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tldr: Simply explained without demonstrations: Quaternions are hypercomplex numbers of the form w + xi + yj + zk Where w, x, y, and z are real and i^2 = j^2 =

tldr: Simply explained without demonstrations: Quaternions are hypercomplex numbers of the form w + xi + yj + zk Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: pqp' where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u .

coldacid 5y ago
I don't believe that most people will consider that "simply explained".
- marcodiego 5y ago
  Maybe succinctly?
  - coldacid 5y ago
    Yes, it is certainly succinct. But only simple and explained for math majors. :D
deleted 5y ago
[deleted]
handrous 5y ago
> Quaternions are hypercomplex numbers of the form I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. > Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right. As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason. > Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: > pqp' > where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u . Oooooh kay... 1) Where'd w go? Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)? Would we need to bring it back in to apply the rest of this? 2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?, 3) A quaternion is... a point, then? Since we're rotating around it? 4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it?
- marcodiego 5y ago
  >> Quaternions are hypercomplex numbers of the form > >I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. Think it is right. >> >> Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. > >Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right. They are not the same thing. All of these are equal to -1 when squared, but they are different when multiplied by any other of this set and multiplication between them is anti-commutative. >As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason. > These are not added for unknown reasons. They specify rotations of unitary vectors of a canonical base around one another. >> Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: >> >> pqp' >> >> where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u . >> >Oooooh kay... 1) Where'd w go? The coordinate 'w' is the real part. For the vector 'u', such coordinate is 0. For 'p' and 'p'', it is cos(theta/2). >Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)? No. >Would we need to bring it back in to apply the rest of this? No. Just follow the rules to multiply hypercomplex numbers and the rotation will occur. >2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?, The notation '(x, y, z)' is just a shorter notation for "xi + yj + zk". For this problem, the vector 'u' defines the axis of rotation. >3) A quaternion is... a point, then? Since we're rotating around it? You can see a quaternion as a point in R^4 since it has 4 coordinates. Actually it describes an axis of rotation, by its imaginary part, and the angle of rotation. >4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it? > The axis of rotation is actually the vector "u".
  - 5y ago
OccamsRazr 5y ago
tldr tldr (Since your version is for people who already understand undergraduate math, this version is for people who already understand upper-level undergraduate/graduate math): If you try to use a product of 3 rotation matrices that represent rotations around fixed axes to represent rotations in 3D, there will inevitably be gimbal lock because there cannot exist a covering map from a product of 3 circles to SO(3). (Gimbal lock happens at the points where the map locally fails to be a covering map) SU(2) is the universal cover of SO(3). The unit quaternions are a faithful representation of SU(2). Conveniently, the double cover map from SU(2) to SO(3) is dead simple in this representation: a unit quaternion q in SU(2) act on a purely imaginary quaternion (thought of as a vector in R^3) by conjugation. This is the formula you wrote. Since it is a covering map, there is no gimbal lock. Edited to fix explanation.
wedesoft 5y ago
Hypercomplex numbers are different to quaternions. They also have 4 components but the multiplication of hypercomplex numbers is commutative.

coldacid 5y ago

I don't believe that most people will consider that "simply explained".

marcodiego 5y ago
Maybe succinctly?
- coldacid 5y ago
  Yes, it is certainly succinct. But only simple and explained for math majors. :D

deleted 5y ago

[deleted]

handrous 5y ago

> Quaternions are hypercomplex numbers of the form I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. > Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right. As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason. > Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: > pqp' > where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u . Oooooh kay... 1) Where'd w go? Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)? Would we need to bring it back in to apply the rest of this? 2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?, 3) A quaternion is... a point, then? Since we're rotating around it? 4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it?

marcodiego 5y ago
>> Quaternions are hypercomplex numbers of the form > >I'm gonna guess "hypercomplex" means "involves imaginary numbers" due to the rest of this. Think it is right. >> >> Where w, x, y, and z are real and i^2 = j^2 = k^2 = -1 and ij = k, ji = -k, jk = i, kj = -i, ki = j, ik = -j. > >Why even use different letters for i, j, and k, if they're all the same thing? Which thing appears to be i, as in, the square root of -1, that is to say, the most well-known and basic imaginary number, if I'm reading this right. They are not the same thing. All of these are equal to -1 when squared, but they are different when multiplied by any other of this set and multiplication between them is anti-commutative. >As for the second part, I can't even begin to unravel the significance. I know it must not be, but it just seems like arbitrary rules added for... some unknown reason. > These are not added for unknown reasons. They specify rotations of unitary vectors of a canonical base around one another. >> Being u = (x, y, z) = xi + yj + zk a unitary vector, it is possible rotate any vector q by an angle theta around u by doing: >> >> pqp' >> >> where p = cos(theta/2) + sin(theta/2)u and p' = cos(theta/2) - sin(theta/2)u . >> >Oooooh kay... 1) Where'd w go? The coordinate 'w' is the real part. For the vector 'u', such coordinate is 0. For 'p' and 'p'', it is cos(theta/2). >Is this one of those things where there's a (situationally-defined) constant in the formula but we just pretend it doesn't exist most of the time (until it comes time to actually use the math to, like, do anything real)? No. >Would we need to bring it back in to apply the rest of this? No. Just follow the rules to multiply hypercomplex numbers and the rotation will occur. >2) "u =" is just defining something, fine, but (x, y, z) doesn't seem to equal the thing after it at all—I suppose this is a shorthand function notation, though it seems really weird to me to use equality to relate that. Am I right, or is this something else?, The notation '(x, y, z)' is just a shorter notation for "xi + yj + zk". For this problem, the vector 'u' defines the axis of rotation. >3) A quaternion is... a point, then? Since we're rotating around it? You can see a quaternion as a point in R^4 since it has 4 coordinates. Actually it describes an axis of rotation, by its imaginary part, and the angle of rotation. >4) I've got a feeling that theta needs a direction in this hypercomplex space but don't see where it's coming from. Somewhere "off screen", in this explanation? Or is it there but I'm not seeing it? > The axis of rotation is actually the vector "u".
- 5y ago

OccamsRazr 5y ago

tldr tldr (Since your version is for people who already understand undergraduate math, this version is for people who already understand upper-level undergraduate/graduate math): If you try to use a product of 3 rotation matrices that represent rotations around fixed axes to represent rotations in 3D, there will inevitably be gimbal lock because there cannot exist a covering map from a product of 3 circles to SO(3). (Gimbal lock happens at the points where the map locally fails to be a covering map) SU(2) is the universal cover of SO(3). The unit quaternions are a faithful representation of SU(2). Conveniently, the double cover map from SU(2) to SO(3) is dead simple in this representation: a unit quaternion q in SU(2) act on a purely imaginary quaternion (thought of as a vector in R^3) by conjugation. This is the formula you wrote. Since it is a covering map, there is no gimbal lock. Edited to fix explanation.

wedesoft 5y ago

Hypercomplex numbers are different to quaternions. They also have 4 components but the multiplication of hypercomplex numbers is commutative.