Complex Numbers: Rotation and Spiral Similarity Revisited

Spiral similarity consists of rotating a point or image and dilating it.

Suppose we want to rotate a complex number $z$ by $x$ radians around the complex number $w$, and the dilate the point by a factor of d (with respect to $w$ ).

First we can shift the entire thing so that $w$ is at the origin

So $z'=z-w$. We rotate it by x radians by multiplying by $e^{ix}$, and then we dilate (or multiply by d) to get

$z'''=d e^{ix} (z-w)$.

We undo the original translation by adding the original w back in.

$z''''=d e^{ix} (z-w) + w$.

Don't memorize this: just remember we translate, rotate, dilate, and translate back in that order.

Spiral Similarity

Let us have a complex number $z$. Suppose we wanted to map $z$ with the spiral similarity $S(w, k, \alpha)$, for some complex number $w$. To do this, we first subtract $w$ from $z$. Then, we need to be able to both apply a homothety and rotate about the origin. However, we already know how to rotate, so we still need to know how to apply a homothety about the origin.

However, that's simple! To get from $a+bi$ to $ka+kbi$, we just multiply by $k$. Therefore, our formula for spiral similarity is just: $$z' = k(z-w)e^{\alpha i} + w$$

Rotating Complex Numbers

When looking at complex numbers, a question might bug us. How do we rotate them about the origin? However, the answer is simple, even though it might elude us.

Let's take the complex number $z=re^{\theta i}$. To rotate this $\alpha$ radians counterclockwise about the origin. However, to do this, we simply multiply the argument by $\alpha$! Therefore, we just multiply the complex number by $e^{\alpha i}$ to rotate it $\alpha$ counterclockwise from the origin.

Now how do we rotate $z$ counterclockwise by $\alpha$, but this time about another complex number $w$? We first subtract $w$ from $z$ to get $v$, then rotate about the origin, then add $z$ back:

Now, we can see that the image $z'$ of rotating complex number $z$ about complex number $w$ by $\alpha$ is just: $$z' = (z - w)e^{\alpha i} + w$$

That's it. Thanks for reading!