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*(**R*** ^{n}*) 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

**Definition: ***An isometry between metric spaces *(*X*,
*d _{X}*)

*d _{Y}*(

*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*(**R*** ^{n}*).