Is circle formula same in different XY system?

Question

I want to check an element if it's in a circle in my browser. I know the circle formula from my high school:

But I'm not sure if it's same in browser XY system too. Because browsers zero coordinate is on left top of the page and from there to right is positive x axis and to bottom is positive y axis.

For example in this picture point c have positive x and y. Please optimize the circle formula for this system and also give me the correct formula for a circle centered in c = (200,200) with 150 radius.

Answer 1

For any affine change of variables $x \to a_1 x + a_2 y + a_3$ and $y \to b_1 x + b_2 y + b_3$ the equation of the circle $(x-x_0)^2 + (y-y_0)^2 = r^2$ translates into $$ ( a_1 x + a_2 y + a_3 - x_0)^2 + (b_1 x + b_2 y +b_3 - y_0)^2 = r^2 $$ Expanding $$ \begin{eqnarray} & & x^2 ( a_1^2 + b_1^2) + y^2 (a_2^2 + b_2^2) + 2 x y ( a_1 a_2 + b_1 b_2) \\ && + 2 x ( a_1 (a_3 - x_0) + b_1 (b_3-y_0) ) + 2 y ( a_2 (a_3 - x_0) + b_2 (b_3-y_0) ) \\ && + (a_3-x_0)^2 + (b_3 - y_0)^2 = r^2 \end{eqnarray} $$ Now, choosing $a_1$, $a_2$ and $b_1$ and $b_2$ so that $$ a_1^2 + b_1^2 = a_2^2 + b_2^2 = 1 \qquad a_1 a_2 + b_1 b_2 = 0 $$ and completing squares will again arrive at $(x-\hat{x}_0)^2 + (y-\hat{y}_0)^2 = \hat{r}^2$.

Answer 2

Even with the $y$-coordinate flipped, the equation for your circle is still $$ (x-200)^2+(y-200)^2=150^2\tag{1} $$ In the normal coordinates, this would be the circle of radius $150$ and center $(200,-200)$ so the equation with flipped $y$-coordinate would be $$ (x-200)^2+(-y-(-200))^2=150^2\tag{2} $$ Equation $(2)$ is the same as equation $(1)$ since $(-1)^2=1$.