Circle equation in homogeneous coordinates

Question

Can someone give me a derivation why the circle equation is expressed in homogeneous coordinates like this (as described in Hartley):

$$ (x-a\cdot w)^2 + (y-b\cdot w)^2 = r^2\cdot w^2 $$

Answer 1

The Cartesian equation of the circle of radius $r$ and center $(a, b)$ in the $(x, y)$-plane is $$ (x - a)^{2} + (y - b)^{2} = r^{2}. \tag{1} $$ To "homogenize" in projective coordinates $(w, x, y)$, multiply out, then multiply each monomial by the appropriate power of $w$ to make each term have the same total degree in $(w, x, y)$.

Here, (1) becomes $$ x^{2} - 2ax + a^{2} + y^{2} - 2by + b^{2} = r^{2}, $$ whose terms have degree at most two in $(x, y)$, so homogenizing gives $$ x^{2} - 2awx + a^{2}w^{2} + y^{2} - 2bwy + b^{2}w^{2} = r^{2}w^{2}, $$ or $$ (x - aw)^{2} + (y - bw)^{2} = r^{2}w^{2}. $$