Distance is always nonnegative, but it is often helpful to define an oriented distance. Similarly, oriented volume is helpful e.g. to understand determinants. I'm working on geometric problems involving arcs of circles where an oriented angular distance (or arc measure) would be helpful, but am struggling to come up with a consistent definition.
So far, I have:
For a given circle, fix an arbitrary orientation around the circle. For two points $a, b$ on the circle, define oriented arc measure $\overset{\huge \frown}{ab}$ such that $$\begin{align*} \overset{\huge \frown}{aa} &= 0 \\ \overset{\huge \frown}{ac} &= \overset{\huge \frown}{ab} + \overset{\huge \frown}{bc} \text{ if } a,b \neq c\\ \overset{\huge \frown}{ab} &= \pi \text{ if } ab \text{ is a diameter}\\ \overset{\huge \frown}{ab} &= -\overset{\huge \frown}{ba} \text{ if } ab \text{ is not a diameter}\\ \overset{\huge \frown}{ab} & \text{ is invariant under rotation around the circle's center}. \end{align*}$$
Questions:
- Is this definition consistent and well-defined? I tried to ensure it covered all special cases.
- If not, can it be repaired?
- Is there a similarly intended definition or measure in the literature?
This is called directed angle, which has properties similar to the above (e.g. $\angle AB = -\angle BA$). The key is to take angles mod $\pi$ (some authors, when dealing with arcs and not angles, use mod $2 \pi$.)
See Kedlaya's Geometry Unbound for properties.