As we are working to set a new Take5 experiment with SPOT5, here are some explanations of how it is possible to change the repeat cycle of a satellite from 26 days to 5 days, by just changing the satellite altitude by a couple of kilometres. There is nothing complicated behind that, just some simple arithmetic.
A phased orbit is an orbit for which the satellite repeats the same trajectory periodically. From its orbit at an altitude of 822 km, SPOT5, like its predecessors, has a cycle of 26 days. Every 26 days, it overflies the same places on earth. In 26 days, SPOT5 makes 369 revolutions around the earth. In 24 hours, a SPOT satellite runs through 369/26=14.19 orbits. Lowering its altitude by 2 km, the satellite slows a little, but the length of the circle it has to run along is reduced. It takes a little less time to make a revolution around the earth. The satellite does exactly 14.2 orbits per day.
Here are some of the orbits of SPOT4 (Take5), with some of the sites observed in France and North Africa during the experiment. The satellite started with the Cyan track, then the green one on the day after, then the yellow one on the next day and so on. 5 days later, it came back to the cyan orbit. You may see that it was possible to acquire a site on on the green track from the adjacent one on the cyan track.
14.2 orbits per days, is equivalent to 71 orbits in 5 days. After 71 orbits and 5 days exactly SPOT4 was always at the same place during the Take5 experiment, and its cycled was changed from 26 to 5 days.
I have been also asked how the initial 26 days repeat cycle of SPOT5 was defined. The CNES engineers who designed it wanted to make it possible to observe each point on the earth from the vertical. As the SPOT satellites had a field of view of 116 km using both instruments, with a 26 days repeat cycle we had 116x26x14.19 = 43000km, just a little more than earth equator length. However, it was quickly seen that users did not ask for exactly vertical images and that the instruments were programmed mostly independently looking in different directions. However, the 26 days cycle was kept for all the SPOT satellites just as the High Speed Trains rail separation is related to the width of the hindquarters of a horse.
Finally, nothing would prevent from using the SPOT satellites from a 5 days repeat cycle orbit, which would not really change the ability to use the images how they are used now, but would allow new possibilities thanks to the possibility to observe users from constant viewing angles.
It is a little funny to observe that SPOT6 and SPOT7 do not use the initial SPOT orbit, and only fly at an altitude of 694 km but still with a 26 days phased orbit, this time obtained with 379/26=14.58 orbits per day. However, the justification cannot be the field of view, as this field of view is only 60 km. But just by rising the orbit by a few kilometers, a 5 days orbit could be obtained