. You might invent imaginary numbers for rotation [0].

Alternatively, consider the multiplication (x + yi)(a + bi), as the value (a + bi) performing a transformation on (x + yi). We want (x + yi)(a + bi) = xy - by + ayi + bxi. If we consider (x + yi) to be a 2 dimensional matrix (with basis 1 and i), we can write the above equation as a matrix multiplication:

    [ x   y ] [  a   b ] = [ ax-by ; ay + bx]
                  -b  a

Notice that

    [  a   b  ] 
      -b  a

is just a rotation matrix multiplied by the scalar (a^2 + b^2).

[0] That is, define a group (in the group-theory sense) of functions of rotations, denoted as xi for real numbers x, and a group of group of functions for sliding, denoted x for real numbers x. You might then notice that you can combine these groups in a field structure, that happens to have 1i * 1i = -1, and that the subfield of elements with no i component happens to be isomorphic the the reals.

Yay skew symmetric matrixes.

It's also fun to introduce e as a matrix operator for 3d rotations.

It's useful for kinematics and having compact representations for axis angle notations. It's a little far in my head but at some point it felt like a ha-ha moment with the Euler identity, in the "2d version".

In the three-dimensional case, for any practical purpose, use quaternions. https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotati...

(At least if you’re ever needing to compose rotations; to apply a quaternion to a big array of 3-vectors, go ahead and convert it to a 3x3 matrix first, which should end up slightly more efficient.)

3x3 rotation matrices are really hard to keep normalized properly, whereas quaternions are trivial to normalize (just divide by the norm).

If you need to compress a unit quaternion down to 3 numbers, instead of taking the logarithm (which is computationally expensive and a pain to deal with) use the stereographic projection.

Quaternions are a useful tool for manipulating rotations in a lot of common applications. But that wasn't my point here. Also quaternions are hard to grasp by humans. I find axis-angle much more palatable in general.

Imaginary numbers can be represented with a 2x2 skew symmetric matrix with no stretch of the imagination at all. And 3x3 skew symmetric matrixes represent rotations most compactly with only 3 actual variables. Instead of 4 for quaternions, 9 for "classic rotation matrixes", or the need to tell which is the order of the angles if you're given 3 euler angles.

There are interesting applications of Lie Algebra on SO(3) [1], notably in computer vision where a global energy is minimized across two successive rgb-d "shots" in order to recover the infinitesimal rotation [2]. It's going to be easier to minimize energy on something that is most compactly defined, and always amounts to a valid rotation.

[1] https://en.wikipedia.org/wiki/Rotation_group_SO(3)#Lie_algeb... [2] https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf

From what I understand that would be (sorta; I’m not an expert in Lie theory) the logarithm of a rotation. I find the stereographic projection to be a more useful way to compress an arbitrary rotation down to 3 dimensions, for most purposes.

Yes, precisely. exp() is the mapping from so(3) to SO(3), so you can use log to go the other way around.

Here is the relationship with the other representations [1]

Where it becomes interesting for our rigid-body transform application (or recovery of it, in the case of computer vision), is with Twist coordinates (6 element vector) which will map to a 4x4 Transform, again using the matrix exp() operator [2].

[1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representat... [2] https://en.wikipedia.org/wiki/Screw_theory#Twists_as_element...

Alternatively if you don't like coordinates, you may notice that multiplication by a unit complex number is linear and preserves the complex norm (i.e. modulus), so it defines an orthogonal transformation of the plane.

Topologically, unit complex numbers are the unit circle which is connected, and 1 is part of this circle so all of these transformations must have determinant 1: these are rotations.

No Euler formula.

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