How do we formally and precisely specify a directed angle?

Question

I haven't been able to find a precise explanation of what a directed angle is. I encountered the concept in Chapter 15 of Spivak's Calculus, where he simply says

In elementary geometry an angle is simply the union of two half-lines with a common initial point. More useful for trigonometry are "directed angles", which may be regarded as pairs $(l_1,l_2)$ of half-lines with the same initial point.

I googled and found this document, which sort of sheds a bit of light. We specify a directed angle as an ordered pair $(\vec{OA}, \vec{OB})$ plus a direction. However, they specify the direction with a picture, as in

Also, apparently it is by convention that if the direction is counterclockwise, then the angle is positive, and if clockwise then it is negative.

When I think about it, in common usage, it seems we say $50^{\circ}$ or $-70^{\circ}$, and this is a measure of magnitude of an angle. We associate the magnitude with a particular angle. We know that if the magnitude is negative, for example, then the angle is being measured counterclockwise.

Is this in fact the notation to specify a directed angle, ie the magnitude?

I found the following definition of a directed angle here

Definition: Given any two non-parallel lines $l$ and $m$, we defined the directed angle $\angle(l,m)$ to be the measure of the angle starting from $l$ and ending at $m$, measured counterclockwise.

Then

Notice that

$$\angle(l,m)+\angle(m,l)=180^{\circ}\tag{1}$$

holds universally. This is kind of nice, but it's a bit annoying to have that $180^{\circ}$ lying around there, and so we will also the all angle measures modulo $180^{\circ}$. That means that $-70^{\circ}=110^{\circ}=290^{\circ}=...$ Once we take mod$180^{\circ}$, $(1)$ becomes the following very important result

Proposition: For any lines $l$ and $m$, $$\angle(l,m)=-\angle(m,l)\tag{2}$$ (In other words, measuring the angle clockwise instead of counterclockwise corresponds to negation).

I'm not sure I follow the calculations. I'm not too familiar with modular arithmetic.

We have

$$180 \mod 180=0$$ Thus $$[\angle(l,m)+\angle(m,l)] \mod 180=0$$

But as far as I can tell

$$[\angle(l,m)+\angle(m,l)] \mod 180=\left [[\angle(l,m)\mod 180]+[\angle(m,l)\mod 180]\right ] \mod 180=0$$

How do we obtain $(2)$?

Also, are these calculations general in the sense that they explain why when we measure angles in a clockwise direction they are negative? I would guess not.

Is the reason rather just convention? The convention being that when we specify a directed angle as an ordered pair $(l,m)$, by default we mean counterclockwise?

Answer 1

$$\angle(l,m)+\angle(m,l)=180^{\circ}\tag{1} \\ \angle (l,m) = 180 - \angle(m,l) \\ [\angle(l,m) \text{ mod } 180] = [180 \text{ mod } 180] - [\angle(m,l) \text{ mod } 180$$

$$\angle(l,m) = -\angle(m,l) \tag2$$

Answer 2

Okay... if I were to try to put this into "rules".

A directed angle is defined by a base ray, and and another ray sharing the vertex. The angle is measured from the base ray in the counterclockwise direction to the secondary ray and may exceed 180.

When it comes to adding and subtracting angles however the sum may be more than 360 or the difference may be less than $0$. This occurs if our manipulations cause us to do a full rotation and then some or if our manipulations cause us to measure in a clockwise direction.

As a full revolution brings us to the beginning and as going (undirected) measure in the clockwise (negative) direction is the same as going in the counter clockwise (positive) direction the full circle minus the measure. We can claim that directed angle measure are modular $360$. That is to say and a measure of $a$ is equal to a measure $a + 360$ and is equal to a measure $a-360$ and that $360 = 0$. If we have a measure $a: 180 < a < 360$ it is convenient to use a negative value of $-180 < a-360= -|360-a| < 0$. (Example: $270^\circ = 270 - 360 = -90$. Or we can think $|360 - 270| = 90$ so $270 = -90$)

The angles in calculus and trigonometry are directed angles where the base ray is understood to be contained in the $x$-axis with the vertex at the origin and the ray directed in the positive direction.

Answer 3

I'd like to post what I've gleaned so far from comments/answers/references.

In Euclidean geometry, an angle is just a vertex and two rays.

However, "angle" is also used to designate the measure of a Euclidean angle, where measure is just the ratio of a circular arc centered at the vertex to the radius of the circular arc. This is, however, ambiguous because there are two possible circular arcs.

Wikipedia says

The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other.

which means the first angle in the picture above is "usually" the one referred to when we say "angle" and are referring to the specific vertex and rays in the picture above.

Moving on, a measure or magnitude is always non-negative, so angles are always non-negative.

Angles that have the same measure (ie the same magnitude) are said to be equal.

Given the ideas above, it seems that under Euclidean geometry, the following angles are all different, but if we use the word "angle" as the measure of a Euclidean angle in the "usual" sense as described by Wikipedia, then the following angles are all equal.

It is possible for two angles to have the same Euclidean angle specification (same vertex, same rays), but to have a measure that differs by an integer multiple of a turn of the chosen circular arc used to measure the angles. These angles are called coterminal, and any angle has an infinite number of coterminal angles.

Next, we move on to the notion of a directed angle, which I believe is closely tied (or is perhaps the same as) the notion of a "signed angle" as I see in Wikipedia.

A convention is used to allow positive and negative angular values to represent orientations and/or rotations relative to some reference. In a 2D Cartesian coordinate system, this reference is the positive x-axis.

We then name the two sides of the Euclidean angle as initial side and terminal side, and we measure from initial to terminal. The directed angle is positive if this measurement is made with a counterclockwise orientation, and it is negative if the measurement is with a clockwise orientation.

Note that the notions of clockwise and counterclockwise depend on the x-axis being positive to the right.

I've been wondering all along what exactly is the specification of a directed angle?

Consider the five examples below

If $l_1$ and $l_2$ are rays (and I am assuming that however we specify the rays it includes the specification of the vertex), then $(l_1,l_2)$ specifies the Euclidean angle $1$.

$2, 3, 4$, and $5$ are directed angles.

Take $2$ for example. Is the specification "$(l_1,l_2)$ counterclockwise"?

If we make it a convention that if we omit the orientation then the default is counterclockwise, then $(l_1,l_2)$ specifies $2$.

On the other hand, when we use angles in, say, calculus, we simply write things like

$$\sin{\pi}, \cos{-1}, \tan{-60^{\circ}}$$

and we know the directed angle being referred to. It seems that this is the case because to a specification such as "$(l_2,l_1)$ clockwise" we attach a number, namely the measure of the angle.

Spivak sort of says this in Ch. 15 of his book Calculus

The sine and cosine of a directed angle can now be defined as follows (Figure 5): a directed angle is determined by a point $(x,y)$ with $x^2+y^2=1$; the sine of the angle is defined as $y$, and the cosine as $x$.

Despite the aura of precision surrounding the previous paragraph, we are not yet finished with the definitions of $\sin$ and $\cos$. Indeed, we have barely begun. What we have defined is the sine and cosine of a directed angle; what we want to define is $\sin{x}$ and $\cos{x}$ for each number $x$. The usual procedure for doing this depends on associating an angle to every number

Anyways, my point is that there does seem to be an actual underlying concept of a directed angle that comes before measuring it and assigning a number to it.

Finally, it also seems that common usage in trigonometry and calculus is based on conventions: Cartesian coordinate system with the x-axis horizontal pointing right as the "initial side", counterclockwise circular arc to the "terminal side" forms a positive angle, clockwise circular arc to the "terminal side" forms a negative angle.

Answer 4

To my knowledge, the best way to define directed angles goes through the following steps.
Since you are requesting for the conceptual definition I am just pinpointing the conceptual steps waving (yet not compromising) on rigorousness.

a) Take two (column) vectors in the $3$-d Euclidean space $$ {\bf v}_1 ,{\bf v}_2 \in \left( {{\mathbb R}^3 ,\left\| {\, \cdot \,} \right\|} \right) $$ and their unitary counterparts $$ {\bf u}_1 = \frac{{{\bf v}_1 }}{{\left\| {{\bf v}_1 } \right\|}},\quad {\bf u}_2 = \frac{{{\bf v}_2 }}{{\left\| {{\bf v}_2 } \right\|}} $$ and assume that they are independent.

b) Then any linear combination of ${\bf v}_1, {\bf v}_2$ will span a $2$-d subspace (plane) that contains the two same vectors. Therefore they define a plane angle between them, whose absolute value is given by $$ \left| {\angle {\bf v}_1 ,{\bf v}_2 } \right| = \arccos \left( {{\bf u}_1 \cdot {\bf u}_2 } \right) $$

c) Now, we would like that the above "measure" be additive, so that $$ \begin{array}{l} \angle \left( {{\bf v}_1 ,{\bf v}_2 } \right) + \angle \left( {{\bf v}_2 ,{\bf v}_1 } \right) = \angle \left( {{\bf v}_1 ,{\bf v}_1 } \right) = 0 \\ \angle \left( {{\bf v}_1 ,{\bf x}} \right) + \angle \left( {{\bf x},{\bf v}_2 } \right) = \angle \left( {{\bf v}_1 ,{\bf v}_2 } \right)\quad \left| {\,{\bf x} = \mu {\bf v}_1 + \lambda {\bf v}_2 } \right. \\ \end{array} $$ To define a sign we shall fix a positive normal direction $\bf n$ of the plane defined by $ {\bf v}_1 ,{\bf v}_2$, and we can do that by imposing that be positive the determinant $$ 0 < \det \left( {{\bf v}_1 ,{\bf v}_2 ,{\bf n}} \right) $$ which means to take the sign according to the sign of the cross product ${\bf v}_1 \times {\bf v}_2$.

Note that this is the same conceptual step as that of passing from the segments $\overline {AB} ,\overline {BC}$ along a given line, to the directed segments $\vec {AB} ,\vec {BC}$ upon having fixed a direction along that line.

d) But we can proceed further and remove the condition that $\bf x$ be coplanar to $ {\bf v}_1 ,{\bf v}_2 $, and ask that for whichever $\bf x \ne \bf 0$ $$ \angle \left( {{\bf v}_1 ,{\bf x}} \right) + \angle \left( {{\bf x},{\bf v}_2 } \right) = \angle \left( {{\bf v}_1 ,{\bf v}_2 } \right)\quad \left| {\,\forall {\bf x} \ne {\bf 0}} \right. $$ To achieve this we shall not only associate a sign but also a direction , i.e. a $1d$ vector. to the angle. We can do that by associating the angle to $$ {\bf u}_1 \times {\bf u}_2 $$ or even better to the association of the dot and the cross product $$ \left( {{\bf u}_1 \cdot {\bf u}_2 ,{\bf u}_1 \times {\bf u}_2 } \right) = \left( {\cos \alpha ,\sin \alpha } \right){\bf n}_{{\bf v}_1 ,{\bf v}_2 } \quad \Rightarrow \quad \alpha = {\rm atan}_{\rm 2} \left( {{\bf u}_1 \cdot {\bf u}_2 ,\left\| {{\bf u}_1 \times {\bf u}_2 } \right\|} \right) $$

Again, this is conceptually the same as that of passing from the segments $\overline {AB} ,\overline {BC}$ on a $2$-d plane, to the directed segments $\vec {AB} ,\vec {BC}$ upon having fixed a sign and direction (a unit vector) to each.

e) Passing to a general $n$-d space, the cross-product generalizes to a wedge product and the association dot and cross product to the geometric product.

The analogy with the undirected / directed line segments continues .