File:Lunar distance over anomalistic month.png

English: Lunar distance when perigee is at syzygy (full moon or new moon, in blue) or at half moon (red). The horizontal lines (extending exactly half a mean anomalistic month to each side of perigee) are the respective averages over one mean anomalistic month, and are almost identical (384,652 km versus 385,349 km).

The curves are calculated using the first four terms of the formula of Chapront and Touzé:

${\displaystyle {\begin{alignedat}{3}{\frac {d}{\mathrm {km} }}=385000.5584&\ -\ 20905.3550\cdot \cos(G_{M})\\&\ -\ 3699.1109\cdot \cos(2D-G_{M})\\&\ -\ 2955.9676\cdot \cos(2D)\\&\ -\ 569.9251\cdot \cos(2G_{M})\\&\ \pm \ \dotsc \end{alignedat}}}$

where ${\displaystyle G_{M}}$ is the mean anomaly (angle from perigee) and ${\displaystyle D}$ is the mean elongation (angle between the sun and the moon). The average of the two curves is what the distance would be without the influence of the sun (containing just the first and last cosine term of the above).

One could make similar curves for when apogee occurs at syzygy or at half moon.

Generally, neither perigee nor apogee occurs exactly at syzygy or half moon, but given enough time the situation can get arbitrarily close to these cases. The situation then almost repeats 14 synodic months later, and repeats even more closely 251 synodic months later (a "saros" and 28 months).