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   iced depresso (icedquinn@blob.cat)'s status on Sunday, 21-Aug-2022 07:02:18 JST iced depresso
   its kind of bothersome that we think of space as being 3D but rotations are 4D somehow :blobcatglare2:
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     SuperDicq (superdicq@minidisc.tokyo)'s status on Sunday, 21-Aug-2022 07:02:17 JST SuperDicq

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     @icedquinn@blob.cat Isn't there yaw, pitch and roll? What's the fourth axis?

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     SuperDicq (superdicq@minidisc.tokyo)'s status on Sunday, 21-Aug-2022 07:36:42 JST SuperDicq

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     @icedquinn@blob.cat That is interesting. I'm only familiar with euclidean space because of my experience with 3D graphics and game development.
     
     I'm not that much into physics but "360 noscope boson" made me laugh. You are right, I forgot gimbal locks are a thing and nature needs to solve that somehow. I didn't know about clifford algebra.

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     iced depresso (icedquinn@blob.cat)'s status on Sunday, 21-Aug-2022 07:36:44 JST iced depresso

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     • SuperDicq
     @SuperDicq yes it is just yaw pitch and roll when expressed on euclidean planes. BUT it is not possible to express rotations as a set of pitch->yaw->roll or else gimbal locks occur in the math.
     
     nature doesn't gimbal lock, and they had to come up with spinors and shit because some particles rotate 360 or 900 degrees and it's apparently important to note that a 360 noscope boson is NOT the same as a 0 rotation boson.
     
     i don't know what the fourth dimension does. in quaternions it's just an imaginary number that is part of the computation until its cancelled to be rendered in 3D again.
     
     in clifford geometric algebra the fourth dimension is a trivector representing the XYZ hyperplane in conjunction with individual positions on the XY, XZ and ZY basis planes.
     SuperDicq repeated this.
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     schwa (spookie@outerheaven.club)'s status on Sunday, 21-Aug-2022 08:25:25 JST schwa

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     • SuperDicq
     @icedquinn
     In order to be able to concatenate transformation matrices, each one of its operations is modified into a homogeneous coordinate system.
     
     While scaling and rotation do share the same dimensions, in their respective transformation matrices, translation does not.
     Why? Well, translation is a sum, the other two are products.
     You see, the translation matrix in product form is pretty heretic. In order to perform a "sum", it takes advantage of matrix math peculiarities. Since a cross product goes in row*column order, you can use a 3*3 identity matrix (aka, product neutral operation), and on the **4th column**, you place the vector you wish to "add" to the original point.
     The 4th column behaves as if scaling a unit vector to the intended characteristics of the translation vector, and adding it to the original point.
     
     HOWEVER, this trick adds a 4th fucking column. You want the innermost dimensions of a cross product to be the same, if you wish to actually perform it. As such, both rotation, and scaling are refactored into a 4th dimension form. This will also take care of rotation's gimbal lock, but it'll become clearer in a sec.
     
     The "w", 4th dimension, whatever you wanna call it, means that each axis is scaled w times. In the end, it means that your world is inside a cube, that cube's scale is w*w. This detail in itself, allows us (in the computer graphics world) to be able to work on "cubes"/objects independently from one another, or to place them all inside a bigger cube, and perform transforms in all of them.
     
     It gives us another **degree of freedom**, or in your case, frees you from the gimbal lock problem.
     
     Anyways, I know this is somewhat off the rails, but I just wanted to add another perspective to the origins of the problem.
     
     @SuperDicq