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In the above picture, I am a bit confused how it turns out parallel lines seems to meet in the artist's potrait. Could someone explain in simple words why the roads which don't intersect in the ambient world do intersect in the painting?

So far, I get the idea that when drawing the painting , a line is drawn from the artist's eye to the object in 3-D space and the point on the canvas is the intersection of the line with it. However, the converging line thing is still tripping me up.

Edit: For future readers, I found this Deviant Art page by Nisio very helpful. You have to click the picture to zoom, but they go over the details of it. Also check out this video (also helpful).

tryst with freedom
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  • A 12-inch ruler placed on a desk in front of you looks much bigger than if the 12-inch ruler were placed a mile away from you. It's the same thing here, except with the ruler being replaced by the width of the road. – angryavian May 17 '22 at 23:29
  • Well I mean I understand and I've experienced this phenomena. I've also seen many videos giving what you said as an explanation but I'm trying to give a deeper explanation on what's going – tryst with freedom May 17 '22 at 23:38
  • Every point on the horizon is the same point at infinity. This means any two distinct lines will appear to converge as they approach it. I think of it as the northern hemisphere being contracted to a single point at the north pole where the equator is the horizon. This is motivated by the relationship between projective spaces and stereographic projection. – CyclotomicField May 18 '22 at 00:21
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    @CyclotomicField The purpose of your comment is unclear to me. There are infinitely many distinct points at infinity -- indeed a whole line at infinity -- in the projective plane. The usual spherical model of a projective plane identifies each pair of antipodal points as a single projective point. – David K May 18 '22 at 00:47
  • @CyclotomicField No. In projective geometry you have many points at infinity. – justt May 18 '22 at 00:47
  • @DavidK https://en.wikipedia.org/wiki/Stereographic_projection#Visualization_of_lines_and_planes – CyclotomicField May 18 '22 at 00:58
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    @CyclotomicField This is one of those places where Wikipedia isn't very good. What they are attempting to describe is the topological equivalence of a projective plane with a circular disk in which opposite points are considered the same point; that is, every diameter begins and ends at the "same" point, but every other diameter begins and ends at a different point. The northern hemisphere did not shrink to a point in this model; it was omitted because every point in it was antipodal to a point that was already projected onto the disk. – David K May 18 '22 at 01:19
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    So it is certainly not true that every point on the horizon is the same point, and not true that any two distinct lines converge as they approach the horizon. And stereographic projection is a really roundabout way to try to explain perspective drawing, if indeed you can explain it that way at all. – David K May 18 '22 at 01:21
  • @Aplateofmomos The red dot on the painting, where the sides of the road appear to intersect, is only at arm's length from the artist; that's how he was able to paint the scene. It's not far away at all. I'm sure that's not what you meant by "near" and "far," but it's not quite clear what you meant. If two parallel lines are never supposed to meet, how could they meet near the artist? – David K May 18 '22 at 02:23
  • I had edited it @Da – tryst with freedom May 18 '22 at 16:33

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One place the mathematics is discussed: Points at infinity where last element in homogeneous vector is $0$?.

More geometrically, an artist's field of view may be modeled either by a sphere of rays or by a projective plane of lines through the eye. We may as well pick the unit sphere centered at the origin of Euclidean three-space.

Parallel lines meet at infinity

An open hemisphere $H$ of the sphere model corresponds to a dense open set $P$ of the projective plane, and the boundary great circle $C$ of $H$ maps to the line at infinity with respect to $P$ by definition. The golden circle shown lies in the equatorial plane $Z = 0$, and the plane $P$ being visualized is $Z = -1$, which corresponds to the lower hemisphere.

Straight lines in $P$ map to the sphere by radial projection from the eye, so their images are great circles. Since great circles intersect on the sphere, the images of lines in $P$ also intersect, even if the lines are parallel as shown. By definition, parallel lines in $P$ do not intersect in $P$. The points of intersection on the sphere therefore line on the boundary great circle, a.k.a., the line at infinity with respect to $P$.

In Penrose's drawing, the artist's canvas might be the plane $Y = 1$ (not shown here). Projecting the great circles to that plane gives a pair of crossing lines, compare the railroad tracks in the linked question.

Added: The diagram below shows the same picture without the sphere, and with the projective lines shown as affine planes. To emphasize,

  • Each colored affine line through the origin (Eye) represents a projective point.
  • Each affine plane through the origin represents a projective line.
  • We're choosing the line at infinity to be the set of gold affine lines lying in the affine plane $Z = 0$.

Parallel affine lines meet at projective infinity

The plane $Z = -1$, by contrast, does not pass through the origin.

  • Its points correspond to projective points: Each point $p$ of the plane $Z = -1$ determines a unique affine line $\ell$ through the origin and $p$, and $\ell$ intersects $Z = -1$ precisely at $p$ rather than being contained in the plane $Z = -1$.
  • Its affine lines (the two parallel blue lines, for example, modeling the edges of Penrose's roadway) correspond to projective lines, represented by the shaded affine planes.
  • The entire affine plane $Z = -1$ corresponds to the "finite" Euclidean part of the projective plane with respect to the projective line at infinity, here chosen to be the affine plane $Z = 0$.
  • The two parallel affine lines in the affine plane $Z = -1$ intersect in the projective plane because the affine planes that represent them intersect along an affine line through the origin, i.e., at a point of the projective plane. The intersection of the two slanted affine planes lies in the affine plane $Z = 0$, a.k.a., the projective line at infinity. In that sense, the parallel affine/Euclidean lines intersect at infinity.
Andrew D. Hwang
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  • Hi Prof! I deleted the question about the boost transformations. I am confused about your approach here. Generally speaking stereographic projection through a sphere lets us associate a plane and a sphere. But the artist in painting objects in 3-D space right? So how can he use that? – tryst with freedom May 20 '22 at 15:34
  • It wouldn't be wrong to view the sphere as scaffolding, but here it's particularly useful because (i) It's a compact set we can visualize via our spatial experience, and (ii) it represents the entire projective plane. (If it matters, the projection here is not stereographic projection as in complex analysis, but central projection from an open hemisphere to the Euclidean plane.) – Andrew D. Hwang May 20 '22 at 15:40
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    I don't think I understood the idea of the answer very well. I think I will revisit it a bit later. Thanks again tho prof. – tryst with freedom May 20 '22 at 20:43
  • I might ought to have added: Any two points on the same ray project to the same point of the sphere; mathematically, two point on a line through the eye "look like" they're in the same location. The sphere represents all directions, i.e., the entire visual field. – Andrew D. Hwang May 21 '22 at 00:59
  • When you say open dense set, what topology are you considering? – tryst with freedom May 22 '22 at 12:34
  • "Open dense" refers to the quotient topology on the projective plane, which is locally homeomorphic to the Euclidean topology on the sphere. When thinking geometrically (as opposed to topologically) I view a point of the projective plane as an antipodal pair of points on the unit sphere. <> If it matters (re: the linked question), those great circles look like straight lines from the center of the sphere: each corresponds to a plane through the eye, see "exactly edge-on." – Andrew D. Hwang May 22 '22 at 13:30
  • So is this rail road bending effect due to how your eye's geometry (approx sphere) rather than something fundamental? – tryst with freedom May 22 '22 at 13:42
  • Maybe it's better to say the intersection is geometrically real in projective space: The artist looking at two parallel lines $L_1$ and $L_2$ "really" sees the planes $P_1$ and $P_2$ through the artist's eye containing those lines. But $P_1$ and $P_2$, being planes in three space, intersect along a line $L$ through the artist's eye (the dashed green line in the diagram). The artist sees this as "the lines intersecting at a point." But this "point of intersection" corresponds to no point in the plane containing $L_1$ and $L_2$. – Andrew D. Hwang May 22 '22 at 14:22
  • That second diagram and explanation is amazing. Have my 50 rep – tryst with freedom May 22 '22 at 20:34
  • The point that artist is actually seeing plane and not line is what I believe to be the most critical point but I have not seen any site or book saying it. – tryst with freedom May 22 '22 at 20:36
  • Could you explain what you mean by Euclidean part of the projective plane? – tryst with freedom May 22 '22 at 20:38
  • Every plane $P_0$ through the origin (such as $Z=0$) represents a projective line; any plane $P \neq P_0$ parallel to $P_0$ (such as $Z=1$ or $Z=-1$) hits every line through the origin and not lying in $P_0$ exactly once. We may view $P$ as a copy of the Euclidean plane (whose points are "finite" by definition). If we do this, the projective line represented by $P_0$ may be viewed as "the line at infinity for the Euclidean plane $P$." – Andrew D. Hwang May 22 '22 at 22:16
  • (This is what Penrose means in the boldfaced quote in your linked question about a projective line being an affine plane through the origin. It's also where we get the description of the Euclidean plane as the set of points $[X: Y: Z]=[X/Z: Y/Z: 1]$ with $Z \neq 0$ fixed, and the line at infinity as the set of $[X: Y: 0]$. :) – Andrew D. Hwang May 22 '22 at 22:22
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    The diagram with the intersecting gray planes is really what makes perspective drawing work, in my opinion. If you put a vertical rectangle of canvas somewhere beyond the "eye" point, line segments painted on that rectangle will look like parts of the dark blue parallel lines if the segments are painted where the gray planes intersect the canvas, below the point where the gray planes intersect each other. – David K May 23 '22 at 18:03
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    I believe there is a lot of rich idea still in this answer. I have to re-read it a few more times to understand, and that is why I had not accepted earlier. Thanks for all the help tho prof! It really is invaluable – tryst with freedom May 25 '22 at 07:37
  • I read this again, now I am confused again. We actually have two eyes, so the model you gave of the one eye thing doesn't translate directly into what goes on irl – tryst with freedom May 30 '22 at 21:24
  • https://physics.stackexchange.com/questions/711361/how-does-viewing-through-two-eyes-give-us-a-sense-of-depth .. this is sooo confusing ... – tryst with freedom May 30 '22 at 21:42
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    The point-projection model here is simple and geometric, accounting only for one eye, and even that not realistically. <> Depth perception is a large enough part of psychology that people devote careers to it. The late neurologist Oliver Sacks wrote (e.g., The Mind's Eye) and spoke about depth perception, and vision more generally. The case of Sue Barry, who grew up with strabismus (non-aligned eyes) and acquired depth perception only as an adult, is particularly remarkable. – Andrew D. Hwang May 31 '22 at 11:55