Representation of Circles by Hermitian Matrices

I borrowed the book Geometry of Complex Numbers by Schwerdtfeger. The material has been fascinating so far, though admittedly it took me about 3 hours to comprehend the material in page 1 of chapter 1. The first subject is this intriguing idea that you can represent a circle as a matrix. More specifically, you you can represent a circle with (complex) center 𝛾 and radius 𝜌 as a matrix

\begin{bmatrix} A & B \\ C & D \end{bmatrix}

Where B = – A𝛾̅, C = -A𝛾, and D = A(𝛾𝛾̅-𝜌²), so that A and D are real numbers, and B and C are complex numbers. For normal circles, you will have A=1, though you can scale the matrix to have other (non-zero) values of A and still have the same circle. (If A is zero, you end up with a straight line, or some other mysterious thing that is not a circle.)

The simple case is the unit circle:

Which is \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}

More exotic is the circle away from the origin, centered at 10+10i, i.e., (10, 10), with radius 3:

\begin{bmatrix} 1 & -10+10i \\ -10-10i & 191 \end{bmatrix}

Here is some code to generate the matrix in Racket, and to put it back:

lang racket

(require math/array)
(require math/matrix)

(define (circle-to-matrix zC r)
  (let ([B (* -1 (conjugate zC))]
        [mzC (magnitude zC)])
    (matrix [[ 1             B                       ]
             [ (conjugate B) (- (* mzC mzC) (* r r)) ]])))

(define (matrix-circle-radius M)
  (let ([A (array-ref M #[0 0])]
        [d (matrix-determinant M)])
    (sqrt (/ d (* -1 (* A A))))))

(define (matrix-circle-center M)
  (let ([C (array-ref M #[1 0])]
        [A (array-ref M #[0 0])])
    (/ C (* -1 A))))
racket@circle.rkt> (circle-to-matrix 0 1)
(array #[#[1 0] #[0 -1]])
racket@circle.rkt> (matrix-circle-radius (circle-to-matrix 0 1))
1
racket@circle.rkt> (matrix-circle-center (circle-to-matrix 0 1))
0

Why is this significant? I think Schwerdtfeger is going to cover that on page 2. :)

1 thought on “Representation of Circles by Hermitian Matrices”

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