Okay, I finished watching the video. I kept waiting with anticipation for math to be done under the dimensions of the Round Earth System, as advertised. However, that did not happen. The video mostly consisted of you reading the equations from the paper.

The honorable thing to do is admit you do not understand the math and concede the point.

The dimensions of the Round Earth System appear nowhere in it. In fact, the author says in the work that they avoided using the RET numbers.

From p. 9:

The value of the angle α is the same for the vectors m, s and z or their corresponding unit vectors, which are used in Eq. (11) **to avoid having to know the actual distances of the moon and the sun from the observer.**

The dimensions for the Round Earth System are nowhere in the math by the authors of that paper, nor is it in your video when describing the matter.

You want the dimensions of the moon and sun to be used? Here they are. You cannot deny it anymore:

m = <Dm*cos(ELm)*cos(AZm), Dm*cos(ELm)*sin(AZm), Dm*sin(ELm)>

s = <Ds*cos(ELs)*cos(AZs), Ds*cos(ELs)*sin(AZs), Ds*sin(ELs)>

where:

Dm is distance from earth to moon = 238,900 miles

Ds is distance from earth to sun = 92,960,000 miles

For my example, ELm=20.89, AZm=136.35, ELs=-0.24, AZs=294.62

m = <-160890, 154062, 85186>

s = <38726600, -84508300, -389400>

s cross m = <-7138932301000, -3236313581600, -7630242937800>

p = m cross (s cross m) = <-899841878721166000, -1835766873255628000, 1620528680300286000>

h = m cross z = <154062, 160890, 0>

p dot h = -433987971757638265212000

|p| = 2608805976556893954

|h| = 222757

(p dot h) / (|p|*|h|) = -0.74680042

taking the abs as per the convention in the paper to get the angle we want...

alpha = arccos(0.74680042) = 41.686 degrees!

The fraction of a degree we get in variance is due to different amounts of precision used in the calculations (round-off error).

So for the last time, stop going on and on about distances. We can easily prove that the distances cancel out of the equation anyway. If you can't follow the math, there are the numbers using the distances to the moon and sun. Stop whining about it already.

Secondly, it is apparent to all that the author needs to project images onto a plane close above the observer's head in order to attempt to describe this. It is *entirely apparent* that the authors cannot "really" explain it.

We may as well just say that the moon and sun are a close distance above the observer's head, as to entertain that.

What is apparent to all is that you simply do not understand the math. You went on and on shouting "DO THE MATH." Well I did the math. It isn't my fault you don't understand it. I'm happy to explain any individual step you can't understand.

This isn't that hard to understand really. A camera takes a snapshot of the 2D image it sees in front of it. There is a 3D world in front of the camera, but the image is 2D. We call this capture of a 3D scene onto a 2D image a "projection". That's what we have done here. That's how you explain this phenomenon. What is happening is a 3D direction (the light that hits the moon coming from the direction of the sun) is resolved by your eyes into a 2D image. Your brain judges the direction of that light based on the 2D image.

Thirdly, I wanted to point out that at the 42 minute mark you claim that the distance from the earth to the sun doesn't matter, and the moon will point in the same direction regardless.

Will a green arrow that points at the sun, located at the height of the moon, as seen from earth, point in the same direction regardless of whether the sun was one foot away from the earth or if it were 100,000,000 miles away? Clearly *not*.

This is your distances complaint AGAIN! I've addressed it several times now. Just above here, I worked the math with all the distances included.

I have shown mathematically that those distances cancel.

I showed in the 3D simulation that those distances do not matter.

Now I've worked the math with the distances included, and guess what? I got the right answer AGAIN!

As long as you can create the proper angles, the distances are not part of this illusion. That's why you can hold a ping-pong ball at arm's length and get the same lighting as the moon has on it.

All-in-all the marks for the "Round Earth explanation" are poor, and I intend to point these things out when I get around to making the Wiki article on the subject.

The only thing poor here is your ability to admit it when you are wrong.