I've been returning to satellite tracking software after a while (I wrote some early AMSAT tools in the early 1980s) and am wondering if there has ever been a resolution of the exact definitions of "apogee height" and "perigee height".
The simple geometric definitions of "perigee" and "apogee" are the points where the spacecraft is the closest to or farthest from the center of the earth. This is easy if you assume the earth is a perfect sphere; all you need is the semimajor axis a and the eccentricity e:
apogee = a * (1+e) - earth_radius perigee = a * (1-e) - earth_radius
But reality is more complicated than that. For a nonequatorial orbit the apogee and perigee usually occur over some point off the equator where the earth's radius is smaller than at the equator. You can correct for this given the inclination and the argument of perigee, which together tell you the latitude at which apogee and perigee occur; one will occur in the northern hemisphere and the other will occur in the southern hemisphere at the opposite latitude.
There's a complication here in that this is geocentric latitude, while we more often use geodetic latitude on a daily basis. Converting geodetic latitude to geocentric is fairly easy, but converting in the other direction is like Kepler's equation: apparently there's no closed form solution so you have to iterate.
But this is a relatively minor detail. The real problem comes when you have a satellite with a relatively high inclination and an argument of perigee close to 0 or 180 degrees; in this situation the satellite can easily be farther from the earth's surface than *either* the calculated apogee or perigee!
The ISS is a case in point at the moment. Using element set 906, which has an argument of perigee of about 329 degrees I calculate an apogee of 408 km and a perigee of 402.4 km assuming an oblate earth and ignoring the distinction between geodetic and geocentric latitude (which is relatively small for this argument of perigee). But near the southernmost point of its orbit, I calculate an altitude of about 421 km, well above both the perigee and apogee heights because the earth's surface through the poles is more elliptical than the ISS's orbit.
So what conventions do people use? How meaningful do people expect these figures to be? For a low altitude orbit like the ISS, the difference in drag between 402 and 421 km is actually quite significant. At the very least it would be nice if everybody used the same convention.