Equation for a circle in homogeneous coordinates

Question

The equation for a circle in homogeneous coordinates is given by $(x - aw)^2 + (y - bw)^2 = r^2w^2$.

I understand that the center of the circle, given by (a, b) in euclidian space is given by (a, b, 1) in projective space, or equivalently (aw, bw, w), but I don't understand how this translates to the above equation. In particular, why there is a $w^2$ term multiplied into $r^2$.

There is another identical question, but the answer doesn't make much sense to me.

Circle equation in homogeneous coordinates

Answer 1

An algebraic curve, with affine polynomial equation $p(x,y)=0$, $\deg p=d$, has a corresponding projective curve with polynomial homogeneous equation $P(X,Y,W)=0$, defined as $$P(X,Y,W)=W^d p\Bigl(\frac XW,\frac YW\Bigr)$$

In practice this means that to, say, the affine cubic curve $y^2=x(x^2+1)$, there corresponds the projective cubic curve $$W^3\frac{Y^2}{W^2}=W^3\frac XW\Bigl(\frac{X^2}{W^2}+1\Bigr)\iff Y^2W=X(X^2+W^2).$$ In other words, each monomial in the affine equation is completed by the power of $W$ so that the total degree of the monomial be $\deg p$.

Answer 2

This might be easier to visualize if you identify a point $(x,y)$ in $\mathbb{RP}^2$ with line through $(x,y,1)$ in $\mathbb R^3$ (less the origin). I.e., the projective plane is the plane $z=1$ in $\mathbb R^3$ with points at infinity and the line at infinity added. The $w$-coordinate of the point’s homogeneous representation is just a $z$-coordinate in $\mathbb R^3$ in this model. A circle in the projective plane then corresponds to a cone in $\mathbb R^3$, which is precisely what the homogeneous equation for a circle describes.