  The superb phase resolution and quality of the OGLE data on LMC and SMC
Cepheids, together with existing data on Galactic Cepheids, are combined to
study the period-colour (PC) and amplitude-colour (AC) relations as a function
of pulsation phase. Our results confirm earlier work that the LMC PC relation
(at mean light) is more consistent with two lines of differing slopes,
separated at a period of 10 days. However, our multi-phase PC relations reveal
much new structure which can potentially increase our understanding of Cepheid
variables. These multi-phase PC relations provide insight into why the Galactic
PC relation is linear but the LMC PC relation is non-linear. This is because
the LMC PC relation is shallower for short (log P < 1) and steeper for long
(log P > 1) period Cepheids than the corresponding Galactic PC relation. Both
of the short and long period Cepheids in all three galaxies exhibit the
steepest and shallowest slopes at phases around 0.75-0.85, respectively. A
consequence is that the PC relation at phase ~0.8 is highly non-linear.
Further, the Galactic and LMC Cepheids with log P > 1 display a flat slope in
the PC plane at phases close to the maximum light. When the LMC
period-luminosity (PL) relation is studied as a function of phase, we confirm
that it changes with the PC relation. The LMC PL relation in V- and I-band near
the phase of 0.8 provides compelling evidence that this relation is also
consistent with two lines of differing slopes joined at a period close to 10
days.
