  Many theoretical models of high temperature superconductivity focus only on
the doping dependence of the CuO2 plane electronic structure. But such models
are manifestly insufficient to explain the strong variations in superconducting
critical temperature Tc among cuprates which have identical hole-density but
are crystallographically different outside the CuO2 plane. A key challenge,
therefore, has been to identify a predominant out-of-plane influence
controlling the superconductivity - with much attention focusing on the
distance $d_A$ between the apical oxygen and the planar copper atom. Here we
report direct determination of how variations of inter-atomic distances within
individual crystalline unit cells, affect the superconducting energy-gap
maximum $\Delta$ of Bi2Sr2CaCu2O8. In this material, quasi-periodic variations
of unit cell geometry occur in the form of a bulk crystalline
'supermodulation'. Within each supermodulation period, we find a $\sim 9\pm1%$
co-sinusoidal variation in local $\Delta$ that is anti-correlated with the
associated $d_A$ variations. Furthermore, we show that phenomenological
consistency would exist between these effects and the random $\Delta$
variations found near dopant atoms if the primary effect of the interstitial
dopant atom is to displace the apical oxygen so as to diminish $d_A$ or tilt
the CuO5 pyramid. Thus we reveal a strong non-random out-of-plane effect on
cuprate superconductivity at atomic scale.
