Observations are reported of different sources of $CP$ violation from an amplitude analysis of $B^+ to pi^+ pi^+ pi^-$ decays, based on a data sample corresponding to an integrated luminosity of $3 ; {rm fb}^{-1}$ of $pp$ collisions recorded with the LHCb detector. A large $CP$ asymmetry is observed in the decay amplitude involving the tensor $f_2(1270)$ resonance, and in addition significant $CP$ violation is found in the $pi^+ pi^-$ S-wave at low invariant mass. The presence of $CP$ violation related to interference between the $pi^+ pi^-$ S-wave and the P-wave $B^+ to rho(770)^0 pi^+$ amplitude is also established; this causes large local asymmetries but cancels when integrated over the phase space of the decay. The results provide both qualitative and quantitative new insights into $CP$-violation effects in hadronic $B$ decays.
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