First of all: A comprehensive outline of the following idea without any mathematical formulas but with detailed explanations can be found here on on 2d-codes.co.uk or, if you happen to speak danish here on http://qrkoder.internet.dk/.

## Teaser

The answer below works (with some modifications). Please click the image to see how the QR code projection looks, when the image is rotated:

And everything here can be used for real applications. At the end of this answer you'll find images of the QR code printed on a real cylinder. But applications are not restricted to this. You can easily adapt the approach to keep you up all night

## The theory

Murta, you wrote

I thought that this would be a nice thing to do with Mathematica, to try to figure out how I could print the QR code into some way that when attached to the cylindrical form, it would be like a normal square, at least **at some angles**.

Exactly the viewpoint, more specifically the perspective projection, is crucial to determine how you have to transform your label so that it is squared again. Let me give an example where I drew something onto a paper-roll which obviously has nothing to do with the transformation used in bills answer:

If I now inspect this roll from a specific viewpoint it looks like a QR code should be recognized again:

The question is what happens here. The theory behind it is pretty easy and the good thing is, it explains what you have to do from *any* (meaningful) viewpoint. Let's use a simple cylinder graphic as example to explain what I mean

```
ParametricPlot3D[{Cos[u], v, Sin[u]}, {u, 0, 2 Pi}, {v, 0, 10},
Boxed -> False, Axes -> False, ViewAngle -> .1,
Epilog :> {FaceForm[None], EdgeForm[Red],
Rectangle[{.4, .4}, {.7, .7}]}]
```

When you finally see the image on your screen, two transformations took place. First, `ParametricPlot3D`

used my formula to transform from cylinder coordinates `{u,v}`

into 3D Cartesian coordinates `{x,y,z}`

. This transformation of the `{u,v}`

plane can easily be simulated by sampling it with `Table`

, doing the transformation to 3D by yourself and drawing lines

```
Graphics3D[{Line[#], Line[Transpose@#]} &@
Table[{Cos[u], v, Sin[u]}, {u, 0, 2 Pi, 2 Pi/20.}, {v, 0, 10, .5}]
]
```

The next thing that happens is often taken for granted: The transformation of 3D points onto your final image plane you are seeing on the screen. This final `ViewMatrix`

can (with some work) be extracted from a *Mathematica* graphics. It should work with `AbsoluteOptions[gr3d, ViewMatrix]`

but it doesn't. Fortunately, Heike posted an answer how to do this.

## Let's do it

OK, to say it with the words of Dr. Faust "Grau, teurer Freund, ist alle Theorie, und grün des Lebens goldner Baum". After trying it I noticed that the last two paragraphs of my first version are not necessary.

Let us first create a 3D plot of a cylinder, where we extract the matrices for viewing and keep them up to date even when we rotate the view.

```
{t, p} = {TransformationMatrix[
RescalingTransform[{{-2, 2}, {-2, 2}, {-3/2, 5/2}}]],
{{1, 0, 0, 0}, {0, 0, 1, 0}, {0, 1, 0, 0}, {0, 0, 0, 1}}};
ParametricPlot3D[{Cos[u], v, Sin[u]}, {u, 0, 2 Pi}, {v, 0, 10},
Boxed -> False, Axes -> False, ViewMatrix -> Dynamic[{t, p}]]
```

Now `{t,p}`

always contain the current values of our projection. If you read in the documentation to `ViewMatrix`

, you see that

The transformation matrix t is applied to the list {x,y,z,1} for each point. The projection matrix p is applied to the resulting vectors from the transformation.

and

If the result is {tx,ty,tz,tw}, then the screen coordinates for each point are taken to be given by {tx,ty}/tw.

Therefore, we can easily construct a function from `{u,v}`

to screen-coordinates `{x,y}`

```
With[{m1 = t, m2 = p},
projection[{u_, v_}] = {#1, #2}/#4 & @@ (m2.m1.{Cos[u], v, Sin[u], 1})
]
```

Let's test wether our projection is correct. Rotate the cylinder graphics so that you have a nice view and execute the `projection`

definition again.

```
Graphics[{Line /@ #, Line /@ Transpose[#]} &@
Table[projection[{u, v}], {u, 0, 2 Pi, .1}, {v, 0, 10}],
Axes -> True, PlotRange -> {{0, 1}, {0, 1}}]
```

Please note that this is no 3D graphics. We transform directly from `{u,v}`

cylinder to `{x,y}`

screen-coordinates. Those screen-coordinates are always in the range [0,1] for x and y.

Now comes the important step: This transformation can directly be used with `TextureCoordinateFunction`

because this function provides you with `{u,v}`

values and wants to know `{x,y}`

texture positions. The only thing I do is, that I scale and translate the texture coordinates a bit so that the QR code is completely visible in the center of the image:

```
tex = Texture[Import["http://i.stack.imgur.com/FHvNV.png"]];
ParametricPlot3D[{Cos[u], v, Sin[u]}, {u, 0, 2 Pi}, {v, 0, 10},
Boxed -> False, Axes -> False, ViewMatrix -> Dynamic[{t, p}],
PlotStyle -> tex, TextureCoordinateScaling -> False,
Lighting -> "Neutral",
TextureCoordinateFunction -> (2 projection[{#4, #5}] + {1/2, 1/2} &)
]
```

Don't rotate this graphics directly, because although it uses specific settings for `ViewMatrix`

, it jumps directly to default settings when rotated the first time. Instead, copy our original cylinder image to a new notebook and rotate this. The `Dymamic`

's will make, that both graphics are rotated.

Conclusion: When I use the following viewpoint to initialize the view point

and then evaluate the `projection`

definition line again and recreate the textured cylinder, I get

which looks as if I just added a QR code layer to the image. Rotating and scaling reveals that it is specific texture projection instead

## Going into real life

When you want to create a printable version of this, you could do the following. Interpolate the QR code image and use the same projection function you used in the texture (note that I used a factor 3 and `{1/3,0}`

inside `ipf`

here. You use whatever you used as texture):

```
qr = RemoveAlphaChannel@
ColorConvert[Import["http://i.stack.imgur.com/FHvNV.png"],
"Grayscale"];
ip = ListInterpolation[
Reverse[ImageData[qr, "Real"]], {{0, 1}, {0, 1}}];
ipf[{x_, y_}] := ip[Mod[y, 1], Mod[x, 1]];
With[{n = 511.},
Image@Reverse@
Table[ipf[3 projection[{u, v}] + {1/3, 0}], {u, -Pi, Pi, 2 Pi/n},
{v, 0, 10, 2 Pi/n}]
]
```

Please note the `Reverse`

since image matrices are always reversed and additionally, that I create now the image matrix for u from `[-Pi,Pi]`

. This was a bug in the last version which created the back-side of the cylinder. Therefore, the perspective was not correct in the final result.

This can now be glued around a cylinder (after printing it with the appropriate height) and with the corrected print version, the result looks

awesome! Here from another perspective

There is a reddit thread about this

– Jacob Akkerboom – 2013-11-27T14:04:33.7972

This is a nice question. I believe guys doing street art in 3D kind of have the same problems ;-)

– halirutan – 2013-06-02T03:10:36.173@halirutan Yes! Maybe they could use MMA too! :) – Murta – 2013-06-02T03:13:27.357

1Related question. – J. M.'s ennui – 2013-06-02T03:13:36.263

1

Another, arguably related question: (5676)

– Oleksandr R. – 2013-06-02T05:25:42.940There are technologies that can encode different images for different angles. I think you can apply them locally, so you can see correct QR squares for a reasonable angle range.

– Silvia – 2013-06-02T06:12:06.5931"I believe that it's equivalent to projecting the texture onto the surface, and then using the projected image." I think you want the inverse: you desire the image such that when it has been projected onto the cylinder, it look square. This should be the inverse of the mapping that takes the square to the cylinder. – bill s – 2013-06-02T09:01:01.947

5

I used Mathematica to find this http://i.stack.imgur.com/fZJe0.jpg

– Dr. belisarius – 2013-06-02T13:53:28.667@belisarius hehe.. Too simple, no fun!.. – Murta – 2013-06-02T14:08:54.517

1

I think it falls into Mirror Anamorphosis. Also see this demo: Cylindrical Mirror Anamorphosis - not exactly what you need but in the same realm.

– Vitaliy Kaurov – 2013-06-02T17:34:23.6938You should marry a package designer. My wife's response to this was, "The shopper's gotta work out just the right angle to photograph the code this way. Why not just introduce a little plastic structural element to make a flat enough surface for it? Something you can clip onto any shopping cart handle?" – Fordi – 2013-06-03T01:24:15.417

1

Could you not just attach a flat advertising section to the handle? Like this

– Tim Dorr – 2013-06-03T02:00:09.3501This is definitely nitpicking, but even with proper projection, you couldn't make a tubular design like one illustrated on your question work! This is because QR codes, like may other codes, require "quiet zone" (essentially background colour) around the actual code. Typically this is explicitly included in encoder output. In the case of QR code, this zone is as wide and high as large alignment blocks on the code.

I hope the best for getting this work, but projections and real-world usage can be tricky. Choose as low "version" (pixel count) code as possible! – kirma – 2013-06-04T21:01:57.973

Not to take away from the awesome underlying problem, but I've seen shopping carts that aren't truly cylindrical (there'll be a flat edge on the side where people pushing the cart can read). – SheetJS – 2013-06-09T15:12:11.867

Yes!.. We know. See @belisarius comment joke link. :) – Murta – 2013-06-09T15:19:42.623