Combining coordinate transformations

[joeljkp]

I have a vector (<1, 0, 0>) that needs to be transformed from an initial 3d rectangular coordinate system M1 through M2 and M3 to a final 3d rectangular system M4.

I'm currently doing this by applying sequential rotations omega, phi, and kappa about the x', y', and z' axes, respectively, for each individual transform, and at the end, the result is the final transformed vector. This is done through a 3x3 transformation matrix for each of M1-M4.

I want to be able to combine these coordinate system rotations, though, so instead of applying M1-M4, I'm simply applying one matrix M.

How is this done?

 

[arildno]

Multiply your matrices M1-M4 together, and you get M.
This is valid, since the matrix product is associative.

 

[tacman]

Combining coordinate transformations is a common practice in mathematics and engineering when working with multiple coordinate systems. In order to combine the transformations, you will need to use matrix multiplication.

First, let's break down the individual transformations that are being applied to the vector (<1, 0, 0>). The rotations omega, phi, and kappa can be represented by the following transformation matrices:

M1 = [cos(omega) -sin(omega) 0]
[sin(omega) cos(omega) 0]
[ 0 0 1]

M2 = [cos(phi) 0 sin(phi)]
[ 0 1 0 ]
[-sin(phi) 0 cos(phi)]

M3 = [1 0 0 ]
[0 cos(kappa) -sin(kappa)]
[0 sin(kappa) cos(kappa)]

M4 = [1 0 0]
[0 1 0]
[0 0 1]

To combine these transformations, we will need to multiply the matrices in the order of the transformations being applied. In this case, we will multiply M1 by M2, then by M3, and finally by M4.

M = M4 * M3 * M2 * M1

Once you have this combined transformation matrix, you can simply multiply it by the vector (<1, 0, 0>) to get the final transformed vector. This method allows you to avoid applying each transformation individually, and instead, apply them all at once.

It is important to note that the order of multiplication matters when combining transformations. In this example, we multiplied the matrices in the order of M1-M4, which means that the vector will first be transformed by M1, then by M2, and so on. If we had multiplied the matrices in a different order, the resulting transformation would have been different.

In conclusion, combining coordinate transformations can be done by using matrix multiplication. This allows for a more efficient way of applying multiple transformations to a vector or point in different coordinate systems.