Mobius transformation mapping three specific points to three specific points

Question

I'm having trouble understanding point (30) on page- 155 of Visual Complex Analysis, the following is given:

Let $C$ be the unique circle through the points $q,r,s$ in the z-plane, oriented so that these points succeed one another in the stated order. Likewise, let $\tilde{C}$ be the unique oriented circle through $\tilde{q} , \tilde{r}, \tilde{s}$ in the $w$-plane. Then the Moebius transofrmation given by (29) maps $C$ to $ \tilde{C}$ , and it maps the region lying to the left of $C$ to the region lying left of $\tilde{C}$

In the above paragraph, what is exactly meant by something being a left of a circle?

Answer 1

It means the dark region whose boundary is the circle.

Imagine you are walking along the circle in the direction indicated by the arrow. Then the region is on your left. On the left figure, the region is the bounded open disk; on the right one, the region is the outside part of the circle.

Answer 2

It explains on pg148:

So it's not really the region to the left of the circle as a whole, it's the region that's to the left of you, while you trace out the circle along its orientation. If you trace out the circle counterclockwise, the interior of the circle is always to the left of you, whereas if you trace out the circle clockwise, it's the exterior which is always to the left of you. This is highlighted in the pictures by the shaded regions.