Equation of Ellipse in Complex Plane

Theorem

Let $\C$ be the complex plane.

Let $E$ be an ellipse in $\C$ whose major axis is $d \in \R_{>0}$ and whose foci are at $\alpha, \beta \in \C$.

Then $C$ may be written as:

$\cmod {z - \alpha} + \cmod {z - \beta} = d$

where $\cmod {\, \cdot \,}$ denotes complex modulus.

Interior

Equation of Ellipse in Complex Plane/Interior

Exterior

Equation of Ellipse in Complex Plane/Exterior

Proof

By definition of complex modulus:

$\cmod {z - \alpha}$ is the distance from $z$ to $\alpha$

$\cmod {z - \beta}$ is the distance from $z$ to $\beta$.

Thus $\cmod {z - \alpha} + \cmod {z - \beta}$ is the sum of the distance from $z$ to $\alpha$ and from $z$ to $\beta$.

This is precisely the equidistance property of the ellipse.

From Equidistance of Ellipse equals Major Axis, the constant distance $d$ is equal to the major axis of $E$.

$\blacksquare$

Examples

Example: Foci at $1$ and $i$, Major Axis $4$

The ellipse in the complex plane whose major axis is of length $4$ and whose foci are at the points corresponding to $1$ and $i$ is given by the equation:

$\cmod {z - 1} + \cmod {z - i} = 4$

Example: Foci at $-3$ and $3$, Major Axis $10$

The ellipse in the complex plane whose major axis is of length $10$ and whose foci are at the points corresponding to $-3$ and $3$ is given by the equation:

$\cmod {z + 3} + \cmod {z - 3} = 10$

and also as:

$\dfrac {x^2} {25} + \dfrac {y^2} {16} = 1$

Example: Foci at $-2 i$ and $2 i$, Major Axis $6$

The ellipse in the complex plane whose major axis is of length $6$ and whose foci are at the points corresponding to $-2 i$ and $2 i$ is given by the equation:

$\cmod {z + 2 i} + \cmod {z - 2 i} = 6$

Example: Foci at $-2 i$ and $2 i$, Major Axis $6$

The ellipse in the complex plane whose major axis is of length $6$ and whose foci are at the points corresponding to $-2 i$ and $2 i$ is given by the equation:

$\cmod {z + 2 i} + \cmod {z - 2 i} = 6$

Example: Foci at $\tuple {2, -3}$ and $\tuple {-2, 3}$, Major Axis $10$

The inequality:

$\cmod {z + 2 - 3 i} + \cmod {z - 2 + 3 i} < 10$

defines the inside of the ellipse in the complex plane whose major axis is of length $10$ and whose foci are at the points corresponding to $\tuple {2, -3}$ and $\tuple {2, -3}$.