More Moon illusion discussion:
On the Art of Seeing
Mike writes:
"What is actually happening is that the viewer's response is DIFFERENT than that which OUGHT to follow from the rules of perspective. If you draw two moons the same size, they ought to be perceived by the viewer as having the same angular size. If you draw a smaller moon in the background, the perspective lines you used in composing the drawing are intended to ensure that the angular size falls off with distance toward the horizon in a linear fashion."
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So far, so good.
"If in fact (after you've erased the perspective lines) the viewer "sees" the background moon AS IF it were too big to be contained within the perspective lines, you have an illusion."
Mike: THAT CAN NEVER HAPPEN, EVER! If they already look like they fit between the red perspective lines, they will remain looking so after removing those lines. To do that illusion, you'd have to invent and add some kind of additional image to the drawing (unknown by anyone what that would be, if anything even can be invented other than by Escher, if even he could). You might as well say "suppose I've poured 2 cups of water into one cup." The supposition is impossible (except for people who will "see:" what they want to see). In this debate, you tend to come up with such things with no obligation to provide serious evidence for them before you incorporate them into your unshakable views. But if you send such a picture, I will change my mind.
But I'll go along with you ANYWAY, for now.
In summary: We don't see anything "out there." All we see is what our inside mind tells us we see. We don't even see what's on the retina. The mind first measures, compares and then "interprets" what is on the retina. Instead of letting us see exactly what's on the retina, it "tells" us to see what it has interpreted from the retina's information. The mind "assigns" sizes and distances to things, based on
* past experience,
* knowledge/learning, and
* from perspective cues on the retina.
Not *just* the latter. If the cues are contradictory, then the mind is tricked or is forced to come up with an interpretation that is an "illusion" despite  what the retina has on it. That's the way we see.
Seeing is not a perfect process. But it's good enough to keep us from tripping over stuff and from realizing that we are -- except for "interpretations of the mind," -- *really* in the dark and only looking at the insides of our own head.
The less we know about something or its actual distance (how can we "see" or visually measure the moon 240,ooo miles away?) -- then the more subject to an illusion (or mistaken interpretation) we are.
On the other hand, the more we know about objects we see, the more that knowledge plays a role in the mind's interpretations of distance and size ("believing is seeing" rather than just "seeing is believing"), and the more accurate our "eye" becomes about measuring reality.
Seeing is a learned [and ongoing learning] process, rather than just biological (i.e., lenses, irises, retinas, etc). The pics will show this.
So -- on with my reply to this in general.
You don't seem to know the meaning of the word "illusion" in perspective. Perspective cues always  play "tricks" on the eye. Some are more extreme or dramatic or based on clues impossible to find in reality (Escher) that the eye cannot deal with without causing an effect that defies, for example, the actual subtended diameters in comparison with another actually identical image, or that shifts conclusions of depth, distance, size so they cannot apply to all parts of the drawing consistently (because, in Escher, for example, they are defying physical law, "object permanence," or the like). But where they apply -- they DO apply consistently. Always. Except for figure and ground shifts (vase/faces) -- which are meant to apply in the same place).
You attach unspoken or mystical attributes to the word "illusion" that it doesn't have.
Now here's my proofs:
If I want to draw three identical streetlight globes screwed to a wall, I will draw this picture "A" below. [If you wish to say that since this is "regular" everyday "ordinary" perspective, and that therefore there is no illusion, that is okay with me, but actually it's dead wrong. There is always an illusion. But let it go for the purposes here. We'll say there is no illusion, throughout this exercise.]
Now I have three equally-sized globes fastened to the wall with 3 round identical fasteners, all shown in perspective.
Now I want to draw another pic, but this time I want to make the middle globe look bigger than the outer two globes.
All I have to do is make the middle globe larger. So I draw it larger -- and ANY size larger that I draw it will do -- no matter WHAT the size I draw it, it will look (in normal interpretation of a perspective picture) "larger" than the other two.
So I now draw this:
Compare back & forth pic A and pic B. The middle globe is bigger than before.
Or -- I could draw yet another still larger-size globe in the middle :
Again, switch back and forth from pic B to C or from pic A to C.
In any of these cases, the middle globe has been made larger than it was originally in pic "A" where it was depicted in correct perspective as being the same size as the other two globes. So in pic B and C, by perspective interpretation of the drawings, the middle globe looks larger than the other 2 globes there.
Now here's the rub. REMEMBER, we are saying that all three pics are drawn in standard everyday normal perspective where (as you prefer) there is no "moon" illusion nor other "optical illusion."
Nothing bizarre, alien or new in some mysterious way that might trigger an electrifying visual illusion or effect has been done to this everyday ordinary perpective exercise except to change the size of the middle globe.
The fact that in picture "C" the middle globe just HAPPENS TO BE THE SAME SUBTENDED DIAMETER OF THE FIRST GLOBE is --- IRRELEVANT. It's just another  increase in the size of one object in the perspective field and the perspective clues have no reason not to work normally. It could have been ANY increase in size.
Unless you wish to claim that making it the same diameter as the first globe does magically change how perspective clues work? But if they work at all for the observer in a normal way it will be seen as described -- by an observer who will not concentrate on trying to measure the flat screen's absolute diameter sizes and wash away the cues surrounding it like we try to wash out the sound of a motor or fan in the room when we try to read, etc.
But now, you could give up the wrong notion that perspective clues do not create illusions. If you do, then there IS an illusion in all three pictures. In the first, by illusion, the 3 globes all look equal, even tho' each is smaller from left to right..
In pic B, the middle globe looks larger than the others, but due to lack of enough distance clues, it is a little ambiguous whether it is clearly "looking" larger than the first globe. The round rivet holding the middle globe to the wall tends to define its size better.
Adding our "man" image -- in perspectively smaller sizes standing next to each globe would completely end the ambiguity. But I won't bother to draw another picture to prove that. You do it.
There is in pic C no doubt  that a normal interpretation of the perspective cues there makes the middle globe largest of the three without ambiguity.That the first and middle globe have the same actual diameters is not grounds to believe there is any "special" illusion here, as it looks bigger than the 3rd furthest globe for the same reasons it looks bigger than both of the other 2 globes: Namely, the "illusion" of perspective interpretation of distance clues.
There is no reason in the picture to presume that any special optical illusion other than perspective itself takes place. The identical retinal diameters of two globes does not provide that reason, unless one reaches for some mystical unknown process that equal subtended diameters in a picture can somehow activate and create a mistakebn interpretation. I won't buy it. I can prove it otherwise, anyway.
If there was any reason to conclude an optical illusion conflict of an Escher type, it would only be due to failure of the perspective information to unambiguously nail down the distance of any object from the others. And that's all. That aspect of insufficient information exists all the time in nature and reality; hence we often mistake the size and distance of objects as a matter of course. The ambiguity in pic B could also exist in nature.
A lack of distance information, misleading information, or a lack of perpective clues is not necessarily an optical illusion. Ambiguity does create mistakes in perception. If you want to call that "illusory," I don't really object much. Often other information (prior knowledge, bias, sound, or expectation) can prevent the ambiguity from producing a mistake. (E.g., headlight or flashlight?)
Learn how to find the lacking distance info, fix that, and the "mistaken" interpretation or "illusion" disappears [or appears]. -- For those not seeing only what they want to see.
with no new pictures:
An exact formula that proves that there is no "special" illusion involved in the moon or any other size-illusion, or any other illusion of the S.A.Y.E. (Subtended Angle at Your Eye), actual or apparent.
When the mind interprets the 2-d retina image, it's an exercise in true Einsteinian relativity. Everything we see is measured and related to everything else around it. It is mutually defining (of sizes and distances): Things are judged as further or nearer -- behind, next to, or in front of -- bigger, the same, or smaller in size -- than each of all the other things seen. The mind then interprets or assigns the appropriate sizes and look to things as best as the distance information will permit.
Therefore, in picture "C," in which the illusion you felt was very strong, the illusion is as follows. (Each starred * sentence is like an exact formula or axiomatic statement):
1* An object that looked bigger (middle globe) actually subtended no greater diameter than the smaller-looking object (the first nearest-looking globe). I.e., It only APPEARED to be bigger (due to cues that locate its perspective distance). Relatively speaking, we MUST and can ALSO SAY:
2* The object that looked smaller (the nearest looking-globe) actually subtended no SMALLER diameter than the bigger-looking object (middle globe). I.e., It only APPEARED to be smaller. Therefore, the "illusion" could be understood to apply to EITHER of the globes that have the actually same diameters (S.A.Y.E.). Thus the illusion could involve either of TWO objects or globes. Which one "owns" or causes the illusion? Which one is producing the mistaken or "apparent" SAYE? Is the middle globe that looks "larger" the mistaken one? Or is the frist (nearest) globe that looks smaller being seen as mistakenly smaller? How do you find out? Or -- maybe the question itself is a false question, based on a wrong assumption that the illusion is "special" or caused from a special occasion or circumstance or "ownable" by only one object or one comparison? Neither the smaller or the larger SAYE is the mistaken or "apparent" SAYE, until you answer this: Would there also be other such "illusions" when the actual dimensions or actual diameters of this object (or any other object in the picture) is compared to ANY other objects' diameters or dimensions in the picture? The answer is: Yes: An alteration of apparent SAYE takes place in every comparison -- IF distance cues are DIFFERENT for the two compared objects. It is irrelevant whether the actual (note, actual) SAYEs are equal to each other, or just a fraction of each other.
3* By natural logical extension, EVERY other globe or object's dimension in the picture is similarly involved in the same KIND of illusion, whether the ACTUAL S.A.Y.E. of the object is equal to, or just a fraction/multiple of dimensions found in these other other objects. Thus, the 3rd furthest-looking globe is similarly involved in an illusion: If its apparent SAYE must be drawn, to meet the demands of perspective, at 1/3 the diameter of the first globe [in order to pictorially "appear" to be a "same-sized" globe as the first globe], then the first two formulii apply to it (and to all other objects), except we substitute different dimensional relationships. To wit, for the third globe we'd say:
            1a* --An object that looked bigger (middle globe) actually subtended no greater than 3 times the diameter than the smaller-looking object (the third furthest-looking globe). I.e., It only APPEARED to be more than 3 times bigger.
            2a* --The object that looked smallest (the furthest-looking-globe) actually subtended no less than 1/3rd the diameter than the bigger-looking object (middle globe). I.e., It only APPEARED to be less than 1/3rd the diameter. Therefore, and so on:
4* Whatever the absolute or actual SAYE of any object, it is ALTERED by the mind's judgement of size to suit its relationship to other objects -- as long as those objects are depicted at a different "distance" from it -- hence the altered or "apparent" S.A.Y.E.s ARE usually different from the actual SAYEs.
5* Only those compared objects depicted as being apparently the SAME distance from the eye will be unaltered and show no mistake or illusion or difference between apparent and actual SAYE.
6* In ALL cases, if you remove ALL the distance clues (i.e., no horizon, convergences, nor any other form of perspective grid) , the SAYE of every object is seen and interpreted both by the eye, and the mind, with its ACTUAL S.A.Y.E. No illusions can take place without distance cues (intended cues or accidental ones). In any perspective scene, distance cues serve as a form of visual "knowledge" which alters our perception [or judgment] of the size (SAYE) of any object -- only when compared to another object (at a different depicted distance).
7* If you then replace ALL the previous distance/perspective clues, then every object gets re-interpreted, as to its relative size, with an altered or "apparent" S.A.Y.E. (except as in #5*) different from the actual SAYE depending on the influence exerted by the perspective/distance cues and requirements. Therefore, the "perspective experience" is an accumulation of dozens or hundreds of so-called "special" illusions in every picture or real scene. It's just arbitrary convenience to pick as "special" only those illusions where 2 of the objects, affected by different surrounding distance clues, when compared, "happen" to have the "same" actual SAYE (rather than being some fraction or multiple of each other's size).
Pictures can be drawn to prove each axiom or formula.
But I haven't drawn them yet.
Picture C is re-attached.

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