1. ## Calculating sphere orientation?

Hi everyone.

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.

2. Your description is too unclear. Here are several examples.

Given a black sphere (binary image so black sphere and white background) I need to define a orthogonal 3D axis on it.
What does an image have to do with a sphere? A sphere is a three-dimensional body; an image is a flat picture. What is a 3D axis? An axis is a line; how is it 3D? Orthogonal to what? Or do you mean axes? Most importantly, what does it mean to "define"?

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.
Do you mean a plan or a plane?

I am not an expert on this, but are you familiar with ways to describe the orientation of a rigid body, such as Euler angles? Do they help?

3. Basically my problem is to rotate the sphere and after it rotation I can set the axes with the exact rotation. In my point of view, the big problem is that I have a black sphere so I don't have any visual feature.

There is a way to solve it?

Thank emakarov, I hope that now it is more clear to understand the problem.

4. Maybe you can rotate it with the axes? I still don't understand what is meant by "set the axes with the exact rotation".

5. Hi.

I will try to put the problem in steps maybe it becomes easier to express myself.

1. A black sphere with axes in the center of it with an arbitrary orientation
2. The sphere is rotated but the axes don't
3. With some method find the rotation of the sphere and rotate the axes equally

So basically I need to find the orientation without using memory.

I hope that now you can understand.

6. 3. With some method find the rotation of the sphere and rotate the axes equally
What does it mean to rotate "equally"?

So basically I need to find the orientation without using memory.
Are you talking about computer memory? How would you do it using memory?

Maybe you can ask the person who rotated the sphere how exactly he/she rotated it and then apply the same rotation to the axes. It is not clear from you description what information is known to you and what isn't. In the worst case you are shown a sphere, then made to turn around and face away while someone rotates the sphere, and then you look back. It is clear that in this case, you can't find out how the sphere was rotated unless it has some marks.

7. Thank emakarov.

In the worst case you are shown a sphere, then made to turn around and face away while someone rotates the sphere, and then you look back. It is clear that in this case, you can't find out how the sphere was rotated unless it has some marks.
Now the thing is, having a plane cutting the sphere in half and passing through the center point of the sphere, so this plane will have the great circle, we can calculate a normal to it. Furthermore, project the normal vector into this plane or calculate a vector that becomes contained in the plane, so that we have two perpendicular vectors. The third one can be calculated by the vectorial product. Having always this plane defined it is possible to calculate the rotation of the sphere. What you think about this? It is feasible?

If what I said before is possible, I have another problem is that the normal vector of the plane can assume two opposite directions, and if we rotate the plane we may lose the previous direction of the normal vector.

I hope that it is clear. Thank you a lot emakarov for helping me and for holding this discussion.

8. Now the thing is, having a plane cutting the sphere in half and passing through the center point of the sphere, so this plane will have the great circle, we can calculate a normal to it.
A normal to the sphere or to the plane? At what point?

Furthermore, project the normal vector into this plane or calculate a vector that becomes contained in the plane, so that we have two perpendicular vectors.
Unless a vector is perpendicular to the plane, it is not perpendicular to its projection to the plane. And if a vector is perpendicular to the plane, its projection is a null vector.

Having always this plane defined it is possible to calculate the rotation of the sphere.
Who defines the plane?

Here is what has been missing in all this discussion. A mathematical problem starts with a description of objects involved. It may be a description of geometric objects and their relationship to each other, followed possibly by a description of a transformation, etc. Then the problem usually asks either to find some parameters or prove some statement.

You should formulate your problem in the same way. Please describe what is given: objects, their parameters, etc. Then ask what has to be computed. The description of the given objects must be unambiguous. For example, you can say, Consider a coordinate system, a sphere of radius 1 with the center at the origin and the big circle that is the intersection of the XY plane with the sphere. It is also OK to say, Suppose some plane cuts the sphere through the center. But this is OK only when the question admits a definite answer, not a whole range of answers. For instance, when the XY plane cuts the sphere, you can ask, Find the equation of the tangent line to the big circle at point (0, 1, 0). On the other hand, asking to find a tangent to some point of the intersection of some plane and some sphere is meaningless because this tangent can be anything.

Formulating a problem precisely is a difficult task. There is a saying that stating a problem is halfway to solving it. It is also usually neither taught nor discussed in textbooks. However, without stating the problem properly there is no mathematics.