A different problem happened when scientists first tried to watch the earth rotate for 24 straight hours. They got super tired...so they called it a day.
That's incredible! Thanks for that. I'm back from that video. Insane timing considering how they're exactly the same. I think it's interesting how pointless knowledge often becomes useful to someone later
I think this is more honest than most youtubers. I love that you kept the erroneous skew in the video, Matt. However I probably would have left it in as a blooper, and "reskewed" it correctly. In the spirit of your honesty I'll admit, that I have only tried doing a rotation with two skew matrices, but never three. Once you learn about skew matrices for the first time, I think that trying this out seems a very intuititive thing to infer, attempt at it, fail at it, let go of with the intent to try again at a later point, and finally forget about it to never ever try it again. Thank you for setting me straight!
I literally screamed in the shower while watching this, excited that someone else realized that modern low-res games try to rotate objects outside of that pixel grid was WRONG!
YES ikr!? this has always bugged me so much. it's not even hard to do in code!! i can understand when it's a web theme or something, because browsers do a lot of shenanigans, but games and pixel art have no excuse!
I'm from the 8-bit days and I approve of it these days though I hated it back in the day when there was fake pixel art made without computers or any understanding of grids and rasters and pixels.
we will correct this now. it has gotten so bad that some old games that use oversized pixels in their big arts have made me doubt if i was playing a real title or an indie game
to be fair i think it looks nicer if they double the resolution then rotate it regularly because then they get the rotation they want instead of something close to it for low res images
8:25 code review: 1) int() doesn't round to a whole number, it truncates. That may actually be what you want, since it means you only move the pixel right or left if it passes the full pixel width threshold. 2) You're using mod to clip the image, which creates the weird wraparound. If the width needs to be accounted for at all, you probably would be better off by doing a pure clip. Drop the information for pixels outside of the range. 3) Put the image into a numpy array, use linear algebra to apply the transform to the entire image instead of calculating a new location for each pixel.
Re 1) it rounds towards zero, which is just a different wording for truncation. So both of you are right. There are a lot of possible ways to round, and towards zero is just one possibility. Granted, it's not a "fair" way to round and it will look a bit glitchy especially when used on a low-resolution image.
2) If the rotation is progressive, you'll want the wraparound to preserve information. If you clip/crop, you'll need to do it from the original, not the most recent.
@@hoejI guess you have a point there. Matt was using Python there (which I gathered the programming context from, as opposed to pure maths) and I suspect that he's aware that `int()` doesn't do traditional rounding, and I suspect he just didn't want to complicate things when talking about it. As it's Python, he could just substitute `round()` for `int()` to get a more traditional rounding. However, he might be restricted to Python 2 (or he might not be aware that it has changed for the better in Python 3), as in that older version of Python `round()`, while rounding, still returns a floating point number, not an integer.
Back in oldschool microsoft paint, you weren't able to rotate stuff, so instead i eyeballed it with 2 skews. Even back then i recall thinking there's probably a perfect way to do it, i just didn't know it. So yea i'm prety much the exact person this video is made for.
I like how your brain works. Is it a special kind of brain? P.s. I used old-school paint to make animations when I was in "basisschool" (don't know what that translates to, but I was around 8-10 I think). The old ms paint had multiple ctrl+z's which could also be reversed. Iirc we had 3 or 4 different frames. My friends quickly took the trick and made their own 😀. At 11 yo I made a scifi shooter, choose your own adventure game, using powerpoint 😅. Using hundreds, maybe thousands of pages. So you could have multiple paths to follow. There was even a moment when a monster showed up in a corridor where it shot a projectile that quickly got closer by automating the page turns. And if you didn't find the gun or you didn't dodge or shoot back in time, you'd die. I think it was a really cool idea and was, within the constraint, a pretty good game😊.
Haha I did exactly that too back when I was a kid and didn't know of any better editing software. And yep, I used only 2 skews too! Learning that a third skew makes the rotation perfect would've blown my mind back then.
I'm a retro game dev, so this technique is right up my alley. I'm shocked I didn't know about this already. Thanks for making this video! I actually wrote my own matrix math transformation routines for translation/rotation/scale in C for SGI workstations/supercomputer in C 30 years ago, for an artificial life virtual reality game that I was developing for SGI. I had found the routines in GL (which soon evolved into OpenGL) at the time to be inadequate for real-time virtual reality, and my own routines solved my problems. Back then I was all about VR on the highest-end systems possible (I had an Onyx RealityEngine2 as my personal computer, provided by the company I worked for, for several months back then), but anymore I am far more interested in developing for the microcomputers and consoles that came out from the 70s until the early 90s and the handhelds up until the early 2000s.
I'm familiar with OpenGL, I'm used to GL just being a shorthand for OpenGL, in this context, what was GL? Just a predecessor to OpenGL, or was it an API similar to OpenGL without the "Open" standard?
@@AssemblyCSharp Yep, like I said, IRIS GL (we called it just GL, much like you're saying folks do today with OpenGL--likely a continuity thing) evolved into OpenGL. It both was a predecessor to OpenGL with a similar API and was a proprietary standard. Sadly, the windowing and mouse and keyboard aspects of the API were removed from OpenGL. IRIS GL first was used with the superior NeWS, which SGI IRIX used for windowing, and then evolved to support the X Window System, when IRIX switched to that. So, one API could be used to create entire 3D windowed apps for deployment on two different windowing systems. There were some other fundamental difference between IRIS GL and OpenGL, including adding a software rendering pipeline so that systems with differing levels of hardware pipeline support (including none at all) could use the exact same OpenGL library with different 3D graphics hardware drivers. However, most of the API, barring some important exceptions, was just changed via name differences.
You can use this information to rotate images in Microsoft Paint. Step 1: Horizontal skew by theta/2 degrees. Step 2: Vertical skew by -arctan(sin theta). Step 3: Same as Step 1. Bug workaround: you have to paste a minus sign into the text field, because you can't type it in
Apparently the people behind MS Paint didn't know this little trick. But who's surprised. Even today windows can't even sort filenames alphabetically. According to Windows "filename, filename 2, filename 3" in alphabetical order is "filename 2, filename 3, filename".
@@awebmatethis only applies to files with an extension since the dot “.” is at 0x2e in ASCII, which is after the space (0x20). «Fi» gets sorted before «Fi 2», but «Fi.txt» after «Fi 2.txt»
Wait, this is the math that Sethbling used in his new Rigid Body Minecraft Datapack. Because Minecraft’s native Block Display Entity only does skews, stretches, and translations. So he could only do rotations via skews
I remember almost discovering this myself as a kid playing with old MS Paint. There weren't free rotation functions in Paint, but there were skew functions. I figured out that I could "rotate" images by skewing them equally horizontally and vertically, and then fiddle about with the aspect ratio a bit. Never thought of applying a third skew!
That’s where I first encountered this. I’d thought about presenting it at MathsJam, but Andrew Taylor did a much better job of it at last year’s conference than I could have, and that’s presumably where Matt encountered it.
Interesting that MS Paint either didn’t see a purpose or realize they have the capability to rotate images but did and could incorporate skew. When there’s so little utility for skew, and by including skew it means they could have rotated. Makes me wonder how computers rotate images these days at its most basic. Maybe it is that skew function
Article from 1986 has always stuck with me. Multiple single direction skews of a picture quickly devolves into chaos with intermittent return of part of original pattern (Chaos, Scientific American 54:12 (1986) 46-57). Phenomenon related to Poincaré Recurrence. Potential video topic.
Matt saying "look up matrices if you don't already know them" reminds me that a great deal of us have already heard of what he's on about but, heck, we love him for the joy he brings to the material we... ought to know better than we do...
I love when an educator for higher level topics makes the reasonable assumption that viewers will already understand the underlying topic to at least some degree. The number of times I've had to skip over a "what is binary?" section in a video about semi-advanced software or computer hardware stuff is way too high. Binary Coded Decimal? Ok, that's a valid topic to cover since there are multiple schemes to make it happen. Floating point representation? Sure, there are a whole host of ways to do that, and which one you choose alters the result of your calculation. Both of those require some knowledge of what binary is, but if I'm to the point that I'm learning about those topics I should have the base level down already anyways.
I was chipping away at a tool for rendering the signed distance functions of shapes in Blender when I ran into trouble with rotation. Little did I know, this treasure of a video had just been uploaded two hours ago and recommended to me by pure chance! It's exactly what I needed, my shapes are rotating perfectly, and you've definitely earned my subscription with this. Thank you so much!
Kind of glad to see this, I remember when I was a kid I would try to do this in MSPaint, because you couldn't do non-90-degree rotates in MSPaint, but I could get very close to getting "clean rotations" with skew operations. I could never get it to be quite exactly right looking, though.
@@nsf001-3the point they’re making is that the physical model on the table (where he mistakenly inverted one of the skews) is literally not a rotation The virtual Lego model that he overlaid in post is correct. Yes, not a “perfect” square, but very close because he performed a proper “rotation” (via 3 specific skews) But if you invert the direction of any one skew, you will get something that looks nothing like a square (see the physical table model), because those particular 3 skews no longer equal a rotation
Only about half way through, so maybe you mention this later, but one huge advantage of this for early/slow computers is that skewing like you did with the plumber can be done entirely with integer math, meaning that there is no need for a floating point unit in the processor. We take this for granted a lot with modern hardware, but so many older computers (let alone microprocessors) couldn't do floating point math at all.
Oh, Mr.Parker Square. At 9:37 you note they have to be multiplied in the correct order, but then mistakenly ordered them ABC instead of CBA! In the end it doesn't change the result because C=A. However when you have the position vector on the right, the matrices right to left are the steps applied to the position. So your example shows the last skew (C) being done first, and the first skew (A) being done last.
Yeah, what I was thinking since if the operations happen A -> B -> C then it would look like C( B ( Ax) ). Thanks for pointing it out, I went in the comments to check if someone else mentioned it or if I was the one who was wrong. Also, somehow the bot account that copied your comment got more likes than you ¯\_(ツ)_/¯
IIRC this also depends on if your matrices are row or column major, and if your vector is interpreted as a row or column. There's consistent conventions for both of these, but *of course* graphics and math ended up with *different* consistent conventions, making everything unnecessarily confusing.
Yes, I just commented on the same issue. In this case, A=C, so the result is the same but it is not helpful for those just learning matrix algebra to see those transforms in the reverse of the correct order.
I came to the comments just for this. I tried doing the matrix multiplication in my head and was surprised by the order Matt put the matrices in. @@SimonBuchanNz There's just one math convention for this. You take the dot product of the row vectors of the matrix on the left by the column vectors of the matrix on the right. AFAIK, this convention is international. The row or column major thing is just an implementation detail of how the matrices are stored in memory and depends on the programming language (Fortran is part of the minority that chose column major). But that doesn't change the math at all. It might merely change the performance of matrix multiplication for large enough matrices.
@@Ceelvain yeah, I meant the math convention is opposite to the graphics convention, IIRC. What I meant about the row vs column major thing was more about if you mess it up, you end up transposing the matrix which also has the effect of reversing the order. And the graphics example matrices for translation, perspective etc are transposed from the math geometry. It's all very confusing to me.
For another way of rotating without rotating, you can utilize the fact that all (simple) rotations are the composition of two reflections. This is actually still useful today, and I utilize it in my own rendering.
@@sion8 It really really really wasn't. Mode 7 enabled affine transformations of the background. The fact that you can get a pseudo 3D effect is by changing the affine transformation matrix throughout the various scan lines so that I'd different on each horizontal line, but it is in no way 3D.
@@JoQeZzZhow exactly is it different from 3D? For 3D you need to be able to divide by the z-direction. Mode 7 clearly allowed for scaling, which you could interpret as "dividing by the z-direction." Add on top of that affine transformations and that seems sufficiently 3D. You have both scaling and slanted views like in the game F-zero. The main limitation was that the hardware could only do this with 1 sprite at a time.
@@blablablablablablablablablbla none of what you said is true. Mode 7 doesn't allow for "dividing the z", it only allows for affine transfers of the background. And it doesn't do it 1 sprite at the time, it can *only* do so with the background. This is why the "background" at the end of SMW is all black, because bowser in his clowncopter is technically the background in hardware. Same for the chronotrigger clock intro. In games like f-zero and super Mario kart the affine matrix of the background was changed after each scanline through DMA, so the only "pseudo 3D" mode 7 can do is perspective things on the vertical axis. Each scan line is an affine transfer. You couldn't do a raycast engine like wolfenstein or doom using mode 7 because those subdivide the screen into vertical stripes, something mode 7 categorically cannot do. Literally none of what you said is true.
7:58 If we want to be *technically* correct, the "int" function is more of a floor function where it always rounds down i.e. int(7.999999) becomes 7 despite it obviously rounding to 8
Back in the MS-DOS days, I designed and co-wrote a powerful graphics library. It implemented rotations on bitmaps using exactly this method. It has certain advantages when working with low resolution. It slides rows or columns of pixels and rounds to a whole number of pixels; so it doesn't have any holes or need to interpolate values. Considering the capabilities of the CPU, it is very fast compared to conventional rotation of a bitmap.
P.S. Another nuance is choosing which axis (x or y) to do first, and which way to put the negative sign, depending on the angle. The code to do the actual copying of pixels was written in 16-bit 8086 assembly language. Higher level logic was written in C++.
On top of the interpolation issues, it avoids the need for expensive switch to floating point processing, and even more expensive use of trig functions. To a person, 12*11 is just as hard to do as 1.2*1.1, but they are way different to a 486 processor.
@@bobthecomputerguyYes, and the amount to shift each row, as you move away from the center, is exactly the same as line drawing algorithm. There were things in the library that didn't use floating point but did use fractions, as fixed-point integers. '486 processor? Jaw-dropping speed, with most operations done in 1 clock cycle. Multiplication takes 40. The '486 does have floating-point built-in. But they were super expensive and uncommon when this library was under development. The timings were generally using the '386 specifications.
When I was kid, the only art software available to me was MS Paint. Since only skew was available, I figured out that you could achieve a rotation by skewing three times. Glad to finally see someone else confirm it!
I've been working on a video game for a lil bit and decided to take a break by watching this video only to realize this exact thing can be used to fix an issue ive been having! Thanks Matt!
To be fair, while the "lo-fi" sprite, "hi-fi" canvas thing is ahistorical for 2D games, it *was* something that showed up in early 3D games. These games generally only had one version of each texture and did nearest neighbour scaling. So when you looked closely at something you would see the pixels of the texture or sprite, scaled/stretched/rotated as necessary to fit their place in the 3D scene. This was exacerbated if you played at a higher than default resolution.
I think it's just that their using pixel art instead of anything high res. I don't think that excludes them from putting them on a high res canvas. Even if the background is also low res. Like Celeste. It's just an art style.
The late Alan W. Paeth is credited as the progenitor of the shear rotate technique. He was a professor at the local college, and generally regarded as an awesome dude.
I've recently been studying image processing and coincidentally our professor asked us to derive the relationship between the skewing and the rotation of the the pixels! How fun.
A fun little observation, I didn't notice when you skewed initially because I had my phone laying on my bed and I was essentially correcting the skew by looking at the video off axis
I think the best part of the edge cases is the fact that it allows the "frame" to maintain its size throughout the rotation. You can see the edge which would normally represent the screen. It was a really clever and easy way to wrap the image that saves you from having to write something that doesn't save those pixels that are outside the frame. If you plan on manipulating image files then perhaps you could look for image processors in faster languages? A dedicated import for dealing with pixels might allow you to streamline the process. Personally always get a laugh out of hearing the runtimes but it might be easier on you and your computer if you plan to continue to tackle topics that are complemented nicely by your programs.
Once in the times of Windows 3.11 I was playing around in Paintbrush (the ancestor of MS Paint), it only had skew and no rotation. I wanted to rotate things, I tried skewing a few times but never really got it right. Well, this video solves the mystery of the 6yr old me ^^ I love it!
This ended up being actually helpful for me! I was creating textures for a Minecraft resource pack and I needed to rotate an existing texture. My software, being free, doesn't have a good rotation function but it does have a normal skew function so I was able to create a good looking rotation! Even though the Minecraft sword physical rotation in the video is mainly just for fun, it was actually the most helpful part for my purposes. So, thank you Matt!
I wonder if a game has implemented this skew algorithm in their pixel art. In hindsight, this looks like the right thing that should have been used and I love it.
This has *absolutely* been used by games, for decades. I know it's been used by a bunch of 2D shooters, although I'm drawing a blank on exactly which I've seen it used in.
Most retro games or games with authentic retro pixel art do something along these lines. Newer (or at least not old) games can get away with not doing pixel perfect rotation, like Terraria, a pixel art game that does not have a pixel grid (uses full-screen native resolution). Other games like Celeste have strict pixel grids, for a more authentic retro look (uses it's own simulated low-res). Old-school games did not have the luxury of choice so they often had to do something clever. In other words. Modern pixel-art games can rotate their very pixels because your monitor is high-res. So now you see less games using this. It's become a style choice.
The Game Boy Advance has built-in hardware to do affine transformations on sprites, and affine transformations are literally what he's talking about when he gets into matrices. Affine math is really nice for this on retro consoles, because you really get all in one: scaling, rotating, and shearing (and technically translation but most games wouldn't use affine for that); the rotating looks really nice for pixel art because it's equivalent to the three shears instead of a naïve rotate; and the math is surprisingly quick when using lookup tables. I'd say most 2D games in the sixteen bit era and beyond use affine transformations for their stuff, assuming they rotate sprites at all (surprisingly rare if you actually look at older games, usually at most they have one or two hand-drawn rotated images)
You can see this in action on the title screen of "Brian the lion" on the Amiga. This was an incredibly impressive large rotation for the Amiga 500 at the time, all due to the power of skewing.
I love the joy you have about this! It really makes me want to learn. Thank you so much for bringing such niche topics to our attention and explaining them to us!
Seems like something that could easily be handled by a GPU shader, so I'd imagine that the main cost would be encoding the video afterwards. Assuming the GPU used has separate components for decoding video, running shaders, and encoding video, and they can be set up as a continuous pipeline (given that effectively describes streaming a game, probably a use-case considered in its design), it would probably run close to real-time. Maybe 2x or 3x faster? Or depending on target resolution, maybe recent hardware can do far better, you'd have to ask someone who's really gone deep into the technical details about encoding performance to get an estimate with narrow error bounds.
As a graphics programmer, I have accidentally multiplied matrices together in the wrong order more times than I can count, and I did learn. The results are usually hilarious
@@mgmchenry Isn't that how the creeper in Minecraft was invented? It was at least some kind of incorrect transformation on what was supposed to be a pig.
Used this recently after watching this video when it came out, rotating low poly lightning messed up the pixels, but could skew it three times to preserve the pixels! :D
I'm literally writing a rendering engine on an 20mhz, 8-bit microprocessor for a 1-bit display right now, which will benefit from this information because it does not have advanced math functions. It can't even divide. So, more applicable to modern life than you might first expect, lol.
SethBling posted a video today (his first Minecraft video in 4 years) showing a rigid body physics engine using just a datapack (so it uses no mods). It uses block display entities, which can't be rotated, but can be skewed, so he used this skew trick to rotate them. So it's even more applicable to modern life.
@@erikkonstas That depends: in SMB1, Mario's shirt was brown, because his hair was brown. In Donkey Kong, Jumpman's hair was blue, because his shirt was blue. Having the shirt, the face, the hair, and the hat be different colors, while still having transparencies (not just being a perfect Spongebob-esque square), would have required a complete character redesign, due to hardware limitations (those limitations being 4 palettized colors; 3 + transparency) per tile, if we're talking about the NES sprites.
Excellent video Matt! Janky code, a botched (and non botched) physical example, and a very fascinating piece of maths! I really appreciate how you embrace failures while also being a stickler for doing things correct for the end product. There will probably be some niche pixel game made the next five years with a special thank you because of this vid.
9:45 Matt: makes a big deal about the fact that the order of multiplication matters for matrices Video: displays the matrices being multiplied in the wrong order
This is what I like most about you Matt. You sometimes screw up. But show it anyway. Which shows that even the most intelligent people make mistakes. But learn from it. And are not afraid of telling others.
I've been participating in a game jam recently, and one of the early resources someone posted was this exact algorithm for generating rotations of pixel art assets that preserve such things as the number of pixels and such. Very on-point lol.
interesting timing of the video, because that's how SethBling implemented his physics simulation datapack in Minecraft. There is no built in function for rotating a block entity but you can skew it on each axis
I know its arbitrary because we later see that A = C, but in your matrix multiplication shouldnt the A matrix be on the right since it is being applied to the vector first? (at least since elsewhere in the video column vectors and left hand multiplication are notated) Great video though! Never seen this specifically applied to pixels in a way that treats each pixel a constant that is just translated three times, very neat.
one big upside for this method, especially on limited/older/retro hardware, is the skew can be done entirely with integer operations, whereas trig for a rotation needs floating point. in systems that don't have dedicated floating point math hardware that can be a LOT slower.
Scalable Vector Graphics (SVG) - 1.0 Specification - W3C Recommendation 04 September 2001 - has it, chapter 7.4 Coordinate system transformations (I’m quoting this early version because I learned about this with this version). Later versions build on top of it. Worth to have a look from mathematicians point of view (😉)
I used similar math and pixel manipulation for my senior design project. We put ultrasonic sensors on a projector to detect the angle of projection between the projector and the wall it was throwing the image to. My job was to take those distances and do a bunch of linear algebra and pixel mapping to adjust the image so no matter how off skew the projector was, a square would be square and a circle would be circular, etc. super interesting project, loved the vid!
0:05 - Me, who has accidentaly rotated my entire display by 45º and have been unable to fix it: jokes on you, triangles man, you have been rotated all along!
I used to use this to rotate images in old MS Paint. You could only rotate by right angles and skew, so if you wanted to rotate by different angles, you'd have to compute manually which 3 consecutive skews you'd need.
Every rotation + translation is just a rotation about a different point (thanks to that #Some3 video about it) And every translation is just two skews, like waddling the image over (according to my brain thinking about it for 2 seconds) So does that mean skews are the one transform to rule them all?
That is actually really interesting, because yes, it seems to imply that you can perform every basic transform only using skews Edit: never mind I forgot about scaling lol
any pair of reflections will also give a rotation or translation. It's interesting, that skews decompose these into three steps, when reflections decompose them into two steps.
I just saw a similar thing done by SethBling in Minecraft, could be a fun thing to look into. He made realistic physics in Minecraft using a datapack (a datapack is similar to a mod but instead of adding extra files it just repurposes code, so it can all be done in the base game without having to download anything else)
I am so glad this showed up in my recommended videos. I love using unconventional math to do something you can do normally but even better I do pixel art for fun and I've never been able to get a good system for rotation, and now I have one. You, my friend, have earned a like and subscribe
I have a feeling that Matt deliberately messes things up. Either to show that making mistakes is ok, or to get as many things named after him as possible, or both.
It was contradictory to first say that one method solves a problem (preserves pixels), then say that mathematically both methods are the same. Surely, the whole point is that the rounding after each skew is what makes the two methods mathematically different (and what solves the pixel preservation problem).
damn, this brought back an old memory of doing this sorta thing intuitively as a kid on a school computer using ms paint, it didn't feel all that special using the skew window to rotate an image like this, that's pretty neat!
I took a computer graphics class back in the late '80s, and it was all about moving things around with transformation matrices. We did rotations, translations, reflections, and such, but I don't think we ever did skews. It's great to see the same techniques used again.
A different problem happened when scientists first tried to watch the earth rotate for 24 straight hours. They got super tired...so they called it a day.
Wow this comment has only 5 likes when I found it
@@CallMeSoumya2063 and 42 when I found it
@@YeshwanthReddy dang
82 now, will make it 83
88. That's bad, maybe?
Crazy that the day after this video released, SethBling released a video where skew rotation was actually useful and integral to his project
this^
i had a massive sense of dejavu when sethbling started talking about block entities.
Was looking for this comment!
That's incredible! Thanks for that. I'm back from that video. Insane timing considering how they're exactly the same. I think it's interesting how pointless knowledge often becomes useful to someone later
Perfect example of a Parker Rotation
Yes!
Just hopping onto what will inevitably be the top comment
At this rate, Matt will be recognized across history as the one to have most math things named after him.
What a time to be alive!
Pretty close to a Lorenz transformation, which is the hyperbolic analogue of rotation.
@@andrewkepert923 You mean the Parker Lorenz transformation?
I think this is more honest than most youtubers. I love that you kept the erroneous skew in the video, Matt. However I probably would have left it in as a blooper, and "reskewed" it correctly.
In the spirit of your honesty I'll admit, that I have only tried doing a rotation with two skew matrices, but never three. Once you learn about skew matrices for the first time, I think that trying this out seems a very intuititive thing to infer, attempt at it, fail at it, let go of with the intent to try again at a later point, and finally forget about it to never ever try it again. Thank you for setting me straight!
Finally someone calls out low-res art for defying the pixel grid!
I literally screamed in the shower while watching this, excited that someone else realized that modern low-res games try to rotate objects outside of that pixel grid was WRONG!
YES ikr!? this has always bugged me so much. it's not even hard to do in code!! i can understand when it's a web theme or something, because browsers do a lot of shenanigans, but games and pixel art have no excuse!
I'm from the 8-bit days and I approve of it these days though I hated it back in the day when there was fake pixel art made without computers or any understanding of grids and rasters and pixels.
we will correct this now. it has gotten so bad that some old games that use oversized pixels in their big arts have made me doubt if i was playing a real title or an indie game
to be fair i think it looks nicer if they double the resolution then rotate it regularly because then they get the rotation they want instead of something close to it for low res images
8:25 code review:
1) int() doesn't round to a whole number, it truncates. That may actually be what you want, since it means you only move the pixel right or left if it passes the full pixel width threshold.
2) You're using mod to clip the image, which creates the weird wraparound. If the width needs to be accounted for at all, you probably would be better off by doing a pure clip. Drop the information for pixels outside of the range.
3) Put the image into a numpy array, use linear algebra to apply the transform to the entire image instead of calculating a new location for each pixel.
Re 1) it rounds towards zero, which is just a different wording for truncation. So both of you are right. There are a lot of possible ways to round, and towards zero is just one possibility. Granted, it's not a "fair" way to round and it will look a bit glitchy especially when used on a low-resolution image.
2) If the rotation is progressive, you'll want the wraparound to preserve information. If you clip/crop, you'll need to do it from the original, not the most recent.
@@uNiels_Heart I guess a maths communicator would see "round toward zero"/truncate as something else than regular rounding.
@@hoejI guess you have a point there.
Matt was using Python there (which I gathered the programming context from, as opposed to pure maths) and I suspect that he's aware that `int()` doesn't do traditional rounding, and I suspect he just didn't want to complicate things when talking about it.
As it's Python, he could just substitute `round()` for `int()` to get a more traditional rounding. However, he might be restricted to Python 2 (or he might not be aware that it has changed for the better in Python 3), as in that older version of Python `round()`, while rounding, still returns a floating point number, not an integer.
For number 3, doing the calc for each pixel MUST be done at some point. Python just hides that from you.
Back in oldschool microsoft paint, you weren't able to rotate stuff, so instead i eyeballed it with 2 skews. Even back then i recall thinking there's probably a perfect way to do it, i just didn't know it. So yea i'm prety much the exact person this video is made for.
How do yo skew in og paint?
I like how your brain works. Is it a special kind of brain?
P.s. I used old-school paint to make animations when I was in "basisschool" (don't know what that translates to, but I was around 8-10 I think). The old ms paint had multiple ctrl+z's which could also be reversed. Iirc we had 3 or 4 different frames. My friends quickly took the trick and made their own 😀.
At 11 yo I made a scifi shooter, choose your own adventure game, using powerpoint 😅. Using hundreds, maybe thousands of pages. So you could have multiple paths to follow. There was even a moment when a monster showed up in a corridor where it shot a projectile that quickly got closer by automating the page turns. And if you didn't find the gun or you didn't dodge or shoot back in time, you'd die. I think it was a really cool idea and was, within the constraint, a pretty good game😊.
@@naga540 ctrl + W
Haha I did exactly that too back when I was a kid and didn't know of any better editing software. And yep, I used only 2 skews too! Learning that a third skew makes the rotation perfect would've blown my mind back then.
@@tiarkrezarYep! I did the same.
I'm a retro game dev, so this technique is right up my alley. I'm shocked I didn't know about this already. Thanks for making this video!
I actually wrote my own matrix math transformation routines for translation/rotation/scale in C for SGI workstations/supercomputer in C 30 years ago, for an artificial life virtual reality game that I was developing for SGI. I had found the routines in GL (which soon evolved into OpenGL) at the time to be inadequate for real-time virtual reality, and my own routines solved my problems.
Back then I was all about VR on the highest-end systems possible (I had an Onyx RealityEngine2 as my personal computer, provided by the company I worked for, for several months back then), but anymore I am far more interested in developing for the microcomputers and consoles that came out from the 70s until the early 90s and the handhelds up until the early 2000s.
I'm familiar with OpenGL, I'm used to GL just being a shorthand for OpenGL, in this context, what was GL? Just a predecessor to OpenGL, or was it an API similar to OpenGL without the "Open" standard?
@@AssemblyCSharp Yep, like I said, IRIS GL (we called it just GL, much like you're saying folks do today with OpenGL--likely a continuity thing) evolved into OpenGL. It both was a predecessor to OpenGL with a similar API and was a proprietary standard. Sadly, the windowing and mouse and keyboard aspects of the API were removed from OpenGL. IRIS GL first was used with the superior NeWS, which SGI IRIX used for windowing, and then evolved to support the X Window System, when IRIX switched to that. So, one API could be used to create entire 3D windowed apps for deployment on two different windowing systems.
There were some other fundamental difference between IRIS GL and OpenGL, including adding a software rendering pipeline so that systems with differing levels of hardware pipeline support (including none at all) could use the exact same OpenGL library with different 3D graphics hardware drivers. However, most of the API, barring some important exceptions, was just changed via name differences.
You can use this information to rotate images in Microsoft Paint.
Step 1: Horizontal skew by theta/2 degrees.
Step 2: Vertical skew by -arctan(sin theta).
Step 3: Same as Step 1.
Bug workaround: you have to paste a minus sign into the text field, because you can't type it in
Apparently the people behind MS Paint didn't know this little trick. But who's surprised. Even today windows can't even sort filenames alphabetically.
According to Windows "filename, filename 2, filename 3" in alphabetical order is "filename 2, filename 3, filename".
@@awebmateIt may be because space is before any of the letters/numbers in ASCII
@@awebmatethis only applies to files with an extension since the dot “.” is at 0x2e in ASCII, which is after the space (0x20). «Fi» gets sorted before «Fi 2», but «Fi.txt» after «Fi 2.txt»
you can also flip, skew by a positive amount, and flip again
the worst is when 10 goes between 1 and 2@@awebmate
Wait, this is the math that Sethbling used in his new Rigid Body Minecraft Datapack. Because Minecraft’s native Block Display Entity only does skews, stretches, and translations. So he could only do rotations via skews
Was looking for a comment about this lmao
@@RowsieFoxsame
Haha we watch the same people! Classic :)
This was exactly what I thought of when I saw this video!
I was thinking the same thing, what timing!
I remember almost discovering this myself as a kid playing with old MS Paint. There weren't free rotation functions in Paint, but there were skew functions. I figured out that I could "rotate" images by skewing them equally horizontally and vertically, and then fiddle about with the aspect ratio a bit. Never thought of applying a third skew!
That’s where I first encountered this. I’d thought about presenting it at MathsJam, but Andrew Taylor did a much better job of it at last year’s conference than I could have, and that’s presumably where Matt encountered it.
Same but mine was Photoshop lol
Interesting that MS Paint either didn’t see a purpose or realize they have the capability to rotate images but did and could incorporate skew. When there’s so little utility for skew, and by including skew it means they could have rotated. Makes me wonder how computers rotate images these days at its most basic. Maybe it is that skew function
@@monhi64The real problem is that MS Paint hadn't been substantially updated between the 90s and last year
What versions of Paint u people are using? You still cant rotate by any angle..
Article from 1986 has always stuck with me. Multiple single direction skews of a picture quickly devolves into chaos with intermittent return of part of original pattern (Chaos, Scientific American 54:12 (1986) 46-57). Phenomenon related to Poincaré Recurrence. Potential video topic.
That seems really nice, I hope he does take the idea!
Matt saying "look up matrices if you don't already know them" reminds me that a great deal of us have already heard of what he's on about but, heck, we love him for the joy he brings to the material we... ought to know better than we do...
hi Urist!
Love the matrix movie 😅
I love when an educator for higher level topics makes the reasonable assumption that viewers will already understand the underlying topic to at least some degree. The number of times I've had to skip over a "what is binary?" section in a video about semi-advanced software or computer hardware stuff is way too high.
Binary Coded Decimal? Ok, that's a valid topic to cover since there are multiple schemes to make it happen.
Floating point representation? Sure, there are a whole host of ways to do that, and which one you choose alters the result of your calculation.
Both of those require some knowledge of what binary is, but if I'm to the point that I'm learning about those topics I should have the base level down already anyways.
It's also nice, after spending all day wrestling with maths we barely understand, to see fun applications for maths which is very familiar
But I don't feel like sitting 4 movies about a war between humans and machines to understand this stuff.
Next video: "Someone improved my skew code by 40,832,277,770%"
Misinformation.
@@piano01 chill it's a joke
ROFL
@@piano01 r/woooosh
Next thing you know, there's a video from Matt that describes his code being improved by 40,832,277,770%
I was chipping away at a tool for rendering the signed distance functions of shapes in Blender when I ran into trouble with rotation. Little did I know, this treasure of a video had just been uploaded two hours ago and recommended to me by pure chance! It's exactly what I needed, my shapes are rotating perfectly, and you've definitely earned my subscription with this. Thank you so much!
Couldn’t you just use the Mapping node on the input vector?
@@palmberry5576Probably, but having my own solution means I'll be able to more easily do stuff like work with rendering 4D visuals in the future.
Kind of glad to see this, I remember when I was a kid I would try to do this in MSPaint, because you couldn't do non-90-degree rotates in MSPaint, but I could get very close to getting "clean rotations" with skew operations. I could never get it to be quite exactly right looking, though.
14:25 The hand waving rationalization of "This is fine." after successful completion of a perfect "Parker Rotation."
Funny he didn't realize it's not a square.
@@TheBlacktom It's as square a square could be given the imprecision
@@nsf001-3the point they’re making is that the physical model on the table (where he mistakenly inverted one of the skews) is literally not a rotation
The virtual Lego model that he overlaid in post is correct. Yes, not a “perfect” square, but very close because he performed a proper “rotation” (via 3 specific skews)
But if you invert the direction of any one skew, you will get something that looks nothing like a square (see the physical table model), because those particular 3 skews no longer equal a rotation
And funny he, a mathematician, got this wrong. 3:51
Only about half way through, so maybe you mention this later, but one huge advantage of this for early/slow computers is that skewing like you did with the plumber can be done entirely with integer math, meaning that there is no need for a floating point unit in the processor. We take this for granted a lot with modern hardware, but so many older computers (let alone microprocessors) couldn't do floating point math at all.
But don't you still need floating point for calculating sin and tan?
@@vibaj16 The point of the skew method highlighted here is that it doesn't use trigonometry, just basic integer math.
@@ex-nerd It still uses trigonometry to figure out the skews.
@@vibaj16 you can precompute the specific skews that you need.
@@lowcostfish then you could just precompute the entire rotation
Oh, Mr.Parker Square. At 9:37 you note they have to be multiplied in the correct order, but then mistakenly ordered them ABC instead of CBA! In the end it doesn't change the result because C=A. However when you have the position vector on the right, the matrices right to left are the steps applied to the position. So your example shows the last skew (C) being done first, and the first skew (A) being done last.
Yeah, what I was thinking since if the operations happen A -> B -> C then it would look like C( B ( Ax) ). Thanks for pointing it out, I went in the comments to check if someone else mentioned it or if I was the one who was wrong.
Also, somehow the bot account that copied your comment got more likes than you ¯\_(ツ)_/¯
IIRC this also depends on if your matrices are row or column major, and if your vector is interpreted as a row or column.
There's consistent conventions for both of these, but *of course* graphics and math ended up with *different* consistent conventions, making everything unnecessarily confusing.
Yes, I just commented on the same issue. In this case, A=C, so the result is the same but it is not helpful for those just learning matrix algebra to see those transforms in the reverse of the correct order.
I came to the comments just for this. I tried doing the matrix multiplication in my head and was surprised by the order Matt put the matrices in.
@@SimonBuchanNz There's just one math convention for this. You take the dot product of the row vectors of the matrix on the left by the column vectors of the matrix on the right. AFAIK, this convention is international.
The row or column major thing is just an implementation detail of how the matrices are stored in memory and depends on the programming language (Fortran is part of the minority that chose column major). But that doesn't change the math at all. It might merely change the performance of matrix multiplication for large enough matrices.
@@Ceelvain yeah, I meant the math convention is opposite to the graphics convention, IIRC. What I meant about the row vs column major thing was more about if you mess it up, you end up transposing the matrix which also has the effect of reversing the order. And the graphics example matrices for translation, perspective etc are transposed from the math geometry.
It's all very confusing to me.
For another way of rotating without rotating, you can utilize the fact that all (simple) rotations are the composition of two reflections. This is actually still useful today, and I utilize it in my own rendering.
6:24 "my code is not so much a blender as it is a juice loosener" 🤣🤣
I nearly laughed out loud inappropriately at this one haha
It's a Simpsons reference. Dr. Nick and Troy McLure do an infomercial for the Juice Loosener, a crappy juicer.
@@russell2952"You got all that juice from just one bag of oranges?!"
I believe this is how Mode 7 on the SNES works. it achieves rotation by skewing various amounts in both the x and y axes.
It could have help to sell the overall visual, but Mode 7 was just very rudimentary 3D rendering.
@@sion8 It really really really wasn't. Mode 7 enabled affine transformations of the background. The fact that you can get a pseudo 3D effect is by changing the affine transformation matrix throughout the various scan lines so that I'd different on each horizontal line, but it is in no way 3D.
@@JoQeZzZhow exactly is it different from 3D? For 3D you need to be able to divide by the z-direction. Mode 7 clearly allowed for scaling, which you could interpret as "dividing by the z-direction." Add on top of that affine transformations and that seems sufficiently 3D. You have both scaling and slanted views like in the game F-zero. The main limitation was that the hardware could only do this with 1 sprite at a time.
@@blablablablablablablablablbla none of what you said is true. Mode 7 doesn't allow for "dividing the z", it only allows for affine transfers of the background. And it doesn't do it 1 sprite at the time, it can *only* do so with the background. This is why the "background" at the end of SMW is all black, because bowser in his clowncopter is technically the background in hardware. Same for the chronotrigger clock intro.
In games like f-zero and super Mario kart the affine matrix of the background was changed after each scanline through DMA, so the only "pseudo 3D" mode 7 can do is perspective things on the vertical axis. Each scan line is an affine transfer. You couldn't do a raycast engine like wolfenstein or doom using mode 7 because those subdivide the screen into vertical stripes, something mode 7 categorically cannot do.
Literally none of what you said is true.
@@blablablablablablablablablbla You can use mode 7 as part of implementing a simple 3d effect, but mode 7 itself isn't a 3d effect.
Accidently learned this with CSS
Underrated comment
Accidently learned CSS with this
Accidentally learned with this CSS
Accidentally CCSed with this learn
But, can you center a div?
7:58 If we want to be *technically* correct, the "int" function is more of a floor function where it always rounds down i.e. int(7.999999) becomes 7 despite it obviously rounding to 8
Back in the MS-DOS days, I designed and co-wrote a powerful graphics library.
It implemented rotations on bitmaps using exactly this method.
It has certain advantages when working with low resolution. It slides rows or columns of pixels and rounds to a whole number of pixels; so it doesn't have any holes or need to interpolate values. Considering the capabilities of the CPU, it is very fast compared to conventional rotation of a bitmap.
P.S. Another nuance is choosing which axis (x or y) to do first, and which way to put the negative sign, depending on the angle.
The code to do the actual copying of pixels was written in 16-bit 8086 assembly language. Higher level logic was written in C++.
Nice!
On top of the interpolation issues, it avoids the need for expensive switch to floating point processing, and even more expensive use of trig functions. To a person, 12*11 is just as hard to do as 1.2*1.1, but they are way different to a 486 processor.
@@bobthecomputerguyYes, and the amount to shift each row, as you move away from the center, is exactly the same as line drawing algorithm.
There were things in the library that didn't use floating point but did use fractions, as fixed-point integers.
'486 processor? Jaw-dropping speed, with most operations done in 1 clock cycle. Multiplication takes 40. The '486 does have floating-point built-in.
But they were super expensive and uncommon when this library was under development. The timings were generally using the '386 specifications.
When I was kid, the only art software available to me was MS Paint. Since only skew was available, I figured out that you could achieve a rotation by skewing three times.
Glad to finally see someone else confirm it!
Im still using Paint lol
"Juice Loosener"
Absolutely champion XD
The whisper quiet rotation.
IT'S WHISPER QUIET!
We got all that juice from just one bag of oranges?
There's a better way?
I've been working on a video game for a lil bit and decided to take a break by watching this video only to realize this exact thing can be used to fix an issue ive been having! Thanks Matt!
1:10 YES!!! Finally somebody says it, it annoys me so much when "lo-fi" sprites are rotated in a very high resolution "canvas"
To be fair, while the "lo-fi" sprite, "hi-fi" canvas thing is ahistorical for 2D games, it *was* something that showed up in early 3D games.
These games generally only had one version of each texture and did nearest neighbour scaling. So when you looked closely at something you would see the pixels of the texture or sprite, scaled/stretched/rotated as necessary to fit their place in the 3D scene. This was exacerbated if you played at a higher than default resolution.
That is not the problem though The problem is rotating on a low resolution canvas as that results in the missarrangement
I think it's just that their using pixel art instead of anything high res. I don't think that excludes them from putting them on a high res canvas. Even if the background is also low res. Like Celeste. It's just an art style.
The late Alan W. Paeth is credited as the progenitor of the shear rotate technique. He was a professor at the local college, and generally regarded as an awesome dude.
The physical pixel illustration made it very practical and approachable. Great idea!
Yes, but he should have called them phyxels. Brand new word. :)
I've recently been studying image processing and coincidentally our professor asked us to derive the relationship between the skewing and the rotation of the the pixels! How fun.
14:29 Parker skew "sharper than ever before" "I think that worked well enough"
A fun little observation, I didn't notice when you skewed initially because I had my phone laying on my bed and I was essentially correcting the skew by looking at the video off axis
I think the best part of the edge cases is the fact that it allows the "frame" to maintain its size throughout the rotation. You can see the edge which would normally represent the screen. It was a really clever and easy way to wrap the image that saves you from having to write something that doesn't save those pixels that are outside the frame. If you plan on manipulating image files then perhaps you could look for image processors in faster languages? A dedicated import for dealing with pixels might allow you to streamline the process. Personally always get a laugh out of hearing the runtimes but it might be easier on you and your computer if you plan to continue to tackle topics that are complemented nicely by your programs.
Once in the times of Windows 3.11 I was playing around in Paintbrush (the ancestor of MS Paint), it only had skew and no rotation. I wanted to rotate things, I tried skewing a few times but never really got it right. Well, this video solves the mystery of the 6yr old me ^^ I love it!
Brilliant. No other word for it. This video will still be doing the rounds forever.
honestly. I expected to see you do this in an excel spreadsheet!
This ended up being actually helpful for me! I was creating textures for a Minecraft resource pack and I needed to rotate an existing texture. My software, being free, doesn't have a good rotation function but it does have a normal skew function so I was able to create a good looking rotation! Even though the Minecraft sword physical rotation in the video is mainly just for fun, it was actually the most helpful part for my purposes. So, thank you Matt!
Matt has, hands down, the best theme music in mathematics.
rip to me thinking it was the camera which was rotated
I wonder if a game has implemented this skew algorithm in their pixel art. In hindsight, this looks like the right thing that should have been used and I love it.
This has *absolutely* been used by games, for decades. I know it's been used by a bunch of 2D shooters, although I'm drawing a blank on exactly which I've seen it used in.
This has been used for decades on 16-bit consoles.
Surely, right?
Most retro games or games with authentic retro pixel art do something along these lines. Newer (or at least not old) games can get away with not doing pixel perfect rotation, like Terraria, a pixel art game that does not have a pixel grid (uses full-screen native resolution). Other games like Celeste have strict pixel grids, for a more authentic retro look (uses it's own simulated low-res). Old-school games did not have the luxury of choice so they often had to do something clever.
In other words. Modern pixel-art games can rotate their very pixels because your monitor is high-res. So now you see less games using this. It's become a style choice.
The Game Boy Advance has built-in hardware to do affine transformations on sprites, and affine transformations are literally what he's talking about when he gets into matrices. Affine math is really nice for this on retro consoles, because you really get all in one: scaling, rotating, and shearing (and technically translation but most games wouldn't use affine for that); the rotating looks really nice for pixel art because it's equivalent to the three shears instead of a naïve rotate; and the math is surprisingly quick when using lookup tables. I'd say most 2D games in the sixteen bit era and beyond use affine transformations for their stuff, assuming they rotate sprites at all (surprisingly rare if you actually look at older games, usually at most they have one or two hand-drawn rotated images)
I left on 0:26, that’s information I needed when i click this video, thank u
I stayed a little longer haha
You can see this in action on the title screen of "Brian the lion" on the Amiga. This was an incredibly impressive large rotation for the Amiga 500 at the time, all due to the power of skewing.
funny enough the developer was "Reflections" not "Skews"
2:43 Thank you for giving the appropriate double take look, this is the icing on the cake I subscribed for
Two videos this week, and pi day next? It really is Matt's Year of RUclips.
thanks for this! transformations are the next unit in my geo classes...definitely will post this to my students!
The Parker rotation 😂
15:13 MatPat, What have you done about the name, I like it.
IIRC, I used this technique for the little tool I made for Project Zomboid, that turned flat textures into the game's native isometric format.
I love the joy you have about this! It really makes me want to learn. Thank you so much for bringing such niche topics to our attention and explaining them to us!
parker rotation of the sword
IKR, but it's great, we all ought to shamelessly give it a go, and keep the fails in.
I remember watching a tutorial on how to rotate some in Paint using the skew tool and it blew my mind
Next video: Someone improved my code by 3hours /3 milliseconds = 360,000,000 %
Lol, tbf if you use matrix multiplication, this would have been so much faster
3 h / 3 ms = 360 000 000 °/.
360 000 000° rotation = ∞ improvement.
Math confirmed
Seems like something that could easily be handled by a GPU shader, so I'd imagine that the main cost would be encoding the video afterwards. Assuming the GPU used has separate components for decoding video, running shaders, and encoding video, and they can be set up as a continuous pipeline (given that effectively describes streaming a game, probably a use-case considered in its design), it would probably run close to real-time. Maybe 2x or 3x faster? Or depending on target resolution, maybe recent hardware can do far better, you'd have to ask someone who's really gone deep into the technical details about encoding performance to get an estimate with narrow error bounds.
Matt just *_skewing_* around with maths is always fun, no matter how obsolete ^^
More teachers should adopt the slogan "You'll have to trust me, and / or learn. 10:03
I had to rewind it cuz I thought he said you'll have to trust me *or* you'll learn. 😊 Like a more polite version of f around and find out lol 😂
As a graphics programmer, I have accidentally multiplied matrices together in the wrong order more times than I can count, and I did learn. The results are usually hilarious
Nullius in verba
@@mgmchenry Isn't that how the creeper in Minecraft was invented? It was at least some kind of incorrect transformation on what was supposed to be a pig.
@@DFPercush what happened with the creeper is that notch got the order of (x,y,z) wrong I believe
Used this recently after watching this video when it came out, rotating low poly lightning messed up the pixels, but could skew it three times to preserve the pixels! :D
"I'm going to keep doing this until my laptop melts, which it's holding up pre--"
Whether by editing or perfect timing, perfectly cut 😂
This was helpful. I have often wondered how software rotates digital images without losing pixels, and now I know.
I'm literally writing a rendering engine on an 20mhz, 8-bit microprocessor for a 1-bit display right now, which will benefit from this information because it does not have advanced math functions. It can't even divide. So, more applicable to modern life than you might first expect, lol.
SethBling posted a video today (his first Minecraft video in 4 years) showing a rigid body physics engine using just a datapack (so it uses no mods). It uses block display entities, which can't be rotated, but can be skewed, so he used this skew trick to rotate them. So it's even more applicable to modern life.
I love the write this in your workbooks now!
Matt, watch out, Nintendo is going to take down this video because you used their 43 year old character without permission
Nintendo is the only law firm with a games division.
@@nicholasvinenIf you don't mind, I'm stealing that one!
@@thedrnailsguythat "joke" has some years of existance
LOL except that's not the real 8-bit Mario, his shirt wasn't blue...
@@erikkonstas That depends: in SMB1, Mario's shirt was brown, because his hair was brown. In Donkey Kong, Jumpman's hair was blue, because his shirt was blue. Having the shirt, the face, the hair, and the hat be different colors, while still having transparencies (not just being a perfect Spongebob-esque square), would have required a complete character redesign, due to hardware limitations (those limitations being 4 palettized colors; 3 + transparency) per tile, if we're talking about the NES sprites.
Excellent video Matt! Janky code, a botched (and non botched) physical example, and a very fascinating piece of maths!
I really appreciate how you embrace failures while also being a stickler for doing things correct for the end product.
There will probably be some niche pixel game made the next five years with a special thank you because of this vid.
9:45 Matt: makes a big deal about the fact that the order of multiplication matters for matrices
Video: displays the matrices being multiplied in the wrong order
This is what I like most about you Matt.
You sometimes screw up. But show it anyway.
Which shows that even the most intelligent people make mistakes.
But learn from it. And are not afraid of telling others.
Can we rotate this image?
We can rotate it at home!
The rotation at home:
I've been participating in a game jam recently, and one of the early resources someone posted was this exact algorithm for generating rotations of pixel art assets that preserve such things as the number of pixels and such. Very on-point lol.
5:12
He has been italicized.
interesting timing of the video, because that's how SethBling implemented his physics simulation datapack in Minecraft. There is no built in function for rotating a block entity but you can skew it on each axis
I know its arbitrary because we later see that A = C, but in your matrix multiplication shouldnt the A matrix be on the right since it is being applied to the vector first? (at least since elsewhere in the video column vectors and left hand multiplication are notated)
Great video though! Never seen this specifically applied to pixels in a way that treats each pixel a constant that is just translated three times, very neat.
one big upside for this method, especially on limited/older/retro hardware, is the skew can be done entirely with integer operations, whereas trig for a rotation needs floating point. in systems that don't have dedicated floating point math hardware that can be a LOT slower.
5:20 bro's in _italics_
Well “skew” is definitely one of those words that start to sound strange after hearing it many times in a short period of time.
Matt, we understand how you rotated the video, but how did you rotate the audio to make it so seamlessly match?
Scalable Vector Graphics (SVG) - 1.0 Specification - W3C Recommendation 04 September 2001 - has it, chapter 7.4 Coordinate system transformations (I’m quoting this early version because I learned about this with this version). Later versions build on top of it. Worth to have a look from mathematicians point of view (😉)
2:24 Skewed Mario giving us the middle finger should be a meme!
I used similar math and pixel manipulation for my senior design project. We put ultrasonic sensors on a projector to detect the angle of projection between the projector and the wall it was throwing the image to. My job was to take those distances and do a bunch of linear algebra and pixel mapping to adjust the image so no matter how off skew the projector was, a square would be square and a circle would be circular, etc. super interesting project, loved the vid!
0:05 - Me, who has accidentaly rotated my entire display by 45º and have been unable to fix it: jokes on you, triangles man, you have been rotated all along!
I used to use this to rotate images in old MS Paint.
You could only rotate by right angles and skew, so if you wanted to rotate by different angles, you'd have to compute manually which 3 consecutive skews you'd need.
"This video is claimed by the Nintendo company"
time to make his horrible code 2384975623847% faster again
Underrated.
Thanks for leaving in past Matt and explaining. I'm sure some things do just die in post, but that was a great one to have remain.
Every rotation + translation is just a rotation about a different point (thanks to that #Some3 video about it)
And every translation is just two skews, like waddling the image over (according to my brain thinking about it for 2 seconds)
So does that mean skews are the one transform to rule them all?
That is actually really interesting, because yes, it seems to imply that you can perform every basic transform only using skews
Edit: never mind I forgot about scaling lol
@@Scum42 @Scum42 ha true... dang. The missing missing link 🙈
any pair of reflections will also give a rotation or translation.
It's interesting, that skews decompose these into three steps, when reflections decompose them into two steps.
I just saw a similar thing done by SethBling in Minecraft, could be a fun thing to look into. He made realistic physics in Minecraft using a datapack (a datapack is similar to a mod but instead of adding extra files it just repurposes code, so it can all be done in the base game without having to download anything else)
3:52 Mario also looks like he's flipping us off!
I am so glad this showed up in my recommended videos. I love using unconventional math to do something you can do normally but even better I do pixel art for fun and I've never been able to get a good system for rotation, and now I have one. You, my friend, have earned a like and subscribe
then at the end of the sketch, the camera zooms out and you were oriented at 45° for the whole video (classic Penn and Teller bit)
In high school, I loved manually doing graphic transformations. I found it fascinating.
I have a feeling that Matt deliberately messes things up. Either to show that making mistakes is ok, or to get as many things named after him as possible, or both.
this reminds me of how people skew in flipaclip animation app before the feature got added by using rotation and size changes
8:06 tiny correction, int does not round, it merely cuts off the values after the decimal point eg. Int(3.99) = 3
that's a type of rounding 🙏
At 04:25, looks like Mario is releasing digested food 😊
It was contradictory to first say that one method solves a problem (preserves pixels), then say that mathematically both methods are the same. Surely, the whole point is that the rounding after each skew is what makes the two methods mathematically different (and what solves the pixel preservation problem).
I'm gonna write my Linear Algebra 1 exam next week and it's nice to see the usages of the stuff we learn! Wish me luck :)
Marvellous how thoroughly you fucked that up. Mad props to you
damn, this brought back an old memory of doing this sorta thing intuitively as a kid on a school computer using ms paint, it didn't feel all that special using the skew window to rotate an image like this, that's pretty neat!
Funny that this principle is exactly what SethBling using in his just released video!
I took a computer graphics class back in the late '80s, and it was all about moving things around with transformation matrices. We did rotations, translations, reflections, and such, but I don't think we ever did skews. It's great to see the same techniques used again.
I have to point out that, while Matt didn't write an excellent image manipulation program, he at least wrote affine image manipulation program.
That's pretty interesting that 3 simple linear transforms can combine to create basically sin/cos functions.