I have recently been introduced to Hilbert's axioms of geometry. Right now I am studying angles. A text I have been using as a guide defines an angle and its opening in the following way:

  • An angle is the union of two half-lines which share a common endpoint and do not lie in the same line. The angle formed by the half-lines $OA$ and $OB$ is denoted by $\angle AOB$.

  • The opening of an angle $\angle AOB$ is the set of points which lie on the same side of the line $OA$ as $B$ and at the same time lie on the same side of the line $OB$ as $A$.

I however have some problems with this definition. First, the definition seems to discard the concept of a straight or a full angle as the half-lines defining the angle must be on separate lines. Secondly the definition also seems to treat an angle and its explementary angle as the same angle. Because of this I have been thinking of an alternative definition:

  • An angle is a pair of half-lines which share a common endpoint. The angle $(OA,OB)$ is denoted by $\angle AOB$ and we call call the half-line $OA$ the left side of the angle $\angle AOB$ and call the half-line $OB$ the right side of the same angle.

As the half-lines may be on the same line, the concept of an opening of an angle doesn't seem to have a obvious replacement.

I am now particularly interested in how we could order these kinds of angles. In the definition of the text that I am using, in the way I have interpreted it, they define an ordering of an angle in terms of its opening: We say that $\angle ABC <\angle PQR$ if and only if there exists a point $S$ in the opening of the angle $\angle PQR$ such that the angles $\angle ABC$ and $\angle PQS$ are congruent.

Attempting to define a similar kind of concept of order has led me to difficulty in the other definition as the concept of an opening of an angle isn't there. Instead of a linear order, I have also been trying to define a cyclic order these kinds of angles (Wikipedia has an article on cyclic orders if they are unfamiliar: https://en.wikipedia.org/wiki/Cyclic_order). However my attempts have not yet succeeded. I would appreciate any ideas for trying to formulate a concept of order for these kinds of angles or other ways to think about angles and their ordering

  • $\begingroup$ "An angle is a pair of half-lines" Mmmm, no; I prefer "is the union of...". "...the left side of the angle..." This assertion needs a definition for "left side". Also, how do you define $0$, $\pi$ or $n\pi$ angles? $\endgroup$ – Ripi2 Oct 3 at 22:34
  • $\begingroup$ @Ripi2 The concept of a left side of an angle is defined to just be the first element of the ordered pair of the half-lines. This could then intuitively correspond to the concept that the angle is then measured clockwise and ends at the other side i.e the right side. Considering the angles $0$, $\pi$ and $n\pi$, we would instead consider them to be measures of an angle; some equivalence class of angles under congruence. $0$ would correspond to a angle whose left and right sides are equal and $\pi$ would be the angle whose sides are opposites. $n\pi$ would be multiples of $\pi$ under addition. $\endgroup$ – SleepySquirrel Oct 4 at 7:57

This is something I've been struggling to formalize for a long time and... I think I succeeded.

First, the definition on an (ordinary) angle as a union of two half-lines doesn't seem the most convienent to me. I prefer to define it as an unordered pair. However, you can obviosly extract the pair from the union i.e. the pair is uniquely determined by the union.

You are absolutely right that the common definition discards zero angle and half-full angle and makes no distinction between the angle and its explementary angle.

Now I will talk about how to (in a sense) generalize the notion of an angle to meet these issues.

You had a good intuition to distinguish what you called left and right side of an angle. I will rather not use these names, but I will introduce the notion of a directed angle as an ordered pair of halflines sharing the common origin.

Now we should be able to distinguish whether we rotate clockwise or anti-clockwise when we start from the first halfine of a directed angle and go (shortest way) to the second halfline (this illustration doesn't necessarily apply to zero and half-full angles). We can't really tell a single directed angle rotates clockwise, for instance. We can, however, tell whether two directed angles (not zero or half-full) both rotate the same direction or not. What we need here is orientation of a plane (you didn't write it but I assume we are doing planar geometry, otherwise it doesn't make much sense). Here's the link to my definition of an orientation:

How to define orientation of ordered plane?

We can assign one of (two) orientations to any directed angle which is not zero or half-full the following way:

For directed angle $(A,B)$ we set $\mathcal{O}((A,B)):=[(L(A),[A],M)]$, where $M$ is a halfplane with boundary $L(A)$ which contains $B$.

The rest is easy. We can now define the interior of the directed angle with respect to one (fixed) orientation $\mathcal{O}$.

Take a directed angle $(A,B)$. If $(A,B)$ is zero angle, then honestly speaking I don't know what is the best choice for the interior, perhaps empty. If $(A,B)$ is not zero nor half-full then we have two possibilities:

  1. $\mathcal{O}(A,B)=\mathcal{O}$. Then the interior with respect to the orientation will be the usual interior of ordinary angle $\{A,B\}$ (which you called the opening)
  2. $\mathcal{O}(A,B)=\mathcal{O}^*$. Then the interior with respect to the orientation will be the complement of $\{o\}\cup A\cup B\cup \mathrm{Int}(\{A,B\})$ i.e. exactly the explementary angle.

Finally, if $(A,B)$ is a half-full angle, then the interior will be the halfplane $M$ with boundary $L(A)=L(B)=L$ such that $\mathcal{O}=[L,[A],M]$ (exactly one of two halplanes will satisfy this condition).

We can go further and define for instance the cyclic order you were asking about. Namely, consider the set $\mathcal{H}$ of all halflines sharing the same origin. We will say that three halflines $A,B,C\in \mathcal{H}$ are in order $A-B-C$ with respect to (fixed) orientation $\mathcal{O}$ whenever $A\neq C$ and $B$ lies in the interior of $(A,C)$ with respect to $\mathcal{O}$.

This order will satisfy usual cyclic order axioms:

  1. If $A-B-C$ then $B-C-A$.
  2. If $A-B-C$ then $\neg C-B-A$.
  3. If $A,B,C$ are pairwise distinct, then $A-B-C$ or $C-B-A$.
  4. If $A-B-C$ and $A-C-D$, then $A-B-D$ and $B-C-D$.

There is much more which can be done about directed angles. We can introduce a relation between directed angles saying that $(A,B)\simeq (C,D)$ iff they are both zero angles or they are both half-full angles or they both aren't these and are congruent and have the same orientation. This relation is an equivalence relation. I call equivalence classes free directed angles. Next we can define addition of free directed angles (adding ordinary free angles is not always possible). And we can prove that the set of free directed angles is an abelian group (commutativity is the hard part). Another thing which can be done is measure of a (free) directed angle with respect to (fixed) orientation. Say we do it in degrees. This measure will take values in $[0,360)$ unlike the ordinary measure which takes values in $(0,180)$. This measure will be additive (modulo 360) and it may serve to order (linearly) the set of free directed angles which is the thing you were asking about. But that's another long story to tell...

  • $\begingroup$ thank you for your thoughtful answer. I'm am still wondering how one would extend this notion of cyclic order of half-lines to angles or their equivalence classes under congruence. Also I'm wondering if we could instead choose one orientation to be fixed for all angles and then define the interior of the angle depending on if the right side of the angle is in the corresponding oriented half-plane i.e. depending on if $B$ of angle $(A,B)$ lies on $M$ where $M$ satisfies $\mathcal{O}=[L(A),[A],M]$. It also seems like addition of all angles is possible without referring to orientation. $\endgroup$ – SleepySquirrel Oct 5 at 16:20
  • $\begingroup$ I'm not sure if I understand your question correctly, but the nature of synthetic geometry is such that there is no fixed orientation. I mean orientation is relative. You can't tell for intance "This directed angle is oriented clockwise" because it's just a matter of drawing a picture. You flip the paper and all axioms hold in the new plane but the same angle is now oriented anti-clockwise. $\endgroup$ – Kulisty Oct 5 at 16:47
  • $\begingroup$ If I've understood your concept of orientations of half-planes correctly (I very well might have not; please say if it seems like so), then there should be two possible orientations. We pick one of these, call it $\mathcal{O}$. Then according to what you said, for all half-lines $A$ there should be exactly one half-plane $M$ such that $\mathcal{O}=[L(A),[A],M]$. Thus this choice of $\mathcal{O}$, while arbitrary, lets us uniquely determine a half-plane from a half-line and thus the opening of an angle with respect to $\mathcal{O}$. $\endgroup$ – SleepySquirrel Oct 5 at 17:21
  • $\begingroup$ Yes, exactly. The interior (opening) of a directed angle with respect to $\mathcal{O}$ will be the usual interior or its complement (without borders) depending on the orientation of this directed angle. It might also be one of the halfplanes for half-full angles. $\endgroup$ – Kulisty Oct 5 at 17:42

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