calculate quaternion by coordinate 2 points of object in two positions

Question

Object rotate in 3D. I have rotation center and coordinate of 2 point in rotate and not rotate position. Calculate quaternion that rotate object from first position to two position.

Answer 1

We have to vector OA' and OB' in first position of object
OA and OB in second position
Q vector part quaternion Q₀ scalar part

fig.1
COMMENT
OAOB - dot vector product
OAxOB - cross vector product

Quaternion of rotation object from first positon to second was:
1 case
If OAOB'-OBOA' not equal zero, then Q=Q₀[(OA-OA')x(OB-OB')]/[OA*OB'-OA'OB],
where Q₀^-2=([(OA-OA')x(OB-OB')]/[OAOB'-OA'OB])²+1

If OAOB'-OBOA'=0, then
2 case
if [OA'+OA]x[OB'+OB] not equal zero and OB-OB' not equal zero,
then exist m that OA-OA'=m(OB-OB'). Search it.
Q=n*(mOB'+OA'),
where n=Q₀(OB-OB')²/(2*OB*[OA'xOB'])
Q₀^-2=(OB-OB')⁴*(OB'+OA')²/(2*OB*[OA'xOB'])²+1

If OAOB'-OBOA'=0 and [OA+OA']x[OB+OB']=0
3 case
If OB+OB'not equal zero, then
Q=(OB+OB')/|OB+OB'|, Q₀=0 else

4 case
OB+OB'=0
Q=([OB']x[OA'])/|[OB']x[OA']|, Q₀=0

From quaternion we can calculate rotation matrix or Euler angles

Answer 2

Look at answer to similar question How to find correct rotation from one vector to another?

Your difference that second vector "up" is not perpendicular to first "dir". Just make it and problem solved