Is there an equivalent notion of winding number for non-closed curves? e.g. one that could define some notion of winding number for the curve below?
For example, I could draw a line segment starting at the origin $(0,0)$ that extends through the curve's starting point $(0,2)$ continuing upwards. Every time the curve crosses the line counter-clockwise: +1, clockwise: -1.
Is there is a different name for this that isn't winding number? Existing formalization?
From A. F. Beardon, Complex Analysis: The Argument Principle in Analysis and Topology (1979, reprinted by Dover 2020), page 91 of the 1979 edition:
Definition 7.2.1 Let $\gamma \colon [a, b] \to \mathbb{C}$ be any curve and suppose that $w \notin [\gamma]$. We define the index $n(\gamma, w)$ of $\gamma$ about $w$ by $$ n(\gamma, w) = \frac{\theta(b) - \theta(a)}{2\pi}, $$ where $\theta$ is any branch of $\operatorname{Arg}(\gamma - w)$ on $[a, b]$. If $\gamma$ is closed then $n(\gamma, w)$ is an integer.
The index $n(\gamma, w)$ is sometimes called the winding number of $\gamma$ about $w$, for it represents the number of times that a point $z$ moves around $w$ as it moves from $\gamma(a)$ to $\gamma(b)$ along $\gamma$.
From D. J. H. Garling, A Course in Mathematical Analysis, Volume III: Complex Analysis, Measure and Integration (2014), page 652:
Suppose that $\gamma \colon [a, b] \to \mathbf{C}$ is a path and that $w$ does not belong to the track $[\gamma]$ of $\gamma.$ Then $\gamma - w$ is a path in $\mathbf{C}^*,$ and so there exists a continuous branch $\theta$ of $\operatorname{Arg}(\gamma - w)$ on $[a, b].$ The winding number $n(\gamma, w)$ of $\gamma$ about $w$ is defined to be $$ n(\gamma, w) = \frac{\theta((\gamma - w)(b)) - \theta((\gamma - w)(a))}{2\pi} = \frac{\theta(\gamma(b) - w) - \theta(\gamma(a) - w)}{2\pi}. $$ It follows from Theorem 21.1.3 that this is well defined. In fact, we shall be principally concerned with the case where $\gamma$ is a closed path, so that $\gamma(a) - w = \gamma(b) - w,$ and $n(\gamma, w)$ is an integer.
From Ian Stewart & David Tall, Complex Analysis (second edition 2018), page 158:
Definition 7.6 The winding number of a path $\gamma \colon [a, b] \to \mathbb{C} \setminus \{0\}$ round the origin is $$ w(\gamma, 0) = \frac{\theta(b) - \theta(a)}{2\pi} $$ for a continuous choice of argument $\theta$ along $\gamma.$
By Theorem 7.4(ii) the winding number is well defined; that is, any continuous choice of argument gives the same value.
For any closed path, the winding number is an integer, because $\theta(b) - \theta(a)$ is an integer multiple of $2\pi.$
[…]
The winding number is additive:
Theorem 7.8 Let $\gamma_1$ and $\gamma_2$ be two paths in $\mathbb{C} \setminus \{0\}$ such that the end point of $\gamma_1$ is the start of $\gamma_2.$ Then $$ w(\gamma_1 + \gamma_2, 0) = w(\gamma_1, 0) + w(\gamma_2, 0) $$
Like Beardon, they give examples using additivity and non-integer values of the winding number.
From Gordon Thomas Whyburn, Topological Analysis (second edition 1964), pp.56-60:
Let $\phi(x)$ be any mapping of a metric space $X$ into the complex plane $Z.$ Let $\phi(x) = E$ and let $p \in Z - E.$ If there exists a continuous complex valued function $u(x)$ on $X$ such that $$ \label{3575488:eq:dagger}\tag{$\dagger$} e^{u(x)} = \phi(x) - p, \qquad x \in X, $$ we will call \eqref{3575488:eq:dagger} an admissible exponential representation of $\phi(x) - p$ on $X,$ and $u(x)$ will be called a continuous branch of the logarithm of $\phi(x) - p$ on $X.$
[…]
(1.2) If $X$ is an interval $ab,$ a ray, a line, or a plane, $\phi(x) - p$ always has an admissible exponential representation \eqref{3575488:eq:dagger} on $X.$ Further, $u(x)$ is determined up to an additive constant.
[…]
We now limit our considerations to the case in which $X$ is an interval or a simple arc $ab.$
${\rm D{\small EFINITION.}}$ For any admissible exponential representation $e^{u(x)}$ of $\phi(x) - p$ on $ab$ we define $$ \mu_{ab}(\phi, p) = u(b) - u(a). $$ When no confusion is likely to result, some or all of the symbols $ab, \phi$ and $p$ in the expression $\mu_{ab}(\phi, p)$ may be omitted.
[…]
(1.3) If $\phi(b) = \phi(a),$ then $(1/2\pi{i})\mu(\phi, p)$ is integer valued and continuous in $p.$ Thus it is constant in each component of $Z - E.$
The commonly used definition of winding number depends on a choice of origin. This winding number is the number of rotations that a person standing at the chosen origin makes if they must look directly at a point whilst it traverses the curve from start to end. Since the imagined person stands at one spot, I shall call this the "standing winding number". For a given choice of origin, consider a polar coordinate system $(r, \theta)$ with that origin, and let $\big( r(t), \theta(t) \big)$ be a parameterisation of the curve, with $0 \leq t \leq T$. Then:
$$\text{standing winding number} := \frac{\theta(T) - \theta(0)}{2\pi}$$
The standing winding number is well-defined on all curves, whether open or closed, about any point that does not itself lie on the curve.
The standing winding number of a closed curve is always an integer, regardless of the choice of origin, but the particular integer value may vary depending on this choice. Specifically, it is constant on the connected components of the complement of the set of points on the curve. That is, imagine placing your finger on any point on the plane and moving it to any other point. If you can do so without your finger crossing over any part of the curve, then the standing winding number of the curve is the same about both of those points.
Indeed, an open curve will be seen to have integer standing winding number when the choice of origin is such that the start and end of the curve lie on a line of the form $\theta = \text{constant}$. That is, if the person standing at the origin walks directly towards the endpoint of the curve that is closest to them, continues walking in that direction, and eventually ends up at the other endpoint of the curve, then that curve has integer standing winding number about that point.
This "$\theta = \text{constant}$" criterion is precisely the criterion which must be satisfied in order for a curve to have integer standing winding number about a point, and since it is trivially satisfied for all closed curves (since the starting and ending points are the same point), this is why all closed curves have integer standing winding number.
We can devise another notion of winding number which does not depend on choice of origin. It is the number of rotations that a person walking along the curve makes if they always face their direction of motion whilst traversing the curve from end to end. For that reason, I shall call it the "walking winding number". We consider the curve in any Cartesian (square) coordinate system $(x, y)$, with a unit speed parameterisation $\mathbf{r}(t) = \big( x(t), y(t) \big)$ such that $\mathbf{r}(0)$ and $\mathbf{r}(T)$ are the endpoints. Consider the velocity vector $\dot{\mathbf{r}}(t) = \big( \dot{x}(t), \dot{y}(t) \big)$. We can define a continuous function $\alpha(t)$ that describes the direction of that vector (the direction that our imaginary person is facing) at time $t$. It is such that $\dot{\mathbf{r}}(t) = \big( \cos\alpha(t), \sin\alpha(t) \big)$, and its derivative $\frac{\mathrm{d}\alpha}{\mathrm{d}t}$ is unique, since the angle $\mathrm{d}\alpha$ by which the direction of motion changes after time $\mathrm{d}t$ does not depend on the choice of coordinate system. Thus, $\alpha(t)$ itself is unique up to a constant. Then:
$$\text{walking winding number} := \frac{1}{2\pi} \int_{t=0}^{T} { \frac{\mathrm{d}\alpha}{\mathrm{d}t} \mathrm{d}t } = \frac{\alpha(T) - \alpha(0)}{2\pi}$$
The walking winding number is unique and well-defined for any curve (whether open or closed), and does not depend on any choices such as coordinate system orientation or origin. Like the standing winding number, it is always an integer for closed curves, but an open curve may have integer winding number. For example, the open curve described in polar coordinates $(r, \theta)$ by $r = \theta$ has a walking winding number of:
