I have a fixed Quaternion t that is defined as Quaternion([0.5, -0.5, -0.5, 0.5]);. The order here is w,i,j,k.
In my program I have an function that I would like to modify. The function takes q as an input and calculates t*q.
I would like to calculate a new fixed tprime such that q*tprime gives me the same result as t*q.
I've read about the non commutative properties of Quaternions. But I haven't been able to find a solution to the problem. Is there a general solution? Or is this depend some properties of t ? Thanks.
In the ring of quaternions every non-zero element has an inverse. So if $q\neq0$ the only solution $t'$ of the quaternion equation $$qt'=tq\qquad(*)$$ is (multiply $(*)$ by $q^{-1}$ from the left hand side and apply associativity) $$ t'=q^{-1}tq.\qquad(**) $$ Obviously this $t'$ also is a solution — we get $(*)$ as a consequence of $(**)$ by multiplying $(**)$ from the left hand side by $q$.
A consequence of this is that $t'$ depends on $q$. This cannot be helped. When you think how composition of rotations corresponds to the product of quaternions, this should not be a surprise. However, the quaternions $t$ and $t'$ have the same real part (check if you don't see it right away), meaning that their resepective angles of rotation are equal, only the axes are different.