  We present, both exactly and approximately, a complete set of mappings
between the vacuum (or fundamental) leptonic mixing parameters and the
effective ones in matter with non-standard neutrino interaction (NSI) effects
included. Within the three-flavor neutrino framework and a constant matter
density profile, a full set of sum rules is established, which enables us to
reconstruct the moduli of the effective leptonic mixing matrix elements, in
terms of the vacuum mixing parameters in order to reproduce the neutrino
oscillation probabilities for future long-baseline experiments. Very compact,
but quite accurate, approximate mappings are obtained based on series
expansions in the neutrino mass hierarchy parameter \eta \equiv \Delta
m^2_{21}/\Delta m^2_{31}, the vacuum leptonic mixing parameter s_{13} \equiv
\sin\theta_{13}, and the NSI parameters \epsilon_{\alpha\beta}. A detailed
numerical analysis about how the NSIs affect the smallest leptonic mixing angle
\theta_{13}, the deviation of the leptonic mixing angle \theta_{23} from its
maximal mixing value, and the transition probabilities useful for future
experiments are performed using our analytical results.
