Coordinate system handedness
From CGAFaq
Introduction
You must have looked at yourself in the mirror and noticed that your hands seem to swap places. Raise your right hand and the image in the mirror raises his/her left hand and vice versa. In fact, you can see a whole world through that mirror. Step in that mirror now. That world seems very similar in every possible way. Things are in the same scale, the distances between them are exactly the same and even the angles everywhere are just like in ours. Clearly these two worlds are connected through mirrors, which act as portals to the other world. Given those similarities, what more could be there to differentiate this place from our world ? A lot. Imagine driving on the right in England, or eating your favorite pizza with fork in the right hand instead of left. Imagine waking up at the morning and smashing on the wall on your way to the toilet.
Reflections
Let’s assume an n-dimensional space. Pick your favorite hyperplane in this space and reflect the whole world with respect to it. Reflection is your way to get to the other world. If you reflect twice or in general an even number of times, then you are back in the world you originally were before the reflections. If you reflect an odd number of times, then you will be in just the other world than where you started. There is a little catch, however. You might find the reflected world becomes rotated and translated. Your mind fastly concludes that we can probably use reflections to construct rigid transformations. You can detect if a transformation matrix changes handedness: this happens only if the determinant of the matrix is negative.
The answer
What is the difference between right and left handed coordinate systems? You must have guessed by now: they are reflections of each other. Where do the names come from? This is plain convention and is determined from the following rules:
- Right hand rule in 2D: Using your right hand with the palm facing you, point your thumb in the direction of X axis. Then straighten up your index finger. If your index finger agrees with the Y axis of your coordinate system, then it is right handed. Otherwise it is left handed
- Right hand rule in 3D: Using your right hand, point your thumb in the direction of X axis and your index finger in the direction of Y axis. Then bend your middle finger to form a 90° angle with the two other fingers. If the middle finger agrees with the Z axis of your coordinate system, then it is right handed. Otherwise it is left handed.
- Right hand rule in nD: From now on, you run out of dimensions, so we use math. Take the first n-1 vectors of your basis and calculate a generalized cross product. If the dot product between the result vector and your last vector is positive, the coordinate system is right handed. Otherwise it is left handed.
