Hello to all,(adsbygoogle = window.adsbygoogle || []).push({});

I am currently studying computer graphics and I have came up with the following problem. Consider that we have three coordinate systems, lets say CSA1, CSA2 and CSA3 that have the same origin and differ by a rotation. That is to CSA2 connects to CSA1 by R12 and CSA3 to CSA1 by R13. Assume that Ai represents a base for the corresponding coordinate system CSAi. That is Ai comprises from 3 unit orthogonal vectors that are the base of CSAi. Then A2 connects to A1 via:

A2=R12*A1 (eq. 1)

and moreover

A3=R13*A1 (eq. 2)

eq. 1 becomes A1=inv(R12) * A2 (eq. 3) (inv stands for inverse) and eq. 2 using eq. 3 becomes

A3=R13*inv(R12)*A2 (eq. 4)

now we also know from theory that the coordinates (P1, P2, P3) of a point P in the three coordinates systems are connected via:

P2=inv(R12)*P1 (eq. 5)

P3=inv(R13)*P1 (eq. 6)

eq. 5 becomes P1=R12*P2 and substituting it in eq. 6 we have that

P3=inv(R13)*R12*P2 (eq. 7)

moreover applying the same rule in eq. 4, we get that the coordinates of P in the second and third coordinate system are connected via:

P3=inv(R13*inv(R12))*P2 that is

P3=R12*inv(R13)*P2 (eq. 8)

equations 7 and 8 gives us that

inv(R13)*R12=R12*inv(R13)

which is obviously wrong.

can anyone help me and show me my mistake?

Many thx,

zokos

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Rotation of coordinate system mistake or paradox?

**Physics Forums | Science Articles, Homework Help, Discussion**