Math: How to convert a left-handed transformation matrix to a right handed one

Question

I have a transformation matrix (rotation + translation) which was constructed in a right-handed coordinate system where X is right, Y is up and Z is into the screen.

If I want to apply this same transformation but in a left-handed coordinate system where X is right, Y is down and Z is out of the screen (is this left handed?) what alterations should I make to the existing transformation matrix.

Thanks

I’m voting to close this question because Stack Overflow is for _programming_ question.s — tripleee, Feb 18 '22 at 06:00

score 8 · Accepted Answer · answered Dec 17 '13 at 18:05

a right-handed coordinate system where X is right, Y is up and Z is into the screen

Then it's a left-handed coordinate system: X is the thumb, Y is the forefinger, Z is the middle-finger. Your right hand should not be able to do what you describe without breaking a few bones :-p

a left-handed coordinate system where X is right, Y is down and Z is out of the screen

This one is indeed a left-handed coordinate system too.

As for the transformation linking these two, this is a simple 180° rotation around the X axis (beware of the wrist sprain while trying this one), which expresses as the following 3x3 matrix:

(1  0  0)
(0 -1  0)
(0  0 -1)

You can pre-multiply your transformation with this one!

Hope this helps!

score -1 · Answer 2 · answered Feb 18 '22 at 05:59

@mbrenon's answer is correct. Just to be clear.

Check my answer on this question Changing a matrix from right-handed to left-handed coordinate system

As for your question, let M_ori be transformation matrix you have, M_dst is what you want in target corrdinate system.

$M_{dst}=\begin{bmatrix}1 & 0 & 0 &0 \\0 & -1 & 0 & 0\\ 0& 0& -1 &0 \\ 0& 0 & 0 & 1 \\\end{bmatrix}\cdot M_{ori} \cdot \begin{bmatrix}1 & 0 & 0 &0 \\0 & -1 & 0 & 0\\ 0& 0& -1 &0 \\ 0& 0 & 0 & 1 \\\end{bmatrix}^{-1}$

Math: How to convert a left-handed transformation matrix to a right handed one

2 Answers2