I have question about 4D-rotation using quaternion. I'm planning to use quaternion rotation in my game development

this is as far as I understand. assume the target point of rotation P in 3Dspace using quaternion Q

   P = (0, a, b, c) = 0 + ia + jb + kc    Q = w + xi + yj + zk

quaternion rotation can be divided into pair of orthogonal plan A and B these planes can be rotated by multiplying each quaternion

  • multiplying (cosθ + I sinθ) from left rotates WX-plane by θ、YZ-plane by θ
  • multiplying (cosθ + j sinθ) from left rotates WY-plane by θ、XZ-plane by θ
  • multiplying (cosθ + k sinθ) from left rotates WZ-plane by θ、XY-plane by θ

we don't need the rotation including W-axis in 3D space, so we can exclude the rotation by multiplying conjugate quaternion from right

(qy (qx p qx*) qy* ) qz* = (qz qy qx) p (qx* qy* qz*) = (qz qy qx) p (qz qy qx)*

I understand these, but the next sentence, I don't understand

combination of rotation quaternion of each plane (qz qy qx) can be written in next way. Suppose any rotation axis in 3D space s R、rotation around the axis as θ

q = cos(α/2) + iRxsinα + jRysinα + kRzsinα

I don't get how this equation was calculated from. I would appreciate for any help. thank you. and sorry for my bad English.


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