A Right Handed Spherical Coordinate System

This post introduced the following questions.

What direction does $\vec{e}_{\phi}$ point? Modern convention dictates that $\vec{e}_{\phi}$ should point in the direction of increasing $\phi$, but what is the reason for this convention? If there are only two unique configurations for three mutually perpendicular unit vectors in light of the ambiguous orientation of each unit vector, it seems likely that the direction of $\vec{e}_{\phi}$ dictates whether the unit vectors in spherical coordinates form a right or left handed coordinate system in the same sense that the Cartesian unit vectors do.

Unique Configurations

First of all, are there only two unique configurations for three mutually perpendicular unit vectors? In this post, I defined an orientation as a type of coordinate system and concluded that all 48, unique orientations are either right handed or left handed coordinate systems. Among these 48 unique orientations, there are only two unique configurations for three mutually perpendicular unit vectors: left-handed and right-handed.

Ordering of Coordinates

In order to determine whether a coordinate system is right handed or left handed, the order in which unit vectors are considered is important. This order is determined by how coordinates are listed, as in $(x,y,z)$. In spherical coordinates, the typical ordering is $(r,\phi,\theta)$. This is a completely arbitrary order now that I think about it. But, I might as well use the same ordering convention that most people use. Since I am using $(r,\phi,\theta)$, a spherical coordinate system is right-handed if:

  1. Thumb of right-hand points in the direction of the first unit vector, $\vec{e}_r$.
  2. Index finger of right-hand points in the direction of the second unit vector, $\vec{e}_{\phi}$.
  3. Middle finger of right-hand points in the direction of the third unit vector, $\vec{e}_{\theta}$.

If this these rules cannot be met, the spherical coordinate system is left-handed.

Recommendation

If one starts with a right-handed Cartesian coordinate system, it makes sense but is not absolutely necessary to use a right-handed spherical coordinate system. The reason this is not absolutely necessary is that $(r,\phi,\theta)$ is, after all, an arbitrary ordering of coordinates. One pitfall to avoid is mixing conclusions from, for example, a right-handed spherical coordinate system with a left-handed spherical coordinate system.

Choosing a Direction for $\vec{e}_{\phi}$

I plan to use right-handed Cartesian and right-handed spherical coordinate systems for upcoming posts. If $\vec{e}_r$ happens to be in the $x-y$ plane, then $\vec{e}_{\phi}$ points in the direction of increasing $\phi$. I am using the same conventions for how the angles $\phi$ and $\theta$ are defined in a spherical coordinate system, because I see no reason to depart from these conventions.

References

[1] https://www.utdallas.edu/atec/midori/Handouts/coord_system.htm

[2] https://bridge.math.oregonstate.edu/papers/spherical.pdf

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