I have been around a problem that I cannot figure out a solution (if there is one) which is related with sphere orientations and rotations. I already searched in many places, including here in this forum but without success.

Let me introduce my problem. Given a black sphere (binary image so black sphere and white background) I need to define a orthogonal 3D axis on it. At a first glance seems easy, we can define the 3D axis in the middle of the sphere because it is a well known point. The problem rises when I need to rotate the sphere, because the previous 3D axis must rotate accordingly with the rotation of the sphere. So basically, I'm trying to find the sphere (shape) orientation along rotation movements. In the book that I'm reading, Volumetric Image Analysis, this problem is called orientation ambiguities and it is a section in the middle of Moments chapter, but unfortunately the book goes just till the ellipsoids, resolving problems of 2 ambiguities. The sphere case has 3 ambiguities because the shape of it cannot define any of the axis implicitly.

I thought in the following: with the point in the middle of the sphere, I make a plan that contains that point and with this plan, I can have the normal vector. With this normal vector I can project it on the plane giving another vector orthogonal with the previous one. To find the 3rd vector I just need to calculate the product between these vectors. But when the sphere is rotated I don't know how to define again the plane to give the same vectors.

Could you please help me?