Complex formulation of affine transformation

Question

Am I correct in thinking that the affine transform

$\begin{bmatrix}x'\\y'\\1\end{bmatrix} = \begin{bmatrix}a&c&e\\b&d&f\\0&0&1\end{bmatrix}\begin{bmatrix}x\\y\\1\end{bmatrix}$

can be represented as

$z' = c_1 z + c_2\bar{z} + c_3$

for $z = x+iy,\ c_1 =\frac{a+d}{2} + i\frac{b-c}{2},\ c_2 = \frac{a-d}{2} + i\frac{b+c}{2},\ c_3 = e + if$ ?

Can transformations in this form be combined as easily as in the matrix form?

Answer 1

$$z(t) = x(t) + i y (t)$$

where $x(t), y(t) \in \mathbb{R}$

$$z'(t)=x'(t)+iy'(t)$$

Assuming $a,b,c,d \in \mathbb{R}$, then we have

\begin{align} c_1z+c_2\bar{z}+c_3 &=\left( \frac{a+d}{2} +i \frac{b-c}2\right)\cdot \left(x+iy \right) + \left( \frac{a-d}{2} +i \frac{b+c}2\right)\cdot \left(x-iy \right)+c_3 \\ &= ax+idy +ibx+cy+c_3 \\ &= ax+cy + i(bx+dy)+e+if \\ &=ax+cy+e+i(bx+dy+f)\\ &= x'+iy' \end{align}

Also, given $c_1$ and $c_2$, we can solve for $a,d$ from $\Re (c_1)$ and $\Re(c_2)$ . We can solve for $b,c$ from $\Im(c_1)$ and $\Im (c_2)$.

Answer 2

The composition of the transforms $f(z) = a_1z+a_2\bar{z}+a_3$ and $g(z) = b_1z+b_2\bar{z}+b_3$, is $g(f(z)) = c_1z+c_2\bar{z}+c_3$, with

$$c_1 = b_1a_1+b_2\bar{a_2} \\ c_2 = b_1a_2+b_2\bar{a_1} \\ c_3 = b_1a_3 + b_2\bar{a_3} + b_3$$