Using TLC's example (in comment to Igor),
$x=(1,−0.3,0.2)$, $y=(0.5,1,−0.7)$, I compute the path
should turn at $z=(1,1,-0.61)$ ($\frac{11}{18}=0.61$), whence the path length is 1.8.
Unless I am mistaken, this does not unfold to a straight line. [True but irrelevant; see below.]
Update. Now showing the range of $z=(1,1,t)$, $t\in(-1,-0.2)$ (green), which yields the same distance $d(x,y)=1.8$.
