Topology of Plane Sets of Points 83implies that z' -=f. z" and conversely, i.e., the mapping g: IR -+ L defined
by z = a + bt is one-to-one.
The set (2.4-2) will be called the graph of line L, or for simplicity, theline L. Now consider a second mapping L': z = a^1 + b^1 T, -oo < T < +oo.
(a) The two mappings L and L' represent the same oriented line iffb' = >.b and a' - a= μb, where ).. andμ are real constants and ).. > 0. In
fact, if the lines have the same graph and have the same orientation, let z 1
and z2 be two distinct points of the graph with z 1 -< z 2 • Then we must havez1 = a + bti = a' + b' 71
z 2 = a + bt 2 = a' + b' Tzwith t1 < t 2 and 71 < Tz. Subtracting (2.4-3) from (2.4-4), we get
b( iz - i1) = b' ( Tz - 71)
so that(2.4-3)
(2.4-4)b' = )..b (2.4-5)where).. = (t 2 - t 1 )/(Tz - T1) > 0.
By using (2.4-5) in (2.4-3), we obtain a+ bt 1 = a' + AbT 1 , or
a' - a= (t 1 - A7 1 )b = μb (2.4-6)
where μ = t1 - A71 is real. We may also note that since a' E L' we must
have a' E L, so for a certain t 0 1, a' = d + bt 0 , or, a' - a = bto. ·
Conversely, if (2.4-5) and (2.4-6) hold, we have
z = a^1 + b^1 T =a+ μb + >.br =a+ b(μ + >.r) =a+ bt
where t = μ + AT. If ti = μ + AT1 and t2 = μ + >.r2, it follows that
t 2 - t1 = >.( 72 - 71), and since ).. > 0, T1 < Tz implies that t1 < t2, and
conversely. Hence the two mappings z = a + bt and z = a' + b' 7 represent
the same oriented line. Clearly, if b' = >.b with ).. < 0, and a' - a = μb,
then the mappings represent the same line with opposite orientations.
(b) The condition a' - a f μb for every real μ means that the point a^1
does not belong to L. If the condition b' = )..b (>. -=f. 0) still holds, then
L and L' are parallel with the same orientation if ).. > 0, with opposite
orientation if ).. < 0.
The condition b' = )..b may also be written as
b' b'
b = >. or Im b = 0 (2.4-7)and, alternatively, as
b' v
b - b =^0 or bb' - bb1
= 0 (2.4-8)