hi,
I wonder wheather any rotaional velocity of a mass object can be expressed by three speeds around three axises.
I cannot somehow believe it. Consider object that rotates around x, around y, around z, and yet around some arbitrary direction vector.
So is it possible to express any rotational velocity by three quaternions?
In 3d, you can represent angular velocity as a 3d vector. The object rotates in the plane perpendicular to the vector at a rate equal to its magnitude.
yes, but I wonder wheather absolutely any rotating tendence can be expressed by three of them.
yes, but I wonder wheather absolutely any rotating tendence can be expressed by three of them.
I have actualy proved it to myself.
Lets break 3 quaternion characteristics to 3 base axises and 3 angular speeds scalars.
This composes a single rotation matrix against a base matrix (the identity rotation matrix I).
We compose a rotation matrix H by s degress around y.
We now perform I*H=D ; D*H=C ; C*H=E ;......
thus we are rotating by s degrees in a time step (second), rotating by quaternion that costructed H matrix, - the quaternion (0,1,0,s);
We now have to extend rotating by this quaternion along with one other quternion , quaternion (1,0,0,2s).
We construct H matrix from previous quaternion (0,1,0,s) and construct G matrix by the second quaternion (1,0,0,2s).
We now perform I*H*G=D ; D*H*G=C ; C*H*G=E ;..... . and we can reduce to J=H*G thus i*J=D ; D*J=C ; C*J=E ; ....
This means that whatever amount of descibing queternions adds up to describe tendence, they cen be incomposed to a single rotation matrix that transforms after itself the object timestep by timestep. A strange J/s unit.
But this yields problem of order of composited quaternion rotations. But I gess this is the responsibility of expressing last added quaternion corectly, rather than physical dilema.
But this yields problem of order of composited quaternion rotations. But I gess this is the responsibility of expressing last added quaternion corectly, rather than physical dilema.
And this is what gives so many permutations of Euler or Tait-Bryan angles.
