Equation of Circle in Complex Plane
Theorem
Formulation 1
Let $\C$ be the complex plane.
Let $C$ be a circle in $\C$ whose radius is $r \in \R_{>0}$ and whose center is $\alpha \in \C$.
Then $C$ may be written as:
- $\cmod {z - \alpha} = r$
where $\cmod {\, \cdot \,}$ denotes complex modulus.
Formulation 2
Let $\C$ be the complex plane.
Let $C$ be a circle in $\C$.
Then $C$ may be written as:
- $\alpha z \overline z + \beta z + \overline \beta \overline z + \gamma = 0$
where:
- $\alpha \in \R_{\ne 0}$ is real and non-zero
- $\gamma \in \R$ is real
- $\beta \in \C$ is complex such that $\cmod \beta^2 > \alpha \gamma$.
The curve $C$ is a straight line if and only if $\alpha = 0$ and $\beta \ne 0$.
Examples
Radius $4$, Center $\tuple {-2, 1}$
Let $C$ be a circle embedded in the complex plane whose radius is $4$ and whose center is $\paren {-2, 1}$.
Then $C$ can be described by the equation:
- $\cmod {z + 2 - i} = 4$
or in conventional Cartesian coordinates:
- $\paren {x + 2}^2 + \paren {y - 1}^2 = 16$
Radius $2$, Center $\tuple {0, 1}$
Let $C$ be a circle embedded in the complex plane whose radius is $2$ and whose center is $\paren {0, 1}$.
Then $C$ can be described by the equation:
- $\cmod {z - i} = 2$
or in conventional Cartesian coordinates:
- $x^2 + \paren {y - 1}^2 = 4$
Radius $2$, Center $\tuple {-3, 4}$
Let $C$ be a circle embedded in the complex plane whose radius is $2$ and whose center is $\paren {-3, 4}$.
Then $C$ can be described by the equation:
- $\cmod {z + 3 - 4 i} = 2$
or in conventional Cartesian coordinates:
- $\paren {x + 3}^2 + \paren {y - 4}^2 = 4$
Radius $4$, Center $\tuple {0, -3}$
The inequality:
- $\cmod {z + 3 i} > 4$
describes the area outside the circle whose center is at $-3 i$, whose radius is $4$.
Annulus: $1 < \cmod {z + i} \le 2$
The inequality:
- $1 < \cmod {z + i} \le 2$
describes the inside of the annulus whose center is at $-i$, whose inner radius is $1$ and whose outer radius is $2$.
This annulus does not include its inner boundary, but does include its outer boundary.
Circle Defined by $z \paren {\overline z + 2} = 3$
The equation:
- $z \paren {\overline z + 2} = 3$
is a quadratic equation with $2$ solutions:
- $z = 1$
- $z = -3$