5 ms·

Implementing strict three-point perspective

133 points by TheRealPomax 5y ago

jiofih 5y ago
> Except that’s not “true” three point perspective Is “strict 3-point perspective” a known concept or the author just invented it? > That’s the easy-for-computer-graphics version of three point perspective It was devised for people, not computers, over 600 years ago. I can’t find any other references or why you’d want this odd infinite representation of space, it makes little sense except as a mathematical curiosity.
- TheRealPomax 5y ago
  In drawing, it's just "three point perspective", and it's this. In computer graphics, every resource you look for explains three point perspective as a plane intersection (because then computers can draw it using a camera), which is not the same thing as pen and paper three point perspective. The word "strict" here is just the natural language version of the word, not a technical term. We're implementing three point perspective following the pen and paper rules, not the "let's subtly change how it works so a 3D camera can do this" rules =) > I can’t find [...] why you’d want this odd infinite representation of space. Like I say in the post: you almost certainly don't ever want it. When drawing three point perspective using pen and paper we typically pick vanishing points that aren't even on the paper itself (you tape down your paper and mark them on your desk instead) to get sweet looking faux architectural drawings while effectively working on a "crop" where the effects of exponential space are subtle, instead of super obvious, so it never gets weird. (heck, even adding secondary vanishing points that are further apart for working at different scales for different parts of your picture so people will never see the effect of exponential space is pretty common) Similarly, you can get something "close enough" in any 3D software with a wide angle camera positioned close to your subject, so unless you've very explicitly setting out to do exponential space graphics, there is literally no reason for you to ever need, let alone implement this. But it is fun to work out what the real behaviour is if we try to implement strict three point perspective on a computer, because we like programming puzzles, and (also as mentioned) there aren't any pages on the web that I've been able to find that cover this extremely niche projection so now there is at least one.
  - j4yav 5y ago
    Do you have animations generated from this? I would love to see rotations in motion. I feel like there is some game that could be created from needing to manipulate objects in this space.
    - TheRealPomax 5y ago
      https://twitter.com/TheRealPomax/status/1400974346160533508 https://twitter.com/TheRealPomax/status/1400974346160533508
- robbrown451 5y ago
  Agree, none of this makes sense. Three point perspective is best explained by the concept of a pinhole camera with a planar film surface, and simply assumes that we are most interested in the theoretical directions of the 3 major axes, X Y and Z. For it to really duplicate the geometry you see with your eyes, you need to look at the resulting photo from the spot where the pinhole is (related to the paper, and presumably turning it upside down first). This of course means you need to close one eye, or show a different image to each eye). But computer graphics, to my knowledge(see 1 below), almost never thinks of it in terms of vanishing points, this is a convenient concept (essentially, a shortcut) for humans who are drawing on paper. Do computers ever even calculate where the vanishing points are on the drawing plane? (other than niche uses, such as an art composition app or the like?) I have never seen computer graphics software "care" about the concept of vanishing points, such as by having a variable that represents said point. I feel weird having such a negative reaction to this article since I have used the author's bezier library for ages and have a lot of respect for his writing regarding beziers and related curves. [1] I implemented view controls in CAD systems 25 years ago that are still in use today, and which concentrated especially on perspective views, so I have some knowledge of the subject. Also I learned perspective drawing skills in my industrial design education prior to that, and previous to that was into photography and mechanical drawing and obsessed over such geometrical stuff, starting 40 years ago now.
  - TheRealPomax 5y ago
    Computers graphics based on linear algebra can't do true vanishing points, so... no? This isn't a tutorial on how to implement a useful three point perspective, this is an analysis of how three point perspective behaves if we don't make any computing concessions and examine the full space. You're never going to use that in 3D graphics, it looks terrible and I can't even think of a fun game mechanic that could be based on it. Just use a wide FOV camera in your software of choice and you'll get something much better. But it is a programming exercise that is worth running through. Remember, when we draw perspective on paper, we never draw all the way up to the vanishing points, we keep them far away enough that every straight line we draw still behaves like a straight line. Things don't get crazy until you get close enough to the vanishing points for the exponential mapping to become really pronounced, and starts doing really wild things. So obviously for an analysis of the space I'm going to draw something that is intentionally close enough to the vanishing points to show that insanity off =)
- captainmuon 5y ago
  Isn't regular point perspective that all axis-aligned lines meet at the one of the three points, whereas here all lines whatsoever eventually converge there? This reminds me of the mathematical trick to make a "hollow earth" work, with us living on the inside: All rays of light are bent. You can never see the curvature of the earth (or it seems that we are living on the outside). The stars that seem to be at ~infinity are at the center of the sphere, and the sun is a ball of fire orbiting around the center, and so on.
  - TheRealPomax 5y ago
    Almost: all equal ratio lines (x=z+c, x=y+c, y=z+c where c is some constant) head off in a straight line towards their respective horizons, but all other non-axis-aligned lines converge at the vanishing points.
- zozbot234 5y ago
  It looks like the OP has come up with their own variation on some sort of curvilinear perspective (which is a known concept in visual design, and may even be useful to model, e.g. distortion introduced by a lens) but this is not how actual 3D projection works.
  - kragen 5y ago
    Well, you can do actual 3D projection using a fisheye lens! It might even be a worthwhile thing to do.
  - TheRealPomax 5y ago
    Very much not, that's the whole point of this post. This is what happens when you take the maths associated with 2 and 3 point perspective, and work out what the non-cartesian properties actually mean if you were to implement it "the way it really is" on a computer. At which point you should go "this is silly, let's never do this" because: it's really silly, let's never do this. And now we know why.
whatshisface 5y ago
That last image is not how artists draw rotated cubes in three-point perspective. So, I have to conclude that the author's formalization only works for lines parallel to the axes.
- TheRealPomax 5y ago
  Artists don't draw all the way to the vanishing points, we only draw in the part that looks normal. Look at the figure showing off how straight lines behave again, and you'll see that everything starts off actually looking like normal straight lines, and things don't actually go crazy until we start getting close to our vanishing points. Which is why on paper, you place the vanishing points outside your drawing, so your drawing can stay constrained to the part of exponential space that looks normal. E.g. if you constrain the viewport to the unit area (the triangle between 1/0/0, 0/1/0, and 0/0/1) and scale your scene down to fit inside of that, things will actually look perfectly fine, with a gorgeous perspective (but will also be visually indistinguishable from a wide angle camera positioned closed to your major scene point, so just use that). However, with computers we can trivially see what will happen if you do try to use the full space, rather than only working inside an (incredibly sensible!) crop. The result is pretty wild.
  - cmiles74 5y ago
    The effect is also pretty dramatic and I think this is a big part of illustration work. Comic books in particular use three point perspective to add drama. https://www.youtube.com/watch?v=wyIKZhIAl0k https://www.youtube.com/watch?v=wyIKZhIAl0k The cropping aspect is made pretty explicit in Stan Lee's video. ;-)
  - egypturnash 5y ago
    Artists don't draw all the way to the vanishing points, we only draw in the part that looks normal I am a professional artist and I would just like to confirm this line. 3-point perspective tends to start looking weird once you get outside of a certain sweet spot; I did so many beginner drawings with the vanishing points too close, which resulted in a weirdly exaggerated set of shapes.
- 5y ago
megameter 5y ago
I did set out to make a wireframe two-point perspective dungeon crawl in PICO-8 once. It worked(I recall the math for making the height of walls feel correct at different distances was hard to understand and I probably did resort to the CG approach) but I eventually abandoned it because I was spending way too much time trying to deal with occlusion and optimize it for PICO-8's CPU. Since then the system constraints have changed and I would probably just make a tline() raycaster.
- TheRealPomax 5y ago
  PICO-8 is such a fun piece of software. Getting lost in graphics optimization for it feels like it's more the norm than the exception =D
  - hypertele-Xii 5y ago
    Getting lost in optimization is the opposite of actually making things, and as an artist, it's at the top of my reasons why PICO-8 isn't fun to use.
    - TheRealPomax 5y ago
      That's a fair point. PICO-8 never appealed to me for artistic expression, it's purely a programming challenge to cram as much functionality as possible into a tiny, tiny program.
      - megameter 5y ago
        My experiences with PICO-8 led me in the direction of making my own fantasy console...which then turned into a more thorough examination of computing itself. I reached the conclusion that what I want is actually a lot of small source mediums that would cooperate well together and be bundled in arbitrary fashion as a single runtime engine, vs the "familiar computer platform but less" model that is intrinsic to the PICO-8 approach, which puts a lot of pressure on the platform to address all needs. So I started working on that problem instead. What I have now is a binary format that might suffice as a common source document type. It's binary because it let me declare more precise meanings for the bits instead of offloading those meanings to a character parser, but it's also text-like because it doesn't enforce a lot of structure, it has notions of "lines" and "strings". I'm now just about to where I can test it on media I might want to write and work in and see how well it edits.
deleted 5y ago
[deleted]
kragen 5y ago
This is not how geometrically-correct pinhole-camera perspective projection works; Pomax is approaching the vanishing points exponentially rather than hyperbolically, which is why straight lines turn into curves. If you work out the geometry of the pinhole camera setup [edit: with a flat imaging plane], you find that you're just doing a division (conventionally, dividing by the Z-coordinate in camera space, though you don't have to set it up that way) instead of an exponential. This is a "projective transformation", and those do map straight lines to straight lines: https://en.wikipedia.org/wiki/Projective_transformation https://en.wikipedia.org/wiki/Projective_transformation If you have a sufficiently wide field of view, even the standard projection transformation gives you things that look pretty "distorted", which is a fun trick often used in art photography. Also fisheye lenses are often designed to do a non-projective transformation and get effects you can't theoretically get with a pinhole camera, like mapping straight lines to curved lines as Pomax does. Pomax's invention is super cool, even though it isn't the usual 3D→2D mapping motivated by pinhole-camera geometry. Weird is good! I wonder if there's a version of it in which straight lines become curves, but polynomial curves of some reasonable degree, or maybe something involving square roots and division, so we can rasterize them without computing a bunch of exponentials? Is that even a reasonable way to seek optimizations in 02021? Pomax may have a good point that projective transformations aren't "strict three-[vanishing]-point perspective" because, instead of having three vanishing points, they have an infinite number of vanishing points, a point others have made in the comments here. The profoundly weird and cool aspect of Pomax's invention is that every line converges to one of those three points.
- klodolph 5y ago
  You can get distorted projections with pinhole cameras too, as long as your sensor is curved. Fisheye is impractical, but cylindrical projections are easy. Common way to do pinhole photography is with an oatmeal box, which is a cylinder. https://sdsu-physics.org/assets/PDFs/oatmeal_pinhole_camera.pdf https://sdsu-physics.org/assets/PDFs/oatmeal_pinhole_camera.... I'm not really trying to nitpick here, just like talking about pinhole cameras. With a flat image plane the pinhole lens is "rectilinear", but the beauty of pinhole cameras is that you can project the image onto any shape you want without affecting focus.
mpetrovich 5y ago
Back in college I wrote a 3D rendering engine using what seems to be a similar method as the OP. I always thought the results looked a little strange... https://github.com/mpetrovich/CUB3D https://github.com/mpetrovich/CUB3D
- TheRealPomax 5y ago
  Neat!
anderskaseorg 5y ago
If you fix your incorrect exponential formula 1.0 - 1.0 / pow(base, step) to the correct inverse-proportional formula 1.0 - 1.0 / (1.0 + log(base) * step) (where I’ve written log(base) just to match the effect of your existing parameter near the origin), you should find that your curved lines become straight. If the “base” parameters are chosen correctly, you should also find that these fixed results match the ones from usual computer graphics.
- TheRealPomax 5y ago
  Except this article isn't interested in "the usual computer graphics". We already know how to do that one, it looks great, it barely has any problems, and on the whole if you want to render things in a cool perspective, use a camera. The whole point of the article is to show what happens if we don't follow conventional computer graphics wisdom and instead implement it based on how it gets taught in art class, and what happens if you break the fundamental "don't get too close to your vanishing points" rule you get taught there. Mind you, it's great advice in general, it just doesn't match up with art class, which was the whole point of this article. That said, if you want to file an issue over on the repo (https://github.com/Pomax/three-point-perspective/issues https://github.com/Pomax/three-point-perspective/issues), I'm sure a new section can be added at the end going "how can we fix this?" and then show this is one of the several ways in which real world 3D graphics handle perspective.
  - anderskaseorg 5y ago
    Art classes don’t teach that objects appear exponentially smaller as they recede into the distance. To make something appear half the size, you need to move it twice as far away, not move it by a constant amount.
    - TheRealPomax 5y ago
      Either you never had art class in high school, or you forgot that it does not get explained that nicely. Most kids, as well as adults taking casual courses, will never learn this. Can we go "hey we can fix all of this with a switch in function"? Absolutely. Should I ammend the article with that? Yeah, also absolutely. Is it how the vast majority of people get taught three point perspective? Very much not. Again, please file an issue so that I don't forget to write that section, because pages on the internet well outlast their life on hackernews, but I work on a million projects and will forget about this if there's no issue =)
pfedak 5y ago
I think a lot of this confusion stems from the author framing this project in a way that is, frankly, nonsense. Vanishing points have nothing to do with the clipping plane, and 3D computer graphics isn't trying to approximate the coordinate transform described in the post, it's trying to emulate real-life optics. The sense in which this definition is "strict" is based on ignoring that N-point perspective is just a tool for drawing lines parallel to the coordinate axes with correct perspective. If the author had framed this as just a fun "what if", making every line converge to the vanishing point, it might have been better received. The observation that some lines look strange is about as meaningful as the same observation on an any given map projection. It's definitely true, but it all depends on the choice of mapping function, nothing inherent to spherical geometry. I choose map projections for a particular reason: as has already been pointing out, there's no connection between this "strict" definition and an exponential coordinate transform. Any function mapping the axes that grows faster than polynomial would produce the desired behavior (essentially, you need the ratio between z and x on a line a*x + b*z = c to approach zero or infinity), but would produce different "line" shapes. This choice is entirely arbitrary, and is in fact equivalent to simply mapping the underlying space by a related transformation and then using a normal projective transformation.
- bscphil 5y ago
  > The sense in which this definition is "strict" is based on ignoring that N-point perspective is just a tool for drawing lines parallel to the coordinate axes with correct perspective. The thing is, the author claims that students in art classes are taught that in n-point perspective, all parallel lines converge at one of n points. In defense of this, they have [1], which says > in basic one-point perspective, lines are either vertical, horizontal or recede toward the vanishing point. In two-point, lines are either horizontal or recede toward one of the two vanishing points. In three-point perspective all lines recede toward one of the three vanishing points. I don't think I've ever heard this before. Maybe what's happening here is that artists are used to thinking about works like [2], in other words cases where the point of the work is to show off 1, 2, or 3 point perspective and so you deliberately minimize lines in the image that are not parallel to the scene's axes. Of course, artists surely realize that this can't possibly apply to all parallel lines in the scene, as the artist's own drawing in [1] illustrates: see my edit here [3]. Still, I find the OP's work interesting as a possible implementation of what it would mean to take this mistaken description of n-point perspective seriously. [1] https://www.craftsy.com/post/three-point-perspective/ https://www.craftsy.com/post/three-point-perspective/ [2] https://upload.wikimedia.org/wikipedia/commons/6/6b/One_point_perspective.jpg https://upload.wikimedia.org/wikipedia/commons/6/6b/One_poin... [3] https://ipfs.io/ipfs/QmTLmrBQz21cgUX3qZJvrhS4zHrENyWyvbNJRrUn9heVs3/cropped.png https://ipfs.io/ipfs/QmTLmrBQz21cgUX3qZJvrhS4zHrENyWyvbNJRrU...
guyoung 5y ago
marked
deleted 5y ago
[deleted]
TheRealPomax 5y ago
For anyone finding this discussion in the future: the article has been completely rewritten since this posting. While perhaps historically interesting for people who find their way here in the future (if any?), note that pretty much none of the comments apply anymore.

133 points by TheRealPomax 5y ago

> Except that’s not “true” three point perspective Is “strict 3-point perspective” a known concept or the author just invented it? > That’s the easy-for-computer-graphics version of three point perspective It was devised for people, not computers, over 600 years ago. I can’t find any other references or why you’d want this odd infinite representation of space, it makes little sense except as a mathematical curiosity.

TheRealPomax 5y ago
In drawing, it's just "three point perspective", and it's this. In computer graphics, every resource you look for explains three point perspective as a plane intersection (because then computers can draw it using a camera), which is not the same thing as pen and paper three point perspective. The word "strict" here is just the natural language version of the word, not a technical term. We're implementing three point perspective following the pen and paper rules, not the "let's subtly change how it works so a 3D camera can do this" rules =) > I can’t find [...] why you’d want this odd infinite representation of space. Like I say in the post: you almost certainly don't ever want it. When drawing three point perspective using pen and paper we typically pick vanishing points that aren't even on the paper itself (you tape down your paper and mark them on your desk instead) to get sweet looking faux architectural drawings while effectively working on a "crop" where the effects of exponential space are subtle, instead of super obvious, so it never gets weird. (heck, even adding secondary vanishing points that are further apart for working at different scales for different parts of your picture so people will never see the effect of exponential space is pretty common) Similarly, you can get something "close enough" in any 3D software with a wide angle camera positioned close to your subject, so unless you've very explicitly setting out to do exponential space graphics, there is literally no reason for you to ever need, let alone implement this. But it is fun to work out what the real behaviour is if we try to implement strict three point perspective on a computer, because we like programming puzzles, and (also as mentioned) there aren't any pages on the web that I've been able to find that cover this extremely niche projection so now there is at least one.
- j4yav 5y ago
  Do you have animations generated from this? I would love to see rotations in motion. I feel like there is some game that could be created from needing to manipulate objects in this space.
  - TheRealPomax 5y ago
    https://twitter.com/TheRealPomax/status/1400974346160533508 https://twitter.com/TheRealPomax/status/1400974346160533508
robbrown451 5y ago
Agree, none of this makes sense. Three point perspective is best explained by the concept of a pinhole camera with a planar film surface, and simply assumes that we are most interested in the theoretical directions of the 3 major axes, X Y and Z. For it to really duplicate the geometry you see with your eyes, you need to look at the resulting photo from the spot where the pinhole is (related to the paper, and presumably turning it upside down first). This of course means you need to close one eye, or show a different image to each eye). But computer graphics, to my knowledge(see 1 below), almost never thinks of it in terms of vanishing points, this is a convenient concept (essentially, a shortcut) for humans who are drawing on paper. Do computers ever even calculate where the vanishing points are on the drawing plane? (other than niche uses, such as an art composition app or the like?) I have never seen computer graphics software "care" about the concept of vanishing points, such as by having a variable that represents said point. I feel weird having such a negative reaction to this article since I have used the author's bezier library for ages and have a lot of respect for his writing regarding beziers and related curves. [1] I implemented view controls in CAD systems 25 years ago that are still in use today, and which concentrated especially on perspective views, so I have some knowledge of the subject. Also I learned perspective drawing skills in my industrial design education prior to that, and previous to that was into photography and mechanical drawing and obsessed over such geometrical stuff, starting 40 years ago now.
- TheRealPomax 5y ago
  Computers graphics based on linear algebra can't do true vanishing points, so... no? This isn't a tutorial on how to implement a useful three point perspective, this is an analysis of how three point perspective behaves if we don't make any computing concessions and examine the full space. You're never going to use that in 3D graphics, it looks terrible and I can't even think of a fun game mechanic that could be based on it. Just use a wide FOV camera in your software of choice and you'll get something much better. But it is a programming exercise that is worth running through. Remember, when we draw perspective on paper, we never draw all the way up to the vanishing points, we keep them far away enough that every straight line we draw still behaves like a straight line. Things don't get crazy until you get close enough to the vanishing points for the exponential mapping to become really pronounced, and starts doing really wild things. So obviously for an analysis of the space I'm going to draw something that is intentionally close enough to the vanishing points to show that insanity off =)
captainmuon 5y ago
Isn't regular point perspective that all axis-aligned lines meet at the one of the three points, whereas here all lines whatsoever eventually converge there? This reminds me of the mathematical trick to make a "hollow earth" work, with us living on the inside: All rays of light are bent. You can never see the curvature of the earth (or it seems that we are living on the outside). The stars that seem to be at ~infinity are at the center of the sphere, and the sun is a ball of fire orbiting around the center, and so on.
- TheRealPomax 5y ago
  Almost: all equal ratio lines (x=z+c, x=y+c, y=z+c where c is some constant) head off in a straight line towards their respective horizons, but all other non-axis-aligned lines converge at the vanishing points.
zozbot234 5y ago
It looks like the OP has come up with their own variation on some sort of curvilinear perspective (which is a known concept in visual design, and may even be useful to model, e.g. distortion introduced by a lens) but this is not how actual 3D projection works.
- kragen 5y ago
  Well, you can do actual 3D projection using a fisheye lens! It might even be a worthwhile thing to do.
- TheRealPomax 5y ago
  Very much not, that's the whole point of this post. This is what happens when you take the maths associated with 2 and 3 point perspective, and work out what the non-cartesian properties actually mean if you were to implement it "the way it really is" on a computer. At which point you should go "this is silly, let's never do this" because: it's really silly, let's never do this. And now we know why.

whatshisface 5y ago

That last image is not how artists draw rotated cubes in three-point perspective. So, I have to conclude that the author's formalization only works for lines parallel to the axes.

TheRealPomax 5y ago
Artists don't draw all the way to the vanishing points, we only draw in the part that looks normal. Look at the figure showing off how straight lines behave again, and you'll see that everything starts off actually looking like normal straight lines, and things don't actually go crazy until we start getting close to our vanishing points. Which is why on paper, you place the vanishing points outside your drawing, so your drawing can stay constrained to the part of exponential space that looks normal. E.g. if you constrain the viewport to the unit area (the triangle between 1/0/0, 0/1/0, and 0/0/1) and scale your scene down to fit inside of that, things will actually look perfectly fine, with a gorgeous perspective (but will also be visually indistinguishable from a wide angle camera positioned closed to your major scene point, so just use that). However, with computers we can trivially see what will happen if you do try to use the full space, rather than only working inside an (incredibly sensible!) crop. The result is pretty wild.
- cmiles74 5y ago
  The effect is also pretty dramatic and I think this is a big part of illustration work. Comic books in particular use three point perspective to add drama. https://www.youtube.com/watch?v=wyIKZhIAl0k https://www.youtube.com/watch?v=wyIKZhIAl0k The cropping aspect is made pretty explicit in Stan Lee's video. ;-)
- egypturnash 5y ago
  Artists don't draw all the way to the vanishing points, we only draw in the part that looks normal I am a professional artist and I would just like to confirm this line. 3-point perspective tends to start looking weird once you get outside of a certain sweet spot; I did so many beginner drawings with the vanishing points too close, which resulted in a weirdly exaggerated set of shapes.
5y ago

megameter 5y ago

I did set out to make a wireframe two-point perspective dungeon crawl in PICO-8 once. It worked(I recall the math for making the height of walls feel correct at different distances was hard to understand and I probably did resort to the CG approach) but I eventually abandoned it because I was spending way too much time trying to deal with occlusion and optimize it for PICO-8's CPU. Since then the system constraints have changed and I would probably just make a tline() raycaster.

TheRealPomax 5y ago
PICO-8 is such a fun piece of software. Getting lost in graphics optimization for it feels like it's more the norm than the exception =D
- hypertele-Xii 5y ago
  Getting lost in optimization is the opposite of actually making things, and as an artist, it's at the top of my reasons why PICO-8 isn't fun to use.
  - TheRealPomax 5y ago
    That's a fair point. PICO-8 never appealed to me for artistic expression, it's purely a programming challenge to cram as much functionality as possible into a tiny, tiny program.
    - megameter 5y ago
      My experiences with PICO-8 led me in the direction of making my own fantasy console...which then turned into a more thorough examination of computing itself. I reached the conclusion that what I want is actually a lot of small source mediums that would cooperate well together and be bundled in arbitrary fashion as a single runtime engine, vs the "familiar computer platform but less" model that is intrinsic to the PICO-8 approach, which puts a lot of pressure on the platform to address all needs. So I started working on that problem instead. What I have now is a binary format that might suffice as a common source document type. It's binary because it let me declare more precise meanings for the bits instead of offloading those meanings to a character parser, but it's also text-like because it doesn't enforce a lot of structure, it has notions of "lines" and "strings". I'm now just about to where I can test it on media I might want to write and work in and see how well it edits.

deleted 5y ago

[deleted]

kragen 5y ago

This is not how geometrically-correct pinhole-camera perspective projection works; Pomax is approaching the vanishing points exponentially rather than hyperbolically, which is why straight lines turn into curves. If you work out the geometry of the pinhole camera setup [edit: with a flat imaging plane], you find that you're just doing a division (conventionally, dividing by the Z-coordinate in camera space, though you don't have to set it up that way) instead of an exponential. This is a "projective transformation", and those do map straight lines to straight lines: https://en.wikipedia.org/wiki/Projective_transformation https://en.wikipedia.org/wiki/Projective_transformation If you have a sufficiently wide field of view, even the standard projection transformation gives you things that look pretty "distorted", which is a fun trick often used in art photography. Also fisheye lenses are often designed to do a non-projective transformation and get effects you can't theoretically get with a pinhole camera, like mapping straight lines to curved lines as Pomax does. Pomax's invention is super cool, even though it isn't the usual 3D→2D mapping motivated by pinhole-camera geometry. Weird is good! I wonder if there's a version of it in which straight lines become curves, but polynomial curves of some reasonable degree, or maybe something involving square roots and division, so we can rasterize them without computing a bunch of exponentials? Is that even a reasonable way to seek optimizations in 02021? Pomax may have a good point that projective transformations aren't "strict three-[vanishing]-point perspective" because, instead of having three vanishing points, they have an infinite number of vanishing points, a point others have made in the comments here. The profoundly weird and cool aspect of Pomax's invention is that every line converges to one of those three points.

klodolph 5y ago
You can get distorted projections with pinhole cameras too, as long as your sensor is curved. Fisheye is impractical, but cylindrical projections are easy. Common way to do pinhole photography is with an oatmeal box, which is a cylinder. https://sdsu-physics.org/assets/PDFs/oatmeal_pinhole_camera.pdf https://sdsu-physics.org/assets/PDFs/oatmeal_pinhole_camera.... I'm not really trying to nitpick here, just like talking about pinhole cameras. With a flat image plane the pinhole lens is "rectilinear", but the beauty of pinhole cameras is that you can project the image onto any shape you want without affecting focus.

mpetrovich 5y ago

Back in college I wrote a 3D rendering engine using what seems to be a similar method as the OP. I always thought the results looked a little strange... https://github.com/mpetrovich/CUB3D https://github.com/mpetrovich/CUB3D

TheRealPomax 5y ago
Neat!

anderskaseorg 5y ago

If you fix your incorrect exponential formula 1.0 - 1.0 / pow(base, step) to the correct inverse-proportional formula 1.0 - 1.0 / (1.0 + log(base) * step) (where I’ve written log(base) just to match the effect of your existing parameter near the origin), you should find that your curved lines become straight. If the “base” parameters are chosen correctly, you should also find that these fixed results match the ones from usual computer graphics.

TheRealPomax 5y ago
Except this article isn't interested in "the usual computer graphics". We already know how to do that one, it looks great, it barely has any problems, and on the whole if you want to render things in a cool perspective, use a camera. The whole point of the article is to show what happens if we don't follow conventional computer graphics wisdom and instead implement it based on how it gets taught in art class, and what happens if you break the fundamental "don't get too close to your vanishing points" rule you get taught there. Mind you, it's great advice in general, it just doesn't match up with art class, which was the whole point of this article. That said, if you want to file an issue over on the repo (https://github.com/Pomax/three-point-perspective/issues https://github.com/Pomax/three-point-perspective/issues), I'm sure a new section can be added at the end going "how can we fix this?" and then show this is one of the several ways in which real world 3D graphics handle perspective.
- anderskaseorg 5y ago
  Art classes don’t teach that objects appear exponentially smaller as they recede into the distance. To make something appear half the size, you need to move it twice as far away, not move it by a constant amount.
  - TheRealPomax 5y ago
    Either you never had art class in high school, or you forgot that it does not get explained that nicely. Most kids, as well as adults taking casual courses, will never learn this. Can we go "hey we can fix all of this with a switch in function"? Absolutely. Should I ammend the article with that? Yeah, also absolutely. Is it how the vast majority of people get taught three point perspective? Very much not. Again, please file an issue so that I don't forget to write that section, because pages on the internet well outlast their life on hackernews, but I work on a million projects and will forget about this if there's no issue =)

pfedak 5y ago

I think a lot of this confusion stems from the author framing this project in a way that is, frankly, nonsense. Vanishing points have nothing to do with the clipping plane, and 3D computer graphics isn't trying to approximate the coordinate transform described in the post, it's trying to emulate real-life optics. The sense in which this definition is "strict" is based on ignoring that N-point perspective is just a tool for drawing lines parallel to the coordinate axes with correct perspective. If the author had framed this as just a fun "what if", making every line converge to the vanishing point, it might have been better received. The observation that some lines look strange is about as meaningful as the same observation on an any given map projection. It's definitely true, but it all depends on the choice of mapping function, nothing inherent to spherical geometry. I choose map projections for a particular reason: as has already been pointing out, there's no connection between this "strict" definition and an exponential coordinate transform. Any function mapping the axes that grows faster than polynomial would produce the desired behavior (essentially, you need the ratio between z and x on a line a*x + b*z = c to approach zero or infinity), but would produce different "line" shapes. This choice is entirely arbitrary, and is in fact equivalent to simply mapping the underlying space by a related transformation and then using a normal projective transformation.

bscphil 5y ago
> The sense in which this definition is "strict" is based on ignoring that N-point perspective is just a tool for drawing lines parallel to the coordinate axes with correct perspective. The thing is, the author claims that students in art classes are taught that in n-point perspective, all parallel lines converge at one of n points. In defense of this, they have [1], which says > in basic one-point perspective, lines are either vertical, horizontal or recede toward the vanishing point. In two-point, lines are either horizontal or recede toward one of the two vanishing points. In three-point perspective all lines recede toward one of the three vanishing points. I don't think I've ever heard this before. Maybe what's happening here is that artists are used to thinking about works like [2], in other words cases where the point of the work is to show off 1, 2, or 3 point perspective and so you deliberately minimize lines in the image that are not parallel to the scene's axes. Of course, artists surely realize that this can't possibly apply to all parallel lines in the scene, as the artist's own drawing in [1] illustrates: see my edit here [3]. Still, I find the OP's work interesting as a possible implementation of what it would mean to take this mistaken description of n-point perspective seriously. [1] https://www.craftsy.com/post/three-point-perspective/ https://www.craftsy.com/post/three-point-perspective/ [2] https://upload.wikimedia.org/wikipedia/commons/6/6b/One_point_perspective.jpg https://upload.wikimedia.org/wikipedia/commons/6/6b/One_poin... [3] https://ipfs.io/ipfs/QmTLmrBQz21cgUX3qZJvrhS4zHrENyWyvbNJRrUn9heVs3/cropped.png https://ipfs.io/ipfs/QmTLmrBQz21cgUX3qZJvrhS4zHrENyWyvbNJRrU...

guyoung 5y ago

marked

deleted 5y ago

[deleted]

TheRealPomax 5y ago

For anyone finding this discussion in the future: the article has been completely rewritten since this posting. While perhaps historically interesting for people who find their way here in the future (if any?), note that pretty much none of the comments apply anymore.