Thanks, L. Chung. I would hate to try explaining the operation of a complicated current mirror. This one is easy.
Even more information about the dropout voltage of an LM317 is the condition for the excellent regulation spec's on the datasheet: a minimum voltage across it (from input to output) of 3.0V. Any voltage that is less then the spec's aren't guaranteed.
Your observation of the curves for dropout are correct. But they apply only to typical LM317s, individual ones vary and could be worse.
Also, the definition of "dropped-out" in the curve's text explains an output voltage drop of 100mV which is pretty darn poor regulation.
On another electronics forum I recommended the LP2951 low-dropout variable regulator. Its typical dropout voltage is 240mV at 20mA. It even has a pin (it is an 8-pin DIL) that goes low during dropout to give a low-batt indication or an indication when power is applied that the voltage is up and running. Unlike most regulators, it is unusual to need some ESR in its output filter cap. I added a 0.22 ohm resistor in series with it to be sure. :lol: