What is the exact and precise definition of an ANGLE?

Solution 1:

"Angle" is a touchy subject. There are (at least) the following three interpretations, applicable depending on circumstances.

(a) The simplest is the following: Given two nonzero vectors ${\bf x}$, ${\bf y}\in{\mathbb R}^n$, $\>n\geq2$, the (nonoriented) angle $\alpha$ between them is the nonnegative number $$\alpha:=\arccos{{\bf x}\cdot{\bf y}\over|{\bf x}|\>|{\bf y}|}\in[0,\pi]\ .$$ In particular: Given two rays emanating from the same point $P$ in space the "enclosed angle $\alpha$" is a number between $0$ and $\pi$ inclusive, and is equal to the length of the shorter arc cut out on the unit circle in the plane of these two rays.

This idea of angle is also at the basis of spherical trigonometry, where the angle between two points ${\bf u}$, ${\bf v}\in S^2$ is considered as distance between these two points. As such it satisfies the triangle inequality.

(b) When some rotation about the origin in ${\mathbb R}^2$ is involved then the group $SO(2)$ steps into action, and it makes sense to talk about oriented angles. An oriented angle is an equivalence class of real numbers modulo $2\pi$. Each such class has a unique representant in the interval $[0,2\pi[\ $, or in the interval $\ ]{-\pi},\pi]$, and the set of these classes is bijectively related to $SO(2)$.

An example: The map $$T:\quad (x,y)\mapsto(x\cos\alpha-y\sin\alpha,\ x\sin\alpha+y\cos\alpha)$$ is a rotation of the euclidean plane about the angle $\alpha\in{\mathbb R}/(2\pi{\mathbb Z})$. When the rotation is physically performed in time and all points are in fact rotated carousel-like $n$ full turns before stopping at the right place the information about $n$ is not present in $T$.

(c) The circle group $SO(2)$ has the full real line ${\mathbb R}$ as its "universal cover". Sometimes it is desirable to work in ${\mathbb R}$ when talking about angles, e.g., when studying the logarithmic spiral $$\sigma:\quad t\mapsto (e^t\cos t,e^t\sin t)\qquad(-\infty<t<\infty)\ .$$

Another important example is the following: You have a closed curve $$\gamma:\quad t\mapsto {\bf z}(t)=\bigl(x(t),y(t)\bigr)\in\dot{\mathbb R}^2\qquad(0\leq t\leq L)$$ which encircles the origin a certain number $n$ of times. In order to compute this $n$ we cumulate the infinitesimal changes of $\phi(t):=\arg\>{\bf z}(t)$ and in the end divide by $2\pi$. Here $\phi(t)$ is only "defined up to $2\pi$", as in (b), but $\phi'(t)$ is well defined, and is given by $$\phi'(t)={x(t)y'(t)-x'(t)y(t)\over x^2(t)+y^2(t)}\qquad(0\leq t\leq L)\ .$$ It follows that the total argument increase $\Delta\phi\in{\mathbb R}$ along $\gamma$ is given by $$\Delta\phi=\int_0^L\phi'(t)\ dt=\int_0^L {x(t)y'(t)-x'(t)y(t)\over x^2(t)+y^2(t)}\ dt\ ,$$ from which we then obtain $n={\Delta\phi\over 2\pi}$. In complex analysis this number appears as $$n(\gamma,0)={1\over2\pi i}\int\nolimits_\gamma{dz\over z}\ .$$