[zorbakim]

The "side-perspective" that produces a slope is a plot of these perceived dimensions. It isn't a real world side view showing a light path.

My issue is that plotting apparent decreasing heights vs. actual/constant interval distances isn't showing curving light. It's merely showing a relationship of perceived height to actual distance. You can do the same thing with perceive distances, which are also influenced by perspective. As things get smaller in the distance, they also appear to get squeezed together. When you plot it like that, the curve becomes straight.

My point is that perspective works like this whether on a flat surface topography or a convex surface. Perspective doesn't cause the appearance of curve when the surface is actually straight. This should be apparent by the two renderings of perspective above: one on a flat surface and one on a curved surface. The "side perspective" of the flat surface model will also plot as a curve if you plot apparent changing height against a constant interval distance. But the bars don't look like they are standing on the surface that curves away even through their 'side-perspective" plot is curved.

I can appreciate what your are trying to say, but I'm saying you're conflating (cross mixing) parameters without consistency and drawing an incorrect conclusion. Perspective can't cause the appearance of curve in the perception of depth.

We seem to have misunderstood each other.
I measured the apparent height in the picture.(y-axis)
but didn't measure the interval distance in the picture(z-axis).
I measure the interval distance in the real world.
I combined the two.
It lets us know how the depth of reality is shown in pictures.

[RonJ]

At sea the earth appears to be perfectly flat.  However you see the spherical nature of things just every so often.  A lot depends on the weather.  After spending half my life (literally) at sea for the last 20 years, the most spectacular visual representation of the spherical earth is the view of Mt Fuji in Japan coming out of the open ocean.  Mt. Fuji is about 12000 feet tall. The top is over two miles above sea level.  That means you can see Mt. Fuji 100 miles out at sea.  At that distance the nearest shoreline is about 40 miles away.  The mountain just appears to grow out of the sea with no other land visible.  Now it takes an unusually clear day for you to see that but sometimes we would get lucky in the winter if it was especially cold.  After a couple of weeks at sea with no other view except the ocean a nice sight like that can really make your day.

[edby]

What I have tried is how the three-dimensional world looks in the two-dimensional visual image.

Right, but what is the shape of the 3-D world? My take is that if the tiles are squares, i.e. each of the four internal angles is 90d, and if we put all the tiles together, this will form a much larger square, with each of the four internal angles also equal to 90d.

Is this your point? But then which part of the large square is curved?

[Bobby Shafto]

We seem to have misunderstood each other.
I measured the apparent height in the picture.(y-axis)
but didn't measure the interval distance in the picture(z-axis).
I measure the interval distance in the real world.
I combined the two.
It lets us know how the depth of reality is shown in pictures.

It's a plot, then. A graph. It's not a depiction of the path of light "in reality."

The plot of apparent height vs. actual interval distance is a curve.
The plot of apparent height vs. apparent interval distance is a slope.

Neither supports your contention that perspective produces the visual effect of a convex surface "in reality." For that, either the surface is really convex or light bends upward to produce the appearance of a convex surface. Perspective doesn't do that. You've created plot of a curve (apparent vs. reality) but that's not showing bending light in reality.

[zorbakim]

What I have tried is how the three-dimensional world looks in the two-dimensional visual image.

Right, but what is the shape of the 3-D world? My take is that if the tiles are squares, i.e. each of the four internal angles is 90d, and if we put all the tiles together, this will form a much larger square, with each of the four internal angles also equal to 90d.

Is this your point? But then which part of the large square is curved?

You are confusing physical with visual.

[zorbakim]

It's a plot, then. A graph. It's not a depiction of the path of light "in reality."

The plot of apparent height vs. actual interval distance is a curve.
The plot of apparent height vs. apparent interval distance is a slope.

Neither supports your contention that perspective produces the visual effect of a convex surface "in reality." For that, either the surface is really convex or light bends upward to produce the appearance of a convex surface. Perspective doesn't do that. You've created plot of a curve (apparent vs. reality) but that's not showing bending light in reality.

You have a serious misunderstanding.
I never said that the path of light is curved.
What many people have misunderstood is that geometrical optics is the nature of light.
But it's not true.
geometrical optics is not the nature of light.
It is only a mathematical interpretation and a tool.

[Bobby Shafto]

I've tried to explain why, but because we are not communicating, maybe pictures will help.

This has all of the perspective features you describe but the surface is flat:



Where is the visual curve?

Here is visual curve:



In this second image, perspective is applied just as in the first, but now the surface is convex.

Perspective doesn't cause that appearance of convexity. If the surface isn't convex, then light must bend to produce that visual effect.  But straight line light and perspective don't produce the visual effect you say it does. The reasoning isn't sound and it isn't born out in either reality or when modeled.