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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

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.

dvt 10y ago
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.
gizmo686 10y ago
>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.
- mtourne 10y ago
  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".
  - jacobolus 10y ago
    In the three-dimensional case, for any practical purpose, use quaternions. https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation 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.
    - mtourne 10y ago
      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_algebra https://en.wikipedia.org/wiki/Rotation_group_SO(3)#Lie_algeb... [2] https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf
      - jacobolus 10y ago
        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.
- chombier 10y ago
  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.

dvt 10y ago

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.

gizmo686 10y ago

>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.

mtourne 10y ago
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".
- jacobolus 10y ago
  In the three-dimensional case, for any practical purpose, use quaternions. https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation 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.
  - mtourne 10y ago
    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_algebra https://en.wikipedia.org/wiki/Rotation_group_SO(3)#Lie_algeb... [2] https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf https://vision.in.tum.de/_media/spezial/bib/kerl13icra.pdf
    - jacobolus 10y ago
      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.
chombier 10y ago
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.