A projected profile is a profile view where the station locations are projected straight down from the plan view. There is no accounting for the twists and turns of the cave's passages, which means that the profile view can twist or cross back over itself.
An extended profile is a profile where the slope distances between the stations are maintained. In other words, the profile is shown as if it "pulled straight," with all the turns and twists taken out. This profile view can be longer in view than the plan view.
An idealized profile is a compromise between the two, and where the cartographer makes subjective decisions about how to handle the passages' turns and twists. In other words, an idealized profile shows some of the twists and turns of the passage, but not all of them.
Each profile has its uses, which is a good thing, because sometimes a projected profile works best, sometimes an extended profile works best, and sometimes an idealized profile works best.
Here the examples from
On Station.