Also, the visualisations are to some extent an artifact of how you arrange the equation: rearranging terms between the left and right side won't change the line

Also, the visualisations are to some extent an artifact of how you arrange the equation: rearranging terms between the left and right side won't change the line but it will change these visualisations. The first example illustrates it nicely: the 'hole' only exists because of a redundant division by (x^2 + y^2) on either side of the equation.