As we have seen, there can be more than one point at each location satisfying various versions of the triangle equality. This makes it awkward to define a line in H(Rn) as the collection of elements satisfying (1). It might be more natural to think of a line (or a segment) as a distance preserving image (a distance preserving map is called an isometry) of a Euclidean line (or segment). This leads to the following definitions as in [5, 6].
Definition: An isometry between metric spaces (X, dX) and (Y, dY) is a function f: X → Y so that
dY(f(a), f(b)) = dX(a, b)
for all a, b in X.
Definition: Let X be a metric space with metric d.
1. A subset S of X is a metric segment with endpoints a ≠ b in X if S is isometric to the interval [0, d(a, b)] in R .
2. A subset L of X is a metric line (or halfline) if there is an isometric embedding f: R → X (f: (0, ∞) → X) so that
f(R) = L (f((0, ∞)) = L).
In the next sections we will see theorems and examples that tell us about segments, lines, and halflines in H(Rn).