Left-right Coordinate System Conversion

After discussing this with a classmate, I wrote a simple derivation and pasted it to the grass for a long time.

Assume that SZ is a scaling matrix, which can reverse Z, that is

SZ = 1 0 0

0 1 0

0 0-1

P = (x, y, z), p' = (X, Y,-Z) that is, P and P' indicate a point at the same position under different chiral conditions, so P = SZ * P', and vice versa, that is, p '= SZ * P

The translation transformation is the same as above.

Now we consider the rotation of the Y axis, that is, yaw. Assume that P1 is the result of the P Transformation in the left-hand system, and assume that the rotation matrix is R, that is, P1 = r * P.

Then, the point of P1 in the right hand system is P1 'p1' = SZ * P1 = SZ * (R * P) = SZ * (R * SZ * p ') according to the associativity, We can get p1' = (SZ * r * sz) * P'. Therefore, if we know that the selection of a coordinate system is R, then, the rotating matrix in another chiral is R' = SZ * r * Sz. The calculation above is equivalent to reverse the M02, M12, M20, and m21 of R.

Regardless of the selection sequence, any choice can be transformed into multiplication of three axial rotation matrices.

That is to say, r = RZ * ry * RX is selected for any axis in Rao. According to the above derivation, the rotation under another chiral should be R' = (SZ * RZ * sz) * (SZ * ry * sz) * (SZ * RX * sz); the combination and the reverse of SZ are itself, then R' = SZ * r * SZ;

Extend the preceding result to a general transformation.

P1 = r * P + T

P1 '= SZ * r * SZ * P' + SZ * t;