In neutral or absolute geometry, and in hyperbolic geometry, there may be many lines parallel to a given line
l
through a point
P
not on line
R
; however, in the plane, two parallels may be closer to
l
than all others (one in each direction of
R
).

Thus it is useful to make a new definition concerning parallels in neutral geometry. If there are closest parallels to a given line they are known as the **limiting parallel**, **asymptotic parallel** or **horoparallel** (horo from Greek: ὅριον — border).

For rays, the relation of limiting parallel is an equivalence relation, which includes the equivalence relation of being coterminal.

Limiting parallels may form two, or three sides of a limit triangle.

A ray
A
a
is a limiting parallel to a ray
B
b
if they are coterminal or if they lie on distinct lines not equal to the line
A
B
, they do not meet, and every ray in the interior of the angle
B
A
a
meets the ray
B
b
.

Distinct lines carrying limiting parallel rays do not meet.

Suppose that the lines carrying distinct parallel rays met. By definition the cannot meet on the side of
A
B
which either
a
is on. Then they must meet on the side of
A
B
opposite to
a
, call this point
C
. Thus
∠
C
A
B
+
∠
C
B
A
<
2
right angles
⇒
∠
a
A
B
+
∠
b
B
A
>
2
right angles
. Contradiction.