What is the relationship between crescent moon and position of the sun? This is a phenomenon that involves the angles between the points of a crescent moon and the apparent  position of the sun.  Step outdoors on a day and time when you can see the sun and a crescent phase  moon.  Mentally construct a line segment AB between the two points of the moon’s crescent (see diagram).  Now construct a straight line which is at 90 degrees to segment AB and which bisects its midpoint (Line 1 in diagram).
Based on what I know about astronomy and geometry, Line 1 should point directly to the center of the sun (i.e. the sun should be in position of SUN 1).  This should be true regardless of the time of day, time of month, or viewing angle from earth.
However, when I actually perform this experiment, the straight line bisecting the midpoint of AB typically appears to lead to a point higher than the actual position of the sun.  In other words, the sun appears to be in the position of SUN 2 and one would have to imagine a curved line (Line 2) to lead directly to the sun. I’m having a difficult time explaining this observation.
Possibilities:

*

*I’m simply wrong in my observation, possibly due to the difficulty of looking near the sun with the naked eye. Concerning this possibility, I would be interested to know if other people observe this phenomenon the way I appear to.

*This is some sort of optical illusion created by an unknown effect, or

*There is a valid astronomical and geometrical explanation for this observation.
Comments?


 A: 
I would be interested to know if other people observe this phenomenon
the way I appear to.

Yes - there are questions about it on this site, and people have blogged about it (this might be of interest). I've noticed this effect myself.

This is some sort of optical illusion created by an unknown effect

I think it is an optical effect (has to do with vision), and while it is, in a way, a perceptual distortion of reality, I wouldn't say it's an illusion in the sense that what you're seeing is not due to atmospheric effects and such, and it's only partially attributable to psychology of human perception.

There is a valid astronomical and geometrical explanation for this observation. Comments?

The explanation that seems the most plausible to me is that it has to do with perspective projection (and thus the way our eyes work). Here's the argument. To us, lines that are parallel in 3D space appear to converge to a vanishing point (imagine a photo of railroad tracks receding into distance). But a set of parallel lines extends indefinitely in both directions; as you turn your head, they appear to converge in the other direction, too. So there are two vanishing points, and you can't see both of them at the same time. As you turn your head, the angle of the parallel lines has to change so that the sense in which they converge eventually flips.
In standard vanishing-point perspective projection, where the image is projected on a flat plane, as the eye/camera changes direction, the lines remain straight, but reorient themselves.
However, in the human eye, the image is projected onto the curved surface of the retina. This is (arguably) better represented by things like curvilinear perspective, or perhaps stereographic projection. Here, if you have a wide enough field of view, straight lines actually curve.
Psychologically, we're generally not very aware of this curvature (and our field of view is not extremely wide). But if you find a long corridor, face the wall, and focus on your peripheral vision, you'll notice that parallel lines formed by the floor/wall and wall/ceiling joints converge on both sides in the opposite senses, so there must be curvature for them to connect in the middle. This effect is more pronounced the bigger the object is.

Since the Sun is so far away, the Moon-Sun line & the eye-Sun line are nearly parallel  (in 3D space), and under perspective projection, their vanishing point is essentially at the center of the Sun's disk. Tracing, visually, the path from the Moon towards the Sun is like looking from one side of the railroad tracks to the other; the perspective-projected direction of the line perpendicular to the shadow terminator plane when looking at the Moon is different from the perspective-projected direction of the same line when you reorient yourself to face the Sun.
If there was a straight, rigid rod connecting the Moon and the Sun that you could see, perhaps your subjective impression of it would be that it appears curved (especially towards the periphery of your vision) - unless you tilted your head so that your local horizon is in the Sun-Moon-eye plane. Or perhaps your brain would "post-process" and interpret the image, so that you wouldn't have a strong sense of the curvature. In any case, without such a line to serve as a reference, the orientation of the Moon appears off.
A: Here's an example of someone else who saw this effect.
Simply put, the Moon and the Earth are quite close together compared to the Sun. So, the Sun appears in a similar place on the sky, whether you are standing on the Earth or the Moon. Our vector to the Sun and the Moon's vector to the Sun are almost parallel.
In this photo, at near Sunset (or is it sunrise?), you can see that the Sun appears to be almost horizontally level with the horizon, and a bit to the right.

You can imagine standing on top of the Moon--the Sun would also appear to be almost horizontally level with the horizon, and a bit to the right, causing the crescent of the moon to point a bit to the right. The crescent of the moon is almost straight up/down because the Sun is horizontal with us, and therefore also horizontal with an imagined observer standing on top of the Moon from our perspective.
The lunar terminator of the moon need not point towards the Sun as it is only the edge of the lit-up moon. The light side of the Moon should be pointing towards the Sun. If you imagine how the overall
light side of the Moon is pointing based on its terminator, you'll find that it makes sense.
During the new moon and full moon, you would be least confused, because you understand intuitively that it's the lit face of the Moon that matters, not the edge. In the new moon, you know the lit face of the Moon faces the Sun. In the full Moon, you know the lit face of the Moon faces the Sun, which is towards the Earth, because the Moon-Earth-Sun make a line.
A: Here is a thought experiment that illustrates a principle that answers the above question.
Imagine there is a meteor in space following a "straight line" (of course we have to pretend it is not influenced by gravity and that the "curvature of space" is unimportant on the scale we are considering).  This meteor, visible in the daytime, is flying in a plane determined by the earth's equator and will pass 100 miles above the earth's surface at its closest point.  You are standing on the equator.  Here is what you would see:  You would note the meteor becoming visible at the horizon (due east or due west depending on which way the meteor is travelling).  It will travel "straight up" to your zenith and then follow a "straight" line down the horizon 180 degrees opposite from where it rose.
Now imagine you are 100 miles north of the equator.  What do you see? You see the meteor still appearing at the horizon due east or west, but it will then slowly follow a rising arc until it is 45 degrees above to the horizon as you look south.  It will then slowly descend to the opposite horizon following a curved arc.  The path not only "looks" curved, but if you took a time lapsed photography of the meteor's passage, you would also see the curve--so it is not just an optical illusion.
This thought experiment reveals that an object or beam of light traveling in a straight line in space will actually trace out a curve to our visual system. So, in the diagram above, the sun may appear in the SUN 2 position because the "straight line" from the lit surface of the moon to the sun is actually Line 2 (remember. . . straight lines in space appear curved to us).
