Lunar Terminator Illusion

When the moon and the sun are both visible in the sky, but not close together, the sunlight often appears to illuminate the moon from a high angle, even when the sun is closer to the horizon than the moon.

As seen in an orthogonal projection, the lunar terminator (the boundary of the illuminated hemisphere of the moon) appears from Earth as a half-ellipse. The major axis of the ellipse is perpendicular to the sun's direction from the moon: the sun lies on an extension of the minor axis.

The illusion occurs when the moon and sun are separated by a wide angle, so that they are perceived relative to the horizon, as if in a panorama. A panoramic photograph is a cylindrical projection. In this projection, most straight lines project as sinusoidal curves. The moon-sun line is curved, unless the moon and sun are on the horizon or directly above one another.

This curve can be seen in the figure below, which shows a cylindrical projection of the sky covering 60 of altitude and 180 of azimuth. Below it is an isometric drawing showing how the moon and the sun project on to the cylinder from the viewpoint.

If you mouse over the figure, it animates to show the radius of the cylinder increasing until it becomes a plane. The projection then becomes a rectilinear one, in which all straight lines remain straight. As the cylinders are tangent to the plane at the moon's position, the angle of the terminator remains constant throughout the animation.

The rectilinear projection is like a wide-angle photograph. Angular displacements are progressively magnified away from the optical centre, as revealed by the grid lines. The angle of the terminator is thus slightly different than it would appear if looking directly up at the moon.

Christopher B. Jones, Sydney, January 2014