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