1

I need to find a general formulae for a square, with its interior included, in terms of complex numbers. Note that your general square should have (general centre, side-length and orientation.)

I do not know how to deal with the orientation i.e. when the square is 45 degrees.

My solution is that let $a,b,c\in\mathbb{R}$ and $a< b$, then the general square and its interior is the intersection of $a\leq\text{Re}(z)\leq b$ and $c\leq \text{Im}(z)\leq c+|b-a|$ but this only has one orientation.

Student
  • 59

1 Answers1

1

A square with center the origin and sides, of length $2a>0$, parallel to the axes: $$\max\{|Re(z)|,|Im(z)|\}\leq a.$$

Rotating the square around the origin with angle $\theta$: $$\max\{|Re(z/e^{i\theta})|,|Im(z/e^{i\theta})|\}\leq a.$$

Translating it: $$\max\{|Re((z-c)/e^{i\theta})|,|Im((z-c)/e^{i\theta})|\}\leq a.$$

Using that $Re(z)=(z+\overline{z})/2$ and $Im(z)=(z-\overline{z})/2i$

$$\max\{|(z-c)/e^{i\theta}+\overline{(z-c)/e^{i\theta}}|,|(z-c)/e^{i\theta}-\overline{(z-c)/e^{i\theta}}|\}\leq 2a$$

OR.
  • 5,941
  • Could you please add some explanations to your answer? i.e, how do you get this solution? – Student Mar 25 '14 at 00:25
  • A geometric way to think about multiplication in the complex numbers involves the polar form. If $z = re^{i\theta}$, then multiplying $z$ to something corresponds to scaling it by $r$, then rotating it by $\theta$. So, for a tilted square, dividing by $e^{i\theta}$ will un-tilt it, so it's back in "standard position", which you have a formula for. – Henry Swanson Mar 25 '14 at 00:59