Equation of Circle in Complex Plane

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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.

Equation of Annulus in Complex Plane-Examples-1 to (z + i) to 2.png


Circle Defined by $z \paren {\overline z + 2} = 3$

The equation:

$z \paren {\overline z + 2} = 3$

describes a circle embedded in the complex plane whose center is at $-1 - i$ and whose radius is $\sqrt 3$.