 Height parallelogram identity

For naive logarithmic $x$-height on an elliptic curve, $h_x(P+Q)+h_x(P-Q)=2h_x(P)+2h_x(Q)+O(1)$ uniformly. The unordered pair of sum and difference on the $x$-line defines a bidegree-$(2,2)$ morphism to ${\mathrm{Sym}}^2{\mathbb{P}}^1$, yielding the estimate. Applying it to $2^{n}P,2^{n}Q$ and taking the defining limit gives the exact identity for the canonical height of an elliptic curve.