Reasoning about Left and Right Handed Coordinate Systems

A coordinate system can be defined by three perpendicular unit vectors. If the coordinate system is Cartesian, which direction does the $+x$ axis point? To resolve this problem, I define an orientation–a coordinate system that is oriented in a certain direction in three-dimensional space. How many unique orientations are there? Here, a unique orientation is an orientation that cannot be rotated by less than 90 degrees into another orientation —admittedly, 90 degrees is an arbitrary number, but at least it is a start. Two orientations related by less than a quarter of a rotation are considered equivalent in contrast to unique. The idea of a unique orientation can be used to avoid counting a numerous amount of orientations formed by rotations. The answer to the posed question is related to the number of ways that three unit vectors can be arranged, if all three unit vectors are perpendicular. Finally, do the unique orientations form left and right handed coordinate systems?

Determining Unique Orientations

One way to answer this question is to draw possibilities. If the $+z$ axis points “up” from the perspective of the observer, then the $x$- and $y$- axes are in a plane. For a given placement of the $+x$-axis in this plane, the $+y$-axis must be 90 degrees away from the $+x$-axis. Otherwise, the $x$ and $y$ axes would not be perpendicular.

It is important to realize that, for a given position of the $+z$-axis, there are many different directions that the $+x$-axis can point. Since I am considering unique coordinate systems involving a quarter of a rotation, the $+x$ axis can point in one of four directions. Then the $+y$ axis can point in one of two directions. So, if the $+z$-axis points up, then there are 1*4*2=8 ways to uniquely arrange the $+x$ and $+y$ axes.

Regardless of the placement of the $+x$ axis, the necessarily perpendicular $y$-axis defines a $+y$ which is 90 degrees away (angular direction matters here) from the $+x$-axis. So this question reduces to “how many ways can the $+x$ and $+y$ axes be 90 degrees apart?” This is illustrated below.

With the current definition of “unique,” system A is unique with respect to system G since the rotation is exactly 90 degrees. In contrast, any coordinate system with the $+x$-axis slightly to the right of the $+x$-axis in A is not unique with respect to A, since that system can be rotated back to A in less than 90 degrees. In fact, in the above diagram, the $+x$-axis rotates around the $+z$-axis in 90 degree increments, and for each rotation of the $+x$-axis, the $+y$-axis is chosen to point in one of two directions along a line perpendicular to the $x$-axis. Hence systems A-H are unique. This is a group of eight coordinate systems.

Rotating the $+z$-axis

The last part of solving this problem is figuring out how many ways the “up” $+z$-axis can rotate while leaving the same the relative orientation of the $+x$ and $+y$ axes. To get all possible unique coordinate systems, there are several ways the $+z$ axis can be oriented relative to an observer: up, right, down, left, forward, & back. That is six options. Hence there are 6*8=48 unique coordinate systems of perpendicular axes. That is a lot more than I expected!

Left and Right Handed Coordinate Systems

How many of these 48 unique coordinate systems are right-hand coordinate systems, and which ones are left-hand coordinate systems? Well, a right-hand coordinate system is one that can be formed with the right hand such that the thumb points in the direction of the $+x$ axis, the $+y$-axis points in the direction of the index finger, and the $+z$-axis points in the direction of the middle finger. In contrast, a left-hand coordinate system is one that can be formed with the left hand with the same association of the axes with the fingers.

Looking at the figure, A is right, and H is left. B is similar to H, so B is left. D is similar to A, so D is right. C is left, and E is left as well. F is right, and G is right.

In summary:

• Right-Handed Coordinate System: A, D, F, G
• Left-Handed Coordinate System: B, C, E, H

Since a hand can be rotated in any direction (ignoring physical limitations, of course!) just like an orientation without changing the relative configuration of the axes, a rotation of the $+z$-axis does not affect the conclusion of whether a given system of A-H is right or left. Therefore, all 48 unique orientations have been designated as either right or left handed coordinate systems.

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