I have a set of 3 distincts points which I know the positions in 3D (say D1, D2, D3).
I also know they are the plane-symmetry of 3 other distinct points in 3D (say B1, B2, B3) which I also know the positions (paired in same notation order).
Regarding the plane of symmetry, all I know is 1 point position (say C).
And what I'm looking for is of course the plane normal (say n=(nx, ny, nz)).
So far I was thinking of iterating to find the plane normal but maybe there is a direct way to resolve for it, no?
NB: I perfectly know the positions of points (up to numerical precision for double), i.e. I'm not trying on a blurry acquired positions, I'm trying to solve inverse kinematic of a robot.
Any single reflected pair $BD$ determines the plane of reflection, since the plane is perpendicular to $BD$ and passes through the midpoint of that segment.
So you have three segments that must be parallel and four points (three midpoints, and $C$) that must all lie on the same plane. That's a good check on your data.