# math

## Quaternion representation of the rotation of a sphere into plane displacement

This is a mathematical problem I'm facing these days.. here you go.

I do have a sphere of known radius which does have a coordinate frame rigidly attached to it. Let's call the coordinate frame attached to the sphere XYZ_{s}. The sphere can be rotated and displaced arbitrarily with respect to a fixed coordinate frame (world) called $XYZ_w$. Let's be XYZ_{w} the coordinate frame of the world where X points the north, Z points the sky and Y is orthogonal to the other 2 axis.

I'm capable of knowing the orientation of the sphere with respect to the XYZ_{w} and I have such orientation represented using a quaternion which we'll call Q^{s}_{w} (that is the quaternion representing the orientation of XYZ_{s} with respect to XYZ_{w}). By knowing Q^{s}_{w} I can easily compute Q^{w}_{s} (that is the quaternion representing the orientation of XYZ_{w} with respect to XYZ_{s}) by computing the quaternion conjugate of Q^{s}_{w}.

Now, let's suppose to place the sphere on a XY_{w} plane, so that by rotating the sphere a displacement of XYZ_{s} with respect to XY_{w} is produced. You can visualize this by thinking of placing a ball on a table and rotating it so that it moves on then table plane.

Now, the problem is, by being able to sample Q^{s}_{w}, getting to know the displacement of XYZ_{s} on XY_{w} after an indefinite rotation.

This is a practical problem araised when using PALLA (as described in chapter 8 of my MoS thesis), a spherical device which contains an accelerometer, a gyroscope and magnetometer, from which doing a sensor fusion I'm able to compute it's orientation with respect to the world. Now, once placed it on a table with the user rotating it to displace it, I'd like to compute how much it is being displaced as described in math language above.

Of course, I'm not looking for the solution to the problem, I'm just looking for hints on possible paths to the solution.

Any suggestions?

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