Topology of Plane Sets of Points 87- Circles. (a) Since lz - al expresses the distance from a to z, ·the
equation of a circle C( a, r) with center at a and radius r 2:: 0 is given by
lz - al= r (2.4-19)
In set notation we have
C(a,r) = {z: lz-al = r, r 2:: O} (2.4-20)
As already noted, the circle C(O, 1) with center at the origin and unit radius
is referred to as the unit circle. The circle C( a, 0) reduces to the point a.
Equation (2.4-19) may also be written as
(z - a)(z -a)= r^2 (2.4-21)
or
zz - az -az + aa - r^2 = 0
An equation of the form
zz + Az + Bz + C = 0 (2.4-22)
will represent a circle iff A = fJ, C is real and r^2 = I BI 2 - C 2:: 0
(Exercises 1.2, problem 25).
(b) A positively oriented circle with center at a and radius r 1s
represented in terms of a real parameter t by the mapping
z - a= reit, 0::::; t::::; 2k7r, k > 0 an integer. (2.4-23)
Clearly, t = arg(z - a) (Fig. 2.5).
Example The equation of the oriented unit circle is
0 ::::; t ::::; 27fA negatively oriented circle with center at a and radius r is defined by
y0Fig. 2.5
z - a= re -it , 0 ::; t ::; 2h' k > 0