Okay… so this draws on a couple of things which can be confusing. 1) perspective projections 2) mapping spheres onto 2D planes.
Usually when we think of a field of vision we imagine some projection that maps the 3D world in front of us to some 2D rectangle image. And that’s all fine and well. We don’t expect the lines in the image to conserve the angles they had in 3D.
I think what the author of the post is saying is that if you use a cylindrical projection that wraps around 360 degrees horizontally, then the lines will appear parallel when you unwrap it. But there’s nothing wrong with this. If it seems like it would be a contradiction, because the lines cross each other at right angles in 3D—it’s because in a z-aligned cylindrical projection, the point where the lines cross will be on one of the singularities that sit on each pole. And if the cylindrical projection is not z-aligned, the lines won’t be parallel, and will cross each other at some angle.
I guess you can also think of this as two projections. There is the two lines on the floor, which are projected up onto the bird’s panoramic view (a sphere), and then the sphere is projected onto a z-aligned cylinder, and then the cylinder is unwrapped to give us our 2D image with the two lines parallel.
Like how if you projected two perpendicular lines up onto the bottom of this globe they might align with say, 0“/180” and 90“/270”, but they would appear parallel on the output cylindrical projection
This is assuming that by “perspective” we mean something like “projection onto a sphere”. Then the lines become great semicircles and it’s true that they are parallel at the horizon, at least in the sense that the great circle representing the horizon meets them each at a right angle.
By “perspective” I mean the fact, that a stick twice as far away appears twice as short. Or z-times as far appears z-times shorter. (Providing that no rotation has been invoked.)
Poor bird thus see all the directions parallel. Which is difficult to imagine, but it just must be so.
Yes, they do. In a distance all directions seems parallel.
Except that I don’t deal with than many directions at once. I never see a bird flying to the West near the horizon, and a bird flying to the East near the horizon at the same time. A bird does see that at once. It sees how they fly apart of each other and fly parallel at the same time. It is counterintuitive for me, but not for the bird, I guess.
I can however, see two birds flying away from me, one to the North, other to the West, both far away. They become smaller and smaller, but the apparent distance between them remains practically unchanged.
I quickly rationalize this as an interesting illusion, at the most.
If you surround the bird B with a ten-meter-radius sphere and map each point A on the ground to the intersection between the line segment AB and the sphere, the x and y axis map to a total of four curves along the lower half of the sphere, all of which are, in fact, parallel at the equator.
This way, only smaller parts of lines are parallel (parallel enough), while in reality—or should I say, on the plane—the biggest part of those lines are parallel.
Mapping on the sphere, even mental, doesn’t account for that. And the bird must know this, because it flies miles and miles.
Most of reality maps to near the equator, therefore the bird’s eye would evolve to have most receptors near the equator and most of its visual cortex would focus there. (Assuming that things don’t become more important to the bird as they grow nearer :P)
Short enough to just post here rater than linking:
Imagine, that you are an intelligent bird with a 360 degrees, panoramic view, flying over a plane equipped with orthogonal x and y axis, clearly visible and – what a coincidence – intersecting just 10 meters beneath you.
I argue, that due to the well known phenomenon of the geometrical perspective, you see in a distance the line which goes North, parallel to the line which goes West. In fact, every direction seems parallel to all other three directions.
Is that right, and why it’s right? How could this be?
Is there an unstated assumption that the panoramic view is accomplished by mapping to a human-evolved ~135 degree field of view? I don’t think this would happen in a brain evolved and trained on panoramic eyes/sensors. It doesn’t happen in reality, where panoramic views exist everywhere and are generally accessed by turning our heads.
Closer objects must appear bigger and this kind of perspective is inescapable for us, cameras or birds.
From here the apparent parallelism of the two, from a single point outgoing lines—follows. How then a 360 degrees vision creature handle this? When the straight road going to the North, is parallel to another straight road going to the West, which is parallel to yet another straight road going to the South? At least in some distance and then to the horizon.
Parallel lines appear to intersect according to perspective. But, the more distant parts of the lines are the parts that appear to intersect. Here, where the lines actually do intersect, the more distant parts are away from the intersection. If these are ideal lines such that one could not gauge distance, and one is only looking downward, such as a projection onto a plane, then they are visually indistinguishable from parallel lines. Whether that’s the same thing as them appearing to be parallel may be … a matter of perspective. But, since this is a bird with 360 degree view, it can see that the lines do not also extend above the bird as parallel lines would, so they do not appear parallel to it.
Actually non-parallel lines which goes out from a point at angle alpha, appear to be parallel far away from your standing point, above the intersection point.
Another problem:
https://protokol2020.wordpress.com/2017/05/07/problem-with-perspective/
Okay… so this draws on a couple of things which can be confusing. 1) perspective projections 2) mapping spheres onto 2D planes.
Usually when we think of a field of vision we imagine some projection that maps the 3D world in front of us to some 2D rectangle image. And that’s all fine and well. We don’t expect the lines in the image to conserve the angles they had in 3D.
I think what the author of the post is saying is that if you use a cylindrical projection that wraps around 360 degrees horizontally, then the lines will appear parallel when you unwrap it. But there’s nothing wrong with this. If it seems like it would be a contradiction, because the lines cross each other at right angles in 3D—it’s because in a z-aligned cylindrical projection, the point where the lines cross will be on one of the singularities that sit on each pole. And if the cylindrical projection is not z-aligned, the lines won’t be parallel, and will cross each other at some angle.
I guess you can also think of this as two projections. There is the two lines on the floor, which are projected up onto the bird’s panoramic view (a sphere), and then the sphere is projected onto a z-aligned cylinder, and then the cylinder is unwrapped to give us our 2D image with the two lines parallel.
Like how if you projected two perpendicular lines up onto the bottom of this globe they might align with say, 0“/180” and 90“/270”, but they would appear parallel on the output cylindrical projection
There is no real paradox here, of course. At least not in reality. Only in a bird’s head perhaps, when he says:
Those two birds in front of me are flying parallely; one is going North, one is going West.
Well if the bird knows they fly apparently parallel, then he’s good.
This is assuming that by “perspective” we mean something like “projection onto a sphere”. Then the lines become great semicircles and it’s true that they are parallel at the horizon, at least in the sense that the great circle representing the horizon meets them each at a right angle.
By “perspective” I mean the fact, that a stick twice as far away appears twice as short. Or z-times as far appears z-times shorter. (Providing that no rotation has been invoked.)
Poor bird thus see all the directions parallel. Which is difficult to imagine, but it just must be so.
Then everyone would. What is the difference between a bird flying 2m above the ground, and a 2m tall human? Do all directions seem parallel to you?
Yes, they do. In a distance all directions seems parallel.
Except that I don’t deal with than many directions at once. I never see a bird flying to the West near the horizon, and a bird flying to the East near the horizon at the same time. A bird does see that at once. It sees how they fly apart of each other and fly parallel at the same time. It is counterintuitive for me, but not for the bird, I guess.
I can however, see two birds flying away from me, one to the North, other to the West, both far away. They become smaller and smaller, but the apparent distance between them remains practically unchanged.
I quickly rationalize this as an interesting illusion, at the most.
If you surround the bird B with a ten-meter-radius sphere and map each point A on the ground to the intersection between the line segment AB and the sphere, the x and y axis map to a total of four curves along the lower half of the sphere, all of which are, in fact, parallel at the equator.
This way, only smaller parts of lines are parallel (parallel enough), while in reality—or should I say, on the plane—the biggest part of those lines are parallel.
Mapping on the sphere, even mental, doesn’t account for that. And the bird must know this, because it flies miles and miles.
Most of reality maps to near the equator, therefore the bird’s eye would evolve to have most receptors near the equator and most of its visual cortex would focus there. (Assuming that things don’t become more important to the bird as they grow nearer :P)
Short enough to just post here rater than linking:
Is there an unstated assumption that the panoramic view is accomplished by mapping to a human-evolved ~135 degree field of view? I don’t think this would happen in a brain evolved and trained on panoramic eyes/sensors. It doesn’t happen in reality, where panoramic views exist everywhere and are generally accessed by turning our heads.
Closer objects must appear bigger and this kind of perspective is inescapable for us, cameras or birds.
From here the apparent parallelism of the two, from a single point outgoing lines—follows. How then a 360 degrees vision creature handle this? When the straight road going to the North, is parallel to another straight road going to the West, which is parallel to yet another straight road going to the South? At least in some distance and then to the horizon.
I have an idea, but first I am asking you. How?
Parallel lines appear to intersect according to perspective. But, the more distant parts of the lines are the parts that appear to intersect. Here, where the lines actually do intersect, the more distant parts are away from the intersection. If these are ideal lines such that one could not gauge distance, and one is only looking downward, such as a projection onto a plane, then they are visually indistinguishable from parallel lines. Whether that’s the same thing as them appearing to be parallel may be … a matter of perspective. But, since this is a bird with 360 degree view, it can see that the lines do not also extend above the bird as parallel lines would, so they do not appear parallel to it.
Actually parallel lines appear to intersect.
Actually non-parallel lines which goes out from a point at angle alpha, appear to be parallel far away from your standing point, above the intersection point.