# How to convert a right handed coordinate system to left handed?

So I have a point p = (2,-5,1) and it is in a "right handed coordinate system" and I want to convert it into a left handed coordinate system. I tried googling it, but all the results have to do with Unity and I am not tying to program anything, I just want to understand the math behind it.

This is the exact question from the slides.

"Given a point P = (2,-5,1) in a right handed coordinate system, what is the point is a left handed coordinate system?"

• I'm not quite sure what you're asking. If you change the handedness of the standard coordinate system by reversing the orientation of the $x$ axis then the point in your question will be $(-2,-5,1)$ in the new coordinate system. Jan 29, 2018 at 18:58
• I updated it, but I don't think its going to help much.. I myself don't understand it either, and all google results are unity.. I figured someone here might. @Ethan Bolker Jan 29, 2018 at 18:59
• Maybe if you posted more information from "the slides" we'd have enough context to help. Maybe not. Jan 29, 2018 at 19:04
• Change the sign of a single coordinate, that performs a mirroring.
– user65203
Jan 29, 2018 at 19:19

By "left-handed" and "right-handed" I understand an orthonormal basis, i.e. unit length basis vectors that are all mutually (pair-wise) perpendicular.

Hold out your right hand and point your index finger at the screen, your middle finger (horizontally) to the left, and your thumb vertically. You have a right-handed coordinate system.

Do the same with your left hand, and you'll notice that the middle fingers point in different directions.

Hold your right hand and left hand out in front of you with the index fingers at the screen, the thumbs up, and the middle fingers pointing at each other. All fingers on the same hand at right angles to each other.

You should see that the difference between the right-handed and left-handed is that the middle fingers point in opposite directions.

Turn your hands towards yourself so you index fingers are tip-to-tip. Now the thumbs point in the same direction and the middle fingers point in the same direction. Only the index figures go against each other.

Go back to the start position, and rotate your hands so your thumbs point towards each other. Index finger and middle finger agree.

This is the difference between a left-handed and right-handed orthonormal coordinate system: one of the axes (fingers) is different. If $$\{{\bf e}_1,{\bf e}_2,{\bf e}_3\}$$ is right-handed then $$\{-{\bf e}_1,{\bf e}_2,{\bf e}_3\},$$ $$\{{\bf e}_1,-{\bf e}_2,{\bf e}_3\}$$, $$\{{\bf e}_1,{\bf e}_2,-{\bf e}_3\}$$ are all left-handed. Moreover, $$\{-{\bf e}_1,-{\bf e}_2,-{\bf e}_3\}$$ will also be left-handed. As Max correctly pointed out in the comments, an odd number of sign changed will change the chirality of the coordinate system, while an even number preserves it.

Also $$\{-{\bf e}_1,{\bf e}_2,{\bf e}_3\}$$, $$\{{\bf e}_1,-{\bf e}_2,{\bf e}_3\}$$, $$\{{\bf e}_1,{\bf e}_2,-{\bf e}_3\}$$ and $$\{-{\bf e}_1,-{\bf e}_2,-{\bf e}_3\}$$ are left-handed.

• Hi Fly by Night, if you do a simple left vs right handed rotation about <e1, e2, e3> you'll notice that <--e1,e2,e3>,<e1,--e2,e3>,<e1,e2,--e3> are indeed left-handed coordinate systems, and <--e1,--e2,--e3> was missed. But the even numbered changes, <--e1,--e2,e3>, <--e1,e2,--e3>, and <e1,--e2,--e3> do not become left handed.
– Max
Aug 15, 2019 at 9:44
• @Max You're right, thank you! Aug 23, 2019 at 19:10
• This answer could do with a helpful summary at the top, it's long and doesn't immediately answer the question. Jul 6, 2020 at 11:24

Simply inverting the $$z$$ coordinate will do the trick.