152 Chapter^3positive or natural orientation of the interval [a, ,B] (as defined by the "less
than" relation) induces on the graph of the arc a certain ordering of its
points called the positive orientation of the arc.
Intuitively, the graph of an arc is regarded as_ traversed in the positive
direction by a moving point (the image oft) as t increases from a to ,8.
However, unless the mapping defined by z = z(t) is one-to-one, the image
of t will move forward and back along the graph as t increases from a to
,8. See Example 3 below.
Remark Sometimes the graph of an arc is called simply "the arc" for
brevity. However, we must emphasize that we have defined an arc as a
continuous mapping of a closed interval, not as a set of points. There is
always an infinite number of arcs having the same graph.
Examples 1. The arcs
(a) z = r:eit, 0 5 t 5 27r(b) z = reit, 0 5 t 5 47r
(c) z = re-it, 0 5 t 5 27r
are all different, yet all have the same graph, namely, the circle lzl
r (Fig. 3.13).
- The arcs
(a) z = t + it^2 , 0 5 t 5 1
(b) z = t^2 + it^4 , 0 :::; t :::; 1
( c) z = -t + it^2 ' -1 :::; t :::; 0
are all different since they are defined by different functions, yet all of
them have the same graph, namely, the portion of the parabola y = x^2
corresponding to the interval 0 5 x :::; 1 (Fig. 3.14).
Note that the image of the point t =^1 / 2 under the mapping (a) is the
point (1/ 2 ,^1 / 4 ), whereas under the mapping (b) is the point (1/ 4 ,^1 / 16 ). Also,
y y y(a) (b) (c)Fig. 3.13