Abel's theorem in problems and solutions - School of Mathematics

166 Problems of Chapter 2
e) See Figure 51; f), g) a turn (the
case (f)) and a double turn (the case (g)), both counterclockwise, along
a circle of radius R, with initial point (Figure 52). Hint. (f)
and (g) h) the semi-circle of
radius R (Figure 53); i) See Figure 54.
FIGURE 52
FIGURE 53 FIGURE 54
245. See Figure 55. By similarity one obtains
When the point moves along the segment from the
position to the position the parameter varies
from 0 to 1. Thus, for example, one can take
obtaining
It is easy to verify that this
formula describes the initial segment for any position of the points and
so, in particular, for or for
246. Since (cf., 221) it follows that: the case (a)
corresponds to the displacement of the curve by the vector corresponding
to the complex number (Figure 56).