# Handedness of 2D coordinate systems

I know that there's always a big debate on the handedness of 3D coordinate systems, but I can't seem to find any information about the handedness of 2D coordinate systems.

Usually, the X axis points to the right, but the Y axis can point up (in the case of many game engines) or down (in the case of screen coordinates and in Godot). What are the standards for these?

• Which one of those is "left-handed" and which one is "right-handed"?

• Are rotations always clockwise? Counter-clockwise? From X to Y? From Y to X?

• "Handedness" requires specifying a third axis. In 3D, if I name my thumb, index, and middle fingers x, y, and z respectively, then only one of my hands can be oriented to match the axes of a given coordinate system (for instance, in Unity my left hand matches and my right doesn't; in 3DS Max it's the opposite). But in 2D, I can make either hand match if I just rotate it — no mirroring required. So it's not clear which term would apply there. – DMGregory Feb 27 '19 at 23:28
• There is no way to rotate it into the other way while saying inside of the 2D plane. If you rotate 180 degrees to change the direction of the Y axis, you also change the direction of the X axis. Changing between X right, Y up, and X right, Y down, requires mirroring vertically, or rotating through a non-existant 3rd dimension (just as I can change between 3D handedness if I could rotate in 4D). – Aaron Franke Feb 28 '19 at 0:45
• Right, but your hand is in 3 dimensions, and you have to choose a rotation to put your thumb and forefinger into the 2D plane in the first place. I can choose a rotation that works for my left hand, and I can also choose a rotation that works for my right hand. So if either hand has a valid choice to map up to the given coordinate system, what would lead us to say the "right hand" is the correct match and not the left? – DMGregory Feb 28 '19 at 0:48
• In my experience, rotations with positive angles typically rotate counter clockwise. I've found that to be the case in high school math and in computer graphics. Once you know that positive rotation is counter clockwise, and you have x = cos(a) and y = sin(a), then the only way to have counter clockwise rotation is to have y as up, not down. – Bram Feb 28 '19 at 1:54

Since rotation relies on trigonometric functions, it's also coordinate system dependent whether it's a CW or CCW system. When the angle is 0°, the vector you get is $$\\vec v(cos(0°),sin(0°))=\vec v(1,0)\$$, and at 90° it's $$\\vec v(0,1)\$$. In the "top-left origin" system this is clockwise, while it's counter-clockwise in the more traditional coordinate system.