Length of Difference Curve Let $\varphi : [a,b] \to \mathbb R^n$ be a curve, and for some partition $\pi = \{ t_0 = a, t_1, \ldots, t_m = b \}$ of $[a,b]$ set
$$
 l(\pi, \varphi) = \sum_{i=1}^m \| \varphi(t_i) - \varphi(t_{i-1})\|
$$
and $L(\varphi) = \sup_{\pi \mbox{ partition}} l(\pi,\varphi)$, the length of the curve. Does it hold that
$$
 L(\varphi_2 - \varphi_1) = L(\varphi_2) - L(\varphi_1) 
$$
for two curves $\varphi_2, \varphi_1 : [a,b] \to \mathbb R^n$?
 A: Just look at an counterexample in $\mathbb R^2$, where 
$\varphi_1(t):=(-t,0),t\in[0,1], \varphi_2(t):=(0,t),t\in[0,1]$. Then $\varphi_2-\varphi_1=(t,t)$, and therefore:
$$
L(\varphi_1)=1,L(\varphi_2)=1,\text{ but }L(\varphi_2-\varphi_1)=\sqrt 2
$$
A: It depends how you define $\varphi_{2}-\varphi_{1}$. If by that you mean the difference in values, that is, $\varphi_{2}(t)-\varphi_{1}(t)$, then there is no reason to suppose it would be true, and a counterexample is easy to find.  
If, on the other hand, your notation means going along the path $\varphi_{1}$ and then following $\varphi_{2}$ backwards, then your result only holds if $\varphi_{1}$ and $\varphi_{2}$ coincide.  
The reason I make this distinction is that it is not uncommon to see the (admittedly confusing) notation $\varphi_{1}+\varphi_{2}$ to denote the union of two paths.
A: Since $L(\varphi_{1} - \varphi_{2}) = L(\varphi_{2} - \varphi_{1})$ (the "difference" curves are congruent), the left-hand side of your proposed identity is symmetric in the two curves. The right-hand side, by contrast, is obviously antisymmetric. Since neither side is generally zero, the proposed identity cannot be correct.
Regarding your comment on Troy Woo's post, $L(\varphi_{2} − \varphi_{1}) \geq \left\lvert L(\varphi_{1}) − L(\varphi_{2})\right\rvert$ does hold: If $v = \varphi_{1}(t) - \varphi_{1}(s)$ and $w = \varphi_{2}(t) - \varphi_{2}(s)$ for some real $s$ and $t$, then
$$
\|v - w\| \geq \bigl| \|v\| - \|w\| \bigr|
$$
by the reverse triangle inequality. Pick an arbitrary partition $\pi$ of $[a, b]$. Summing the preceding inequality gives
$$
l(\pi, \varphi_{1} - \varphi_{2})
  \geq \bigl|l(\pi, \varphi_{1}) - l(\pi, \varphi_{2})\bigr|.
$$
Finally, pick a sequence of partitions $\pi_{k}$ for which $l(\pi_{k}, \varphi_{i}) \to L(\varphi_{i})$ and $l(\pi_{k}, \varphi_{1} - \varphi_{2}) \to L(\varphi_{1} - \varphi_{2})$ to get the stated inequality for lengths.
