Which way does the moon point?

After sunset the sun is below the horizon — so the moon’s lit side should point down at it. Often it points up instead. Drag the sliders and watch where it flips.

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The condition. Put yourself at the origin. Let m be the unit vector to the moon and s the unit vector to the sun. The sun is effectively infinitely far away, so the same hemisphere of the moon is lit wherever you stand; what changes is your angle on it. The lit limb points, on the sky, along the sun direction projected into the plane perpendicular to the moon:

p = s − (s·m)m

For the case where the sun sits directly opposite the moon in azimuth — moon at altitude β, sun at depression α — that reduces to p_z = −sin α + cos(β−α)·sin β, which is exactly zero when β = α, positive above and negative below. So:

the lit side points upward whenever the moon is higher above the horizon than the sun is below it.

And the boundary turns out to be a familiar object. The anti-solar point — the spot directly opposite the sun, where a full moon sits — is at azimuth 180° from the sun and altitude +α when the sun is at depression α. So saying “the moon is higher than the sun is deep” is saying the moon is above the anti-solar point. Which means:

• moon below the anti-solar point → you see it from slightly above → the dark sliver is on top, the lit side points down;
• moon above it → you see it from below → the dark sliver is at the bottom, the lit side points up;
• moon exactly at it → the moon is full, there is no terminator at all, and nothing to point. Try the boundary preset: the readout says undefined rather than flat.

That is why the effect is strongest near full moon and vanishes at half — the paradox is a question about which side of the anti-solar point you are looking from, and near half moon the anti-solar point is nowhere near the moon. Nothing about the moon changed. Only where you are standing relative to a point in the sky.

Two honest notes. The β = α rule is for the coplanar case only — with the azimuth gap away from 180° the boundary moves, which you can find by dragging, but I have not solved that in closed form. And I built this expecting the boundary to be “the limb lies flat”; it is not, and the code told me so. At β = α the projected sun direction collapses to zero length, which is the arithmetic announcing that the moon is full. I had to go back and change what I thought the threshold was.

Sparked by Dima Kogan’s lunar terminator paradox note, which explains the mechanism — you are looking at the moon from below, so you see a slice of its dark underside. He wrote a program because the prose explanations did not land for him; this is the same instinct, with the threshold marked. The geometry here is mine, checked numerically against the closed form before it was drawn.