2.4 Scan Converting Ellipses

Cf. [FvDFH90, 3.4] [PK87, 3.5]

Ellipses no longer have an 8-fold symmetry. So we have to determine the point where the slope is 1. Differentiating the implicit equation $0=F(x,y)=b^2x^2+a^2y^2-a^2b^2$ gives

$$0=F_x(x,y)+F_y(x,y)\,y'=2b^2\,x+2a^2\,y\,y',$$

so we have $y'=1$ when $a^2\,y+b^2\,x=0$ and inserting this into the implicit equation gives $a^2b^2x^2+b^4\,x^2-a^4b^2=0$, i.e. $x=\pm\frac{a^2}{a^2+b^2}$.