Circumference Equation from Complex Numbers

October 9, 2024

This is a derivation of the equation of a circumference from complex numbers.

As we know the equation of a cirfumference centered at w=(a,b)w=(a, b) with radius rr is:

(x−a)2+(y−b)2=r2 (x-a)^2 + (y-b)^2 = r^2

Be the affix of a point zz in the complex plane z=x+iyz = x + iy. The affix of the center of the circumference is w=a+ibw = a + ib.

Let’s substitute zz and ww in the equation of the circumference:

(x−a)2+(y−b)2=r2(x−a)2+(y−b)2=r2(x−a)(x−a)+(y−b)(y−b)=r2x2−2ax+a2+y2−2by+b2=r2x2+y2−2ax−2by+a2+b2=r2 \begin{aligned} (x-a)^2 + (y-b)^2 &= r^2 \\ (x-a)^2 + (y-b)^2 &= r^2 \\ (x-a)(x-a) + (y-b)(y-b) &= r^2 \\ x^2 - 2ax + a^2 + y^2 - 2by + b^2 &= r^2 \\ x^2 + y^2 - 2ax - 2by + a^2 + b^2 &= r^2 \\ \end{aligned}

As we also know, the modulus of a complex number zz is:

∣z∣=x2+y2 |z| = \sqrt{x^2 + y^2}

And also the product of a complex number and the conjugate of another complex number is:

zw‾=(x+iy)(a−ib)=xa−ixb+iya+yb=xa+yb+i(ya−xb) \begin{aligned} z \overline{w} &= (x + iy)(a - ib) \\ &= xa - ixb + iya + yb \\ &= xa + yb + i(ya - xb) \\ \end{aligned}

So we can write the equation of the circumference as:

∣z∣2−2(ax+by)+∣w∣2=r2∣z∣2−2Re(zw‾)+∣w∣2=r2 \begin{aligned} |z|^2 - 2(ax + by) + |w|^2 &= r^2 \\ |z|^2 - 2Re(z \overline{w}) + |w|^2 &= r^2 \end{aligned}

Therefore, we have derived the equation of a circumference from complex numbers.