Page 5 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following:
The equation for a circle in homogeneous coordinates $(x, y, w)$ is of the form
$$(x - aw)^2 + (y - bw)^2 = r^2 w^2$$
This represents the circle with centre represented in homogeneous coordinates as $(x_0, y_0, w_0)^T = (a, b, 1)^T$. It is quickly verified that the points $(x, y, w)^T = (1, \pm i, 0)^T$ lie on every such circle.
The aforementioned section of the textbook is available freely here.
Now, as I understand it, I can verify that the points lie in the circle by using the Euclidean distance:
$$d(x, y, w) = \sqrt{(x - x_0)^2 + (y - y_0)^2 + (w - w_0)^2}$$
However, if I plug in the provided values, then it seems that we would end up with the following two messy expressions:
$$d(1, i, 0) = \sqrt{(1 - a)^2 + (i - b)^2 + 1} \tag{1}$$
$$d(1, -i, 0) = \sqrt{(1 - a)^2 + (-i - b)^2 + 1} \tag{2}$$
It is not clear to me how/whether this tells us that the points lie in the circle?
So how does one verify that the points lie on every such circle? The author states that this is "quickly verifiable".
I would greatly appreciate it if people could please take the time to clarify this.
Plug $x=1$, $y=\pm i$ amd $w=0$ into the equation, to get: $$ (1-0)^2+(\pm i-0)^2 = 0, $$ which is obviously always true.