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July 28th, 2019, 10:20 AM   #1
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Rotation in n dimension

Hello

In 2D, the rotation matrix is 2x2, R(theta), in 3D you have 3 3x3 matrices, Rx(theta), Ry(theta) and Rz(theta), as you can see here:
https://en.wikipedia.org/wiki/Rotation_matrix

In n dimension, you have n rotation matrices, all of them nxn?If so, how do you construct them?

Do they look like this in 4D?
https://ksgamedev.files.wordpress.co...-rotation2.png
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July 29th, 2019, 05:36 PM   #2
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Your first example is a rigid rotation in the plane/space or in more formal language a transformation of coordinates. I'm not in a position to answer the n dimensional case because I don't know, but since I've chosen the tensor route (systems independent of coordinates) I know it's going to pay off.
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July 30th, 2019, 12:26 AM   #3
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Quote:
Originally Posted by ricsi046 View Post
In n dimension, you have n rotation matrices, all of them nxn?If so, how do you construct them?
If you work it out for n = 2, 3, 4 ... you'll see a pattern.

Do they look like this in 4D?
https://ksgamedev.files.wordpress.co...-rotation2.png[/QUOTE]

Could be. I only worked out n=3 once. The trick is that the n-th column of the matrix is image of the n-th standard basis vector (all zeros except 1 in the n-th coordinate).

Quote:
Originally Posted by NineDivines View Post
since I've chosen the tensor route (systems independent of coordinates) I know it's going to pay off.
Can you say more about this? I know tensor products in abstract algebra, but I don't know much about the practical aspects of tensors.
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July 31st, 2019, 01:34 AM   #4
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http://wscg.zcu.cz/wscg2004/Papers_2004_Short/N29.pdf

worth reading this if you are interested in n dimensional rotations, what I needed is the general matrix for main rotations, it's on the top of the 2nd page

took some time to realize what I wanted
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August 2nd, 2019, 08:50 AM   #5
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What I find most illuminating about this is how does one even depict a geometrical figure in the n-dimensional case to be rotated?
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