Formulas
We can derive many formulas working in several coordinate systems. We give in the following pages formulas for:
 Affine \((x, y)\) coordinates
 Affine and Jacobian \((x, w)\) coordinates
 Affine and fractional \((x, u)\) coordinates
 xonly point multiplication ladders
Proofs for these formulas, as well as extra structures and other mathematical considerations, are explained at length in the whitepaper.
In all these formulas, the following apply:

The base curve \(E\) has equation \(y^2 = x(x^2 + ax + b)\). Group \(\mathbb{G}\) consists of the point of \(E\) which are not points of \(r\)torsion.

The neutral is \(N = (0, 0)\). The group law is called "addition" and denoted as such. Group \(\mathbb{G}\) is homomorphic to the subgroup of points of \(r\)torsion \(E[r]\) and thus has the same resistance to discrete logarithm.