Rotating a Camera on a Sphere: Finding a Coherent Direction

[jimesh]

Homework Statement

I have a task to move a camera attached to a robot arm along the surface of a sphere. The requirments are,
* Z axis of the camera coordinate system should always point to the origin of the sphere.
* X-Y plane of the camera coordinate system shold be always in the curvilinear direction... (this is because the camera sholuld not rotate its XY plane since the robot will reach singlaritty position at some point..)

The final result should be the rotation angles of the new coordinate system with respect to a world coordinate system at the origin of the sphere.

Homework Equations

Spherical coordinates (r,theta,phi)
x = r * Cos(theta) * Sin(theta);
y = r * Sin(theta) * Sin(phi);
z = r * Math.Cos(phi));

Solving for spherical coordinate we get points P1(x,y,z), P2(x,y,z), P3(x,y,z) etc.. on the sphere. These are the points where the origin of camera coordinate system should be.

The Attempt at a Solution

1. A world coordinate system at the origin of sphere
2. Create another coordinate system at P1, P2 etc which are identical to world coordinate sysem.
3. Calculate the vector from the origin of P1 to origin of world cordinate system (sphere origin)
4. Calculate a vector along the Z axis of the coordinate system at the point P1, say (0,0,1)
5. Find a rotation quaternion by clculating the normal to this 2 vectors and angle between these 2 vectors.

As a result I got the coordinate system at P1 rotates in such a way that Z axis is pointing to the origin of world coordinate system.

But now the problem is
* The camera (X-Y Plane) will also rotate when I move to next point P2 and do the same calculation using quaternion.
* I want to rotate this plane to a coherent direction (say curvilinear direction) at all points.
I tried to find a reference vector for this to rotate.. but couldn't..

any body has any idea.. ?

 

[jimvoit]

curvilinear direction?

X-Y plane of the camera coordinate system shold be always in the curvilinear direction...Can you explain that in some other way?

 

[piercas]

I would suggest approaching this problem by breaking it down into smaller, more manageable steps. First, let's consider the requirement that the Z axis of the camera coordinate system should always point to the origin of the sphere. This can be achieved by using the spherical coordinates of the points on the sphere, as shown in the equations provided. By using these coordinates, you can calculate the position of the camera's origin at each point on the sphere, ensuring that the Z axis always points towards the origin of the sphere.

Next, let's address the requirement that the X-Y plane of the camera coordinate system should be in the curvilinear direction. This can be achieved by using a reference vector, as you mentioned. One way to find this reference vector is to use the cross product of the vector from the origin of the camera coordinate system to the origin of the world coordinate system and the vector along the Z axis of the camera coordinate system at that point. This will give you a vector perpendicular to both of these vectors, which can serve as the reference vector for the X-Y plane of the camera coordinate system.

Finally, to find the rotation angles of the new coordinate system with respect to the world coordinate system, you can use the rotation quaternion calculated in step 5 of your attempt at a solution. This quaternion can be converted into Euler angles, which will give you the rotation angles around each axis. By repeating this process for each point on the sphere, you will have the complete set of rotation angles needed to move the camera along the surface of the sphere while maintaining the desired orientation.

In summary, the key steps to solving this problem are:
1. Use spherical coordinates to calculate the position of the camera's origin at each point on the sphere.
2. Use the cross product to find a reference vector for the X-Y plane of the camera coordinate system.
3. Use a rotation quaternion to determine the rotation angles of the new coordinate system with respect to the world coordinate system.
4. Repeat this process for each point on the sphere to obtain the complete set of rotation angles needed.