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