# Relevance of Complex Coordinate Geometry [closed]

I have been studying complex numbers (but not Calculus with them). So I have understood

• Arithmetic with Complex Numbers: Add, Subtract, Multiply, Divide, Exponentiate
• Forms of Complex Numbers: Rectangular, Polar, Exponential
• Roots of Unity and their basic properties
• Complex Plane/Argand Diagram

For example, I have learnt that the general equation of a circle is $$az\overline{z}+\overline{B}z+B\overline{z}+c=0$$

And the condition for three points to form an equilateral triangle in the complex plane is: $$z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_1z_3 + z_2z_3$$

I know that as I go forward, I will learn about lines, and ellipses, parallelograms and quadrilaterals in the complex plane, with their related equations, and properties. Such questions tend to get asked in entrance exams and contest math (both of which interest me).

I know that complex numbers get used in:

• Electrical Engineering, and that they are useful in dealing with waves/oscillations
• Solving contest math problems (say tiling with dominoes, or in generating functions)

However, I wanted to know, where else in math will I make use of complex coordinate geometry(circles, lines, ellipses, etc). That is, what are the applications of complex coordinate geometry?

• CHECK : brilliant.org/wiki/complex-numbers-in-geometry : It says handling Circles & Polygons with rotation is simple with Complex Numbers while "rotating Cartesian coordinates involves heavy calculation and (generally) an ugly result" & then goes on to say that "complex geometry is most useful when there is a primary circle in the problem that can be set to the unit circle" & It contains Simple Solutions to Cases which are Cumbersome & Difficult with Cartesian Methods. It talks about "similarities between complex numbers and vectors". That Single Article has a lot of great content !
– Prem
Commented Oct 26, 2022 at 4:02
• Having worked in complex analysis for years, I've never seen that equilateral triangle thing, so I can safely say they aren't ubiquitous. I'd guess such explicit representations are useful for some algebraic geometry calculations, but in practice most of the things one works with in complex geometry of a single variables are conformal transformations of a line (e.g. half planes, circles, disks, etc) or merely up to biholomorphism (e.g. Riemann Surfaces). I also think I've seen some relationships like the ones you mentioned used to find explicit harmonic solutions (i.e. Dirichlet Problem) Commented Oct 26, 2022 at 4:10
• @prem the Brilliant page gives applications to contest math (not surprising since it is mainly a contest site). I am more interested in applications beyond contest math. Commented Oct 26, 2022 at 6:02
• About parallelograms, we can start with this: $b-a=c-d \iff b+d=a+c \iff \frac12(b+d)=\frac12(a+c)$ Commented Oct 26, 2022 at 6:54
• For those equations in particular, I don't know. But complex geometry (geometry in complex coordinates, not necessarily in Euclidean ("flat") space) has a lot of applications in physics (quantum field theory, string theory, etc...) if you want "real-world" applications. Else, as Brevan mentioned, it's at the root of algebraic geometry, but this is a much more abstract topic, which is not very accessible unless you've already studied quite a bit of maths. I know there have been applications to number theory, so maybe you can count that as well. But it takes a while to get there :)
– Azur
Commented Oct 26, 2022 at 11:41