Circle
Complex plane

In the complex plane, a circle with a centre at c and radius r has the equation | z − c | = r. {\displaystyle |z-c|=r.}

In parametric form, this can be written as z = r e i t + c. {\displaystyle z=re^{it}+c.}

The slightly generalised equation p z z ¯ + g z + g z ¯ = q {\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}

for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with p = 1, g = − c ¯, q = r 2 − | c | 2 {\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}, since | z − c | 2 = z z ¯ − c ¯ z − c z ¯ + c c ¯ {\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.
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