h o g 1 (E)
ho g 1 o h-^1 (E)7.5. FAMILY OF ALL QUASICIRCLESh(E)FIGURE 7 .27.5. Family of all quasicircles93We construct next a simple family I: of Jordan curves S c R^2 such that the
boundary f)D of each bounded quasidisk D is bilipschitz equivalent to a curve S
in I:. The construction is a generalization of the construction of von Koch's well-
known snowflake curve.
Given r with 1/4::::; r < 1 /2, we set
s =Jr -1/4
and assign to each segment/= [a, b] C R^2 a polygonal curve 1(r) consisting of the
union of four segments
(7.5.1)/1 = [a, a+ r(b - a)],
12 = [a + r ( b - a), 1 (a + b) - i s ( b - a)],
/3 = [ 1 (a + b) - i s ( b - a) , b - r ( b - a)],
/4 = [b-r(b-a), b].Hence length(rj) = r length(!) for j = 1, 2, 3, 4 and
1° 1(r) consists of four equal segments in/ when r = 1 /4,
2° 1(r) consists of two segments in/ plus a "triangle" with base in/ on the
right side of/ when 1 /4 < r < 1/2.
Now for each parameter 1 /4 < p < 1 /2 we define a family I:(p) of polygons Sn
as follows. First let 8 1 denote the positively oriented unit square with sides
11 = [O, 1], /2 = [1, 1 + i], /3 = [1 + i, i], /4 = [i, O].