write down functions in terms of complex coordinate $z=x+iy$

Question

Indentifying $\Bbb R^2$ with the complex plane $\Bbb C$ via the map $(x,y)→ x+iy$, write down the following functions in terms of complex coordinate $z = x + iy$.

i) the translation by the vector (1,2)

ii) a rotation anticlockwise by $\theta$

iii) a reflection in the x-axis

iv) reflection in the line $y=x+10$

v) reflection in the line $y=x$

vi) inversion in the circle centered at $(0,0)$ with radius r

Here are my answers, I am just looking for verification. Thanks!

ANSWERS:

i) $t(z) = z + 1 + 2i$

ii) $t(z) = e^{i\theta}z$

iii) $t(z) = \overline z$

iv) ?

v) ?

vi) ?

Answer 1

Think about what a reflection about the line $y = x+10$ means. For instance, the point $(0,10)$ is on the line and gets mapped to the same point. Similarly, the point $(1,11)$ stays where it is. So the point $(1,10)$ would be mapped to $(0,11)$ and vice versa. What kind of transformation does this? Well, in general, it seems that you would want to take flip the $x$- and $y$-values, and then add $(-10,10)$; i.e., $$(x,y) \mapsto (y-10,x+10).$$

Well, while the above works, it seems to lack some rigor. So let's think about this more methodically. You may note that the reflection about the line $y = x$ is just $$(x,y) \mapsto (y,x).$$ So if we do a translation of $y = x+10$ by $(0,-10)$, followed by the reflection, then another translation back by $(0,10)$, we have the composition of mappings $$(x,y) \mapsto (x,y-10) \mapsto (y-10,x) \mapsto (y-10,x+10),$$ which is what we found earlier.

How do we write this transformation in terms of complex numbers? We clearly want $$z = x + yi \mapsto (y-10) + (x+10)i.$$ But is there a function of $z$ that does this? Well, one way is to observe $$z + \bar z = (x+yi)+(x-yi) = 2x, \\ z - \bar z = (x+yi)-(x-yi) = 2yi,$$ so that $$(y-10) + (x+10)i = \left(\frac{z - \bar z}{2i} - 10 \right) + \left(\frac{z + \bar z}{2} + 10\right)i = -10 + (\bar z + 10)i.$$

I will skip the next part and go to the last. Note that inversion in the circle with radius $r$ requires that $$|t(z)||z| = r^2,$$ that is to say, the product of the magnitudes of the inverted point and the original point must equal the square of the radius. Moreover, we must also have $$\arg(t(z)) = \arg(z),$$ meaning their angles remain the same. So if $$z = |z|e^{i\arg(z)},$$ what is $t(z)$?