When are two paths in $\mathbb{C}$ equivalent? In studying complex analysis, I've come across the notion of equivalent paths. Specifically, we say that smooth (by which I mean their derivatives are continuous) paths $\gamma: [a,b] \to \mathbb{C}$ and $\sigma: [c,d] \to \mathbb{C}$ are equivalent if there is an increasing, continuous function $\varphi: [a,b] \to [c,d]$ such that $\gamma = \sigma \circ \varphi$.
I was wondering if there were any necessary and sufficient conditions for two paths being equivalent by this definition. I've proved that if $\gamma: [a,b] \to \mathbb{C}$ and $\sigma: [c,d] \to \mathbb{C}$ are equivalent smooth paths, then

*

*$\gamma(a) = \sigma(c)$, $\gamma(b) = \sigma(d)$

*$\{\gamma\} = \{\sigma\}$, where $\{\gamma\}$ is the trace of $\gamma$, $\{\gamma\} = \gamma([a,b])$.

*$V(\gamma) = V(\sigma)$, where $V(\gamma)$ is the total variation of $\gamma$ and is defined as

\begin{equation*}
V(\gamma) = \int_a^b |\gamma'(t)| \; dt.
\end{equation*}
Intuitively, I think that 1 says that $\gamma$ and $\sigma$ start and end at the same points, 2 says that they trace out the same path in $\mathbb{C}$, and 3 says that they have the same length.
In other words, I've proved that 1, 2, and 3 are necessary conditions for $\gamma$ and $\sigma$ to be equivalent, but are they sufficient?
I'm not sure how to define $\varphi$ that ensures it is increasing and continuous, and so that $\gamma = \sigma \circ \varphi$. My idea so far has been to define it like this: For $t \in [a,b]$, define $\varphi(t)$ to be the point in $[c,d]$ so that $\gamma(t) = \sigma(\varphi(t))$. We know that such a point exists given 2, but not that it is unique, so that this definition does not uniquely defined a function $\varphi: [a,b] \to [c,d]$.

In my book (Conway's Function of One Complex Variable), this question really came up for rectifiable paths, but I figured I'd ask this for smooth paths instead, since that, along with piecewise smooth paths, are what I'm more interested in.
 A: Your conditions are not sufficient.  For instance, consider two paths which go from $0$ to $2$ along the real axis: one of them goes from $0$ to $1$, then goes back to $0$, and then goes straight to $2$.  The other goes from $0$ to $2$, then back to $1$, and then to $2$.  These two paths will satisfy your conditions, but are not equivalent.  Intuitively, your condition (3) is too weak: it only says the total length traversed by the paths is the same, but the repetitions along the trace could be in different parts for the two paths.
There are other sorts of ways it can go wrong.  For instance, you can take two paths that go around the same loop (and end where they start), but in opposite directions.  Or you could take two paths which traverse a figure 8 in different ways at the intersection point of the two loops.
For paths which are injective, though, your conditions (1) and (2) alone suffice.  Indeed, if $\gamma:[a,b]\to\mathbb{C}$ is continuous and injective, then it is an embedding since $[a,b]$ is compact and $\mathbb{C}$ is Hausdorff.  It follows that if $\sigma:[c,d]\to\mathbb{C}$ is any other continuous injection with the same image, then the function $\varphi:[a,b]\to[c,d]$ defined by $\sigma^{-1}\circ\gamma$ is a homeomorphism.  If $\gamma(a)=\sigma(c)$ then $\varphi(a)=c$ and so $\varphi$ must be increasing.
