Finding the spin of a sphere given X, Y, and Z vectors relative to sphere

Question

I'm using Electro in Lua for some 3D simulations, and I'm running in to something of a mathematical/algorithmic/physics snag.

I'm trying to figure out how I would find the "spin" of a sphere of a sphere that is spinning on some axis. By "spin" I mean a vector along the axis that the sphere is spinning on with a magnitude relative to the speed at which it is spinning. The reason I need this information is to be able to slow down the spin of the sphere by applying reverse torque to the sphere until it stops spinning.

The only information I have access to is the X, Y, and Z unit vectors relative to the sphere. That is, each frame, I can call three different functions, each of which returns a unit vector pointing in the direction of the sphere model's local X, Y and Z axes, respectively. I can keep track of how each of these change by essentially keeping the "previous" value of each vector and comparing it to the "new" value each frame. The question, then, is how would I use this information to determine the sphere's spin? I'm stumped.

Any help would be great. Thanks!

Answer 1

My first answer was wrong. This is my edited answer.

Your unit vectors X,Y,Z can be put together to form a 3x3 matrix:

A = [[x1 y1 z1],
     [x2 y2 z2],
     [x3 y3 z3]]

Since X,Y,Z change with time, A also changes with time.

A is a rotation matrix! After all, if you let i=(1,0,0) be the unit vector along the x-axis, then A i = X so A rotates i into X. Similarly, it rotates the y-axis into Y and the z-axis into Z.

A is called the direction cosine matrix (DCM).

So using the DCM to Euler axis formula

Compute

theta = arccos((A_11 + A_22 + A_33 - 1)/2)

theta is the Euler angle of rotation.

The magnitude of the angular velocity, |w|, equals

w = d(theta)/dt ~= (theta(t+dt)-theta(t)) / dt

The axis of rotation is given by e = (e1,e2,e3) where

e1 = (A_32 - A_23)/(2 sin(theta))
e2 = (A_13 - A_31)/(2 sin(theta))
e3 = (A_21 - A_12)/(2 sin(theta))

Answer 2

I applaud ~unutbu's, answer, but I think there's a simpler approach that will suffice for this problem.

Take the X unit vector at three successive frames, and compare them to get two deltas:

deltaX1 = X2 - X1
deltaX2 = X3 - X2

(These are vector equations. X1 is a vector, the X vector at time 1, not a number.)

Now take the cross-product of the deltas and you'll get a vector in the direction of the rotation vector.

Now for the magnitude. The angle between the two deltas is the angle swept out in one time interval, so use the dot product:

dx1 = deltaX1/|deltaX1|
dx2 = deltax2/|deltaX2|
costheta = dx1.dx2
theta = acos(costheta)
w = theta/dt

For the sake of precision you should choose the unit vector (X, Y or Z) that changes the most.