# Affine (x, y) Coordinates

Given points $$P_1 = (x_1, y_1)$$ and $$P_2 = (x_2, y_2)$$ in $$\mathbb{G}$$, their sum in the group $$P_3 = (x_3, y_3) = P_1 + P_2$$ is computed with: $\begin{eqnarray*} x_3 &=& \frac{b((x_1 + x_2)(x_1 x_2 + b) + 2a x_1 x_2 + 2 y_1 y_2)}{(x_1 x_2 - b)^2} \\ y_3 &=& \frac{b(2a(x_1 y_2 + x_2 y_1)(x_1 x_2 + b) + (x_1^2 y_2 + x_2^2 y_1)(x_1 x_2 + 3b) + (y_1 + y_2)(3b x_1 x_2 + b^2))}{-(x_1 x_2 - b)^3} \end{eqnarray*}$

These formulas are complete. Denominators can never be zero (because $$x_1 x_2$$ is a quadratic residue, but $$b$$ is not).