problem with 3x3 3d rotation matrices logic - concatenation doesn't work as expected

In basic words I need a simple and fast algorithm to find the solution X from C * X = M where all variable are matrices. More explanations below.

I'm trying to compute one specific matrix but it doesn't really work as expected:

Vz - negative Z-axis vector (or any other)
Vg - current gravity vector
Vc - zero reference vector (for gravity calibration)

M0 - current rotation matrix
C0 - reference rotation matrix
X0 - unknown rotation matrix to find
*t - transposed versions of above matrices

Upong runtime only Vg, M and C are known.

Rules:

1) Vz == Vg * M0
2) Vg == Vz * Mt
3) Vz == Vc * C0
4) Vc == Vz * Ct
5) Vz == Vx * X0
6) Vx == Vz * Xt
7) Vx == Vg * C0
8) M0 == C0 * X0 (wrong!!! see update notes below)
...
?) X0 = ?

I tried to use formula like that:

X0 = M0 * Ct

But the resulting matrix does not satifsy the rules (5) and (6) as expected.

Any ideas what's wrong here?

UPDATE:

The formula I tried (X0 = M0 * Ct) is correct. The question was incorrect as (8) is actually M0 = X0 * C0.

The problem why I thought it doesn't work was because I tried to compute Vx = Vg * C0 - but actually neither Vx = Vg * C0 nor Vg = Vx * Ct are correct.

Thus I'm moving to the next task - that is better to describe as a n开发者_如何学JAVAew question :-)

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If M0 = C0 * X0 (rule 8), then X0 = Ct * M0 (your formula X0 = M0 * Ct is wrong).

If this X0 does not also satisfy the other rules, then your set of rules doesn't have a solution.

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You need to know two things:

• For rotations, transposes are inverses
• Matrix multiplication does not commute.

We know:

  M0 = C0 * X0  (8)

and these are rotations. So:

inverse (C0) = Ct

Pre-multiply (8) by Ct:

Ct * M0 = Ct * C0 * X0
        = X0  

And hence we have X0.