hhEb09'1, I don't seem to see any pictures of crepuscular and/or anticrepuscular rays that show the middle of the sky between the two points. I see diverging, and converging, but none going overhead such that they both originate and terminate farther away than they pass - they both converge and diverge. So yes, they look straight, but they don't satisfy the conditions that SeanF is discussing.

I have never argued that they cannot appear not straight. I've always said that the illusion does exist.

But are you then saying that they can also appear straight? Compared to a meter stick, say?I don't know. What would the definition of relative slope be? If you just use some sort of extrinsic slope, I don't see how that translates into the (perception) slope.I just looked at ten pages of google images thumbnails and I couldn't find any either. I guess middle ground crepuscular rays don't seem to catch the fancy of the photographers. That's another reason to be a scientist--everybody else misses so much.

But I'm pretty sure that a photograph taken of crepuscular rays so the center of the photo was between the sun and antisun would still show the rays as straight lines.

I think by relative slope he means that the altitude of a straight line (as measured using an azimuthal mount) increases and decreases as you pan along the line, and that the relative altitudes of two parallel lines, as measured with an azimuthal mount, can vary a great deal in different ways, depending on the orientation of the lines to the observer.

Hence, to follow the sun-moon line with a Dobsonian mount, you have to vary the altitude non-linearly (in a sinusoid) as you increase azimuth.

However, if you use an equatorial mount you can follow the sun-moon line with just one axis, right ascension, tracing out a straight line on the inside wall of a cylinder.

hhEb09'1 it all comes down to the fact that your brain is bolted to an equatorial mount whilst the rest of us are struggling with a Dobsonian.

clop

Clop, those all appear to be terrific characterizations!
And you started this thread confused? Great work!

Sean, your attempts to define terms appear to be much
more productive than I expected. I had no problem with getting
into minutiae -- I just thought your method couldn't succeed.
But you're making excellent progress! Keep at it!

You, too, hhEb09'1!

-- Jeff, in Minneapolis

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were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

But here you get stuck, I think, since we have no reason to believe that you can apply such mathematically valid definitions to perception. A necessary and sufficient reason for something to be perceived as a straight line is that some neurones fire in the usual place that signals a straight line. No amount of mathematical argument and definition is relevant to that perception.
Last week I spoke to a lady who was being followed around by the upper half of a yellow Confederate soldier. (No, this is true.) She was fully aware that here normal environment, and indeed the world in general, met all the definitional criteria of the statement "a partial, yellow, levitating Confederate soldier is absent". And yet she perceived one.
I do not suggest that hhEb09'1 is hallucinating: I just point out that neurones do what neurones do. And hhEb09'1 has much better grounds for perceiving straight lines than my unfortunate lady had for perceiving her Confederate soldier. If a great circle line were actually projected on the sky, it would form an image on the retina identical to the image of a straight line.
I'm sure the perception of convergence and divergence has some relevance to the fact that you and I see these lines as curved, but I'd maintain that you can't ever argue your way from the real world to the perceptual level in the way that you're attempting.

Grant Hutchison

My own interpretation of what I see (and this is at least a testable hypothesis if I had the resources of a planetarium) goes like this.

We have a constant perceptual awareness of the vertical (otherwise we'd fall over a lot).
We judge slope visually by comparing it to the vertical (or horizontal, which is effectively the same thing).
To be perceived as being of constant slope, a line must cross successive verticals at the same angle.
A great circle arc which is anything other than horizontal or vertical does not cross successive verticals at the same angle, so we have a sense of changing slope, or curvature.
The "straighter" line we think we can draw in the sky is actually a rhumb line or loxodrome: one that does cross successive verticals at the same angle.
By taking our feedback from the vertical in this way, we effectively turn our view of the world into a cylindrical projection laid flat, which is why the sinusoids generated by such a projection seem familiar to those of us who see the illusion.
Within a single field of view, a rhumb line can always be drawn that is very close to identical with the arc of any great circle, so we are generally undisturbed by perceived curvature unless we move our heads through a broad arc and synthesis a view of the world from that scan (which is the situation in which the illusion we're discussing becomes most salient).

But there are other perceptual clues in the world that do keep whispering "straight" when we look at great circles: the fact that they project on to the retina in precisely the way a straight line projects on to the retina, for instance. So our brains must be striking a balance between contradictory feedback.
And I would guess that hhEb09'1 simply strikes a different perceptual balance from the rest of us, for some reason, be it learned or innate.

Grant Hutchison

I did a small experiment yesterday evening. I didn't come to any conclusions, but I'll offer the data in case it inspires any of the poster's here.

At about 6:30 EDT, the Sun was low in the West, and the moon very high in the SE sky. The geometry was such that one could not look directly at the moon and see the Sun at the same time. I was on fairly flat ground.

I expected to see no illusion, partly because I feel that the illusion is more likely to be noticed with the moon lower, and partly because this thread had pre-conditioned me to expect a straight line.

I looked straight at the moon, and mentally traced the perpendicular across the sky. It passed to the right (north) of the Sun! I did it again, and again it passed to the right.

Then I held up my "meter stick" (actually a long strand (~ 18 inch) of wild grass that is straight when pulled taut), positioned it perpendicular to the terminator, and turned my head. The grass exactly intersected the Sun.

Finally, I turned so that the Sun was directly behind me (performed by staring at my shadow on my house) and tilted my head directly back. The moon was not overhead (clearly, since my location is New Jersey); it was off by many (10?, 20?) degrees, but it was obvious that the terminator-perpendicular was parallel with the direction my body was pointed.

So:
a) with the straight blade of grass: no illusion
b) orienting my body with repect to the sun, and looking at the moon to compare to body orientation: no illusion
c) mentally extending the line from terminator across the sky: ILLUSION!

The whole thing makes me wonder if when turning my head (and without a blade of grass or something else to keep my brain oriented), my brain lost its direction relative to the horizontal and therefore the line didn't intersect the Sun.

(In retrospect, I wish I had redone the experiment without the grass again, after having done it with the grass. Tonight it will rain, so I likely won't be able to re-do)

Yes. The point I'm trying to make is that I don't understand how you can "see" a divergence and reconvergence without "seeing" a curve, which is what you seem to be saying happens for you. A thought occurs - you have insisted, have you not, that the illusion does not exist for you?

I am, too. But then, I'm pretty sure the lines on that photograph would also be perfectly parallel - they would not have a greater separation in the middle than at the ends - so it doesn't seem like the picture exactly reproduces what is seen. (Again, it seems from your posts that you have acknowledged the divergence-reconvergence; if not in crepuscular rays specifically, at least in analogous situations).

I honestly don't know why not. If "looks like a circle" doesn't necessarily mean "looks like the set of all planar points equidistant from a given point," then it means nothing at all, and there's no point in discussing what anything "looks like."

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I'd agree (and this is actually my point) that there's no usefulness in trying to reason about what something looks like. It just looks the way it looks, to the person who is reporting that sensation. You can't, by some process of logic, talk hhEb09'1 out of seeing a straight line, any more than hhEb09'1 can talk me into seeing a straight line, any more than my unfortunate lady could talk herself out of seeing that pesky Confederate soldier.
These great circle arcs are identical to straight lines, when projected on to our retinas. So they should look straight. But they also converge and diverge. So they should look curved. Our brains just deal with that, and all we experience at the conscious level is the conclusion reached by some complicated processing. So I guess it really shouldn't surprise us that people can have different perceptions of the same things.

Grant Hutchison

But that's not what I'm saying.

If someone agrees with the statement, "Lines A and B look like they converge," but disagrees with the statement, "Lines A and B look like they're not parallel," there's something wrong.

His perception of whether or not they converge may be different than mine - and his perception of whether or not they're parallel may be different than mine - but his perception of whether or not they converge can not be different than his perception of whether or not they're non-parallel (keeping the same context of "perception," of course).

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Converging is parallel if you allow depth of view. The lines project away from you, so they remain the same distance apart while appearing to converge. But in the view of the piece of paper I would draw the perspective view on, those lines converge, and I have to draw them converging. So while I perceive them as parallel going into the distance, in fact they are on a flat plane and converging.

Perhaps. And perhaps a picture of overhead crepuscular/anti-crepuscular rays would not have a wide enough field of view without distortion to display the effects of perspective. I don't know. But without a picture, we don't know.

Most times, even an equatorial mount will not slew directly from sun to moon, but even then the path of an equatorial mount is not a straight line in space.I don't think so I just wanted to remind everyone that it is straight lines, not great circles, that I am looking at. But what does "look like a straight line" mean? Your definition seems to include comparing lines to another straight line (the horizon), but that doesn't help then--the question then becomes, why is that a straight line.

I disagree with that conclusion, though. Straight lines look like they converge and diverge, but there is no reason that straight lines should look curved.

Here's another attempt at some fun. At my office I have a wall that is about 100 ft long, about 8 ft tall. It is unobstructed as a walkway with a door at each end. I can stand about 10 ft away from it in a particular location between cubicles an see the full length of the wall.

If I look left, I see the floor line and ceiling line run away from me toward the left door. The lines are sloped toward each other. I perceive them as parallel lines receding into the distance. If I were to draw a perspective view of what I see, I would have to translate those two lines into non-parallel sloped lines on the paper, a greater-than sign.

Now if I look right, I see essentially the same thing but reversed, a less-than sign.

Now when I look straight ahead, I can see a variety of things, depending upon how I look.

1. Look straight ahead in normal, focused view. When I do, I perceive mostly about a 90 degree cone of view, i.e. I see the wall with two straight lines, one for the ceiling and one for the floor. I do not notice very well what happens outside about 45 degrees to each side, so the effects of perspective do not warp the ends of the view. Ergo, the lines are and appear parallel.

2. Now I open my perception farther, and rely on my peripheral vision. I have to consciously pay attention to the periphery, at the expense of some focus in the middle. When I do, I begin to see the lines stretch further away to each side. I can simultaneously see about 60 degrees to each side. If I look at the top line (the ceiling line), I see the top line as straight. I also can see the bottom line, but it looks bent. Not lightly curved, but actually bent. The left part goes upward as it projects left, the right part goes upward as it goes right, and the middle part directly in front of me has to merge the two lines. It isn't quite an angle, and if I look at it at all the image fades as my focal point shifts down toward it. But if I maintain my focal point on the ceiling line and use periphery to see the floor line, it is curved within the 20 degrees or so right in front of me so the lines can project away to each side.

3. I can swap the effect from the floor line to the ceiling line by placing my focal point on the floor line and using peripheral vision on the ceiling line.

4. If I place my focal point directly between the floor and ceiling line and then investigate both with peripheral vision, I see both lines as slightly curved. This is stronger for the floor line because it is farther away (given my height), and the curve is subtle, but it is there visually. Yet we know the lines are actually parallel.

When you say "Straight lines look like they converge and diverge," do you mean, "Straight lines look like they converge and diverge - simultaneously, but not, obviously, at the same apparent place - within the same field of vision"?

If so, how is that not the same thing as saying they look "curved"? Isn't converging and diverging at different places pretty much the definition of "curved"?

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That's interesting, I'll add it to the list.

When I just now tried this with a 4 x 24 foot whiteboard (actually, two 12s), the perception changed depending upon whether I had my glasses on or not. That's kinda to be expected with trifocals though.

Map the 3-D view to that 2-D flat plane of paper in the perspective view required for art. The reality is the lines are straight and parallel. The projection on the paper shows them converging at each end and apart in the middle. How do you draw on paper something that changes angular separation?
What is in the middle? Either a sharp bend, or a series of smaller bends. String out an appropriate number of smaller bends and you get a curve.

No, he perceives whatever he perceives: you find these to be logically inconsistent, but perception doesn't have to be logically consistent.

Grant Hutchison

If the view on the paper shows both ends, the view must be greater than 180 degrees right?Why would looking at a straight line and seeing a straight line be logically inconsistent?

That wasn't my conclusion. Lines that diverge and converge are generally curved. SeanF's position appears to be that "diverge + converge" is a sufficient condition for us to perceive curvature. Like you, I disagree with that, since there exists a category of lines (straight lines of effectively infinite extent, which correspond to great circles when mapped to our sphere of view) which are exceptions to that general rule.
However, "diverge + converge" is a very powerful hint to our nervous systems that lines are curve, especially since our brains evolved in a world entirely free from straight lines of apparently infinite extent. To that extent, such lines should look curved. However, other things about them (such as their straightness!) means they should look straight.

I confess I'm becoming increasingly baffled that you seem to be denying so absolutely the power of "diverge + converge" as a generally useful marker for curvature, as well as SeanF's apparent reluctance to accept that lines that are straight in the field of view might appear straight to some people.
I can see no potential for you two ever reaching any agreement, at present.

Grant Hutchison

That was the conclusion that I disagreed withMaybe I look at things too simplistically That's not true. I have never argued against the illusion, nor its power.

Now I'm struggling. Do you believe that divergence + reconvergence is a visual cue for curvature, or that it isn't? You seem to be asserting both.
(Red text reinserted in quote, for clarity of context.)

Grant Hutchison

But the language must be. What I am seeing - and objecting to - is an acceptance of a definition of a curved line as a description of the perception but a denial of the term "a curved line" as a description of the same perception.

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Don't ask me. Ask SeanF.

Grant Hutchison

Ah, I see.

I think there has been plenty of evidence presented that is indeed, I just disagree that it necessarily should be. Rather than "To that extent, such lines should look curved," I would say, "To that extent, such lines may look curved."
My acceptance?

PS: I just searched back, and as near as I can tell, this post of mine is where our discussion of the definition.

You have two problems here, I think.
1) Your definition is not sufficient to specify a curved line, since straight lines do diverge and converge in our view of the real world.
2) Even if this were not the case, there is no requirement for perception to be internally consistent in the way you describe.

Grant Hutchison

We have a linguistic divergence again, too tedious to go into.

So you accept that divergence and reconvergence are powerful cues for a perception of curvature? (I belabour this point because, deliberately or otherwise, you do seem never to have given a definite answer to repeated queries.)

Grant Hutchison

Yes. But you can show converging toward each other at both ends without showing the end points. The converging point is off the paper, but the effects of the convergence are still evident. That is the same situation I get when I look at my long wall. The points of convergence are out of my field of view to each side, but I still see the effects.

grant hutchison, I think you and SeanF are still stuck on the word "perceive". Perceive occurs in the mind. Perceive is the mental construction the mind makes. Perception has many factors, some of which we don't know how they work, only that they do. The fact is that the brain is able to perceive some pattern of light on the retina and convert that via nerve signals into a model that includes depth perception, parallel lines that converge as they go away from you, and other things such as pareidolia. That is the brain processing the signal it receives.

There are at least three levels being discussed.
1. Reality - a 3-D world.
2. Perception - the brain construct of the images received onto the retina and converted to nerve impulses and relayed to the cerebral cortex, then translated into some symbolic thing we call vision.
3. _______ - the 2-D projection (flat paper, cylindrical curve, whatever) that shows what the eye sees prior to any processing by the brain. This has to strip out the clues that the brain interprets and show them how they can be represented on that 2-D projection to convey those clues to the brain. This is the level where parallel lines converge. Reality, no. Brain, no. On paper, yes - you draw them converging. I don't know what to call this. Pre-processed Image? Visual Projection? Thingamabob?

You and SeanF seem to be mixing up level 2 and level 3 with your terminology. SeanF is saying that at level 3, diverging + converging = curve, regardless of what level 2 does with that signal. You are saying that at level 2 the brain can see it as straight or curved. You're not disagreeing on what the brain is doing, but on the level of image you are talking about.

I can't pin down hhEb09'1. Whenever I talk about level 3 he talks about level 2. When I talk about 2, he talks about 1.

Excellent example. "Straight lines look like they converge and diverge" (in level 3), "but there is no reason that straight lines should look curved" (in level 2).

Level 1 is straight, parallel lines receding to the distance in both directions.
Level 2 is straight or curved, your brain's choice.
Level 3 is the intermediate step. At 3, two lines diverge, then smoothly change directions so they converge. Whatever the brain wants to do with that at 2, at level 3 that can only be described as curved lines.

I've given my direct answers, but as you say we have linguistic diverence. They are not powerful cues for me, and I've given my reasons why. That doesn't mean that they can't be powerful cues for someone else.