Computing the canonical height on K3 surfaces

Let $S$ be a surface in $\mathbb P ^2\times \mathbb P ^2$ given by the intersection of a (1,1)-form and a (2,2)-form. Then $S$ is a K3 surface with two noncommuting involutions $\sigma ^x$ and $\sigma ^y$. In 1991 the second author constructed two height functions $\hat{h} ^+$ and $\hat{h} ^-$ which behave canonically with respect to $\sigma ^x$ and $\sigma ^y$, and in 1993 together with the first author showed in general how to decompose such canonical heights into a sum of local heights $\sum _v\hat{\lambda} ^\pm (\,\cdot \,,v)$. We discuss how the geometry of the surface $S$ is related to formulas for the local heights, and we give practical algorithms for computing the involutions $\sigma ^x$, $\sigma ^y$, the local heights $\hat{\lambda} ^+(\,\cdot \,,v)$, $\hat{\lambda} ^-(\,\cdot \,,v)$, and the canonical heights $\hat{h} ^+$, $\hat{h} ^-$.