# Math, Optics, And CNC Combine To Hide Secret Images In Acrylic

Magic mirrors, with an LCD panel hidden behind a partially reflectively mirror, are popular for a reason — they’re a good-looking way to display useful information. A “Magic Window,” however, is an entirely different thing — and from the look of it, a far cooler one.

If you’ve never seen a Magic Window before, don’t worry — it’s partially because you’re not supposed to see it. A Magic Window appears to be a clear piece of glass or plastic, one with a bit of a wave in it that causes some distortion when looking through it. But as [Matt Ferraro] explains, the distortion encodes a hidden image, visible only when light passes through the window. It looks a bit like a lithophane, but it’s projected rather than reflected, and it relies on an optical phenomenon known as caustics. If you’ve ever seen the bright and dark patches cast on the bottom of a swimming pool when sunlight hits the surface, you’ve seen caustics.

As for how to hide an image in a clear window, let’s just say it takes some doing. And some math; Snell’s Law, Fermat’s Theorem, Poisson’s Equation — all these and more are mentioned by [Matt] by way of explanation. The short story is that an image is morphed in software, normalized, and converted into a heightmap that’s used to generate a toolpath for a CNC router. The design is carved into a sheet of acrylic by the router and polished back to clarity with a succession of sandpaper grits. The wavy window is then ready to cast its hidden shadow.

Honestly, the results are amazing, and we marvel at the skills needed to pull this off. Or more correctly, that [Matt] was able to make the process simple enough for anyone to try.

## 45 thoughts on “Math, Optics, And CNC Combine To Hide Secret Images In Acrylic”

1. Krzysztof says:

Watched video, thought to myself “Looks like with some more processing, he could make projection holograms, this already looks like a hologram”. Read the text – holy moly, this IS a hologram!.

It’s not a hologram yet, because the input data is just one 2D photo of a cat. But if he managed to take many pictures from known perspectives, knowing also camera’s focal point, he could create a 3D map (x,y,z). Let’s imagine a space containing rectangles with grayscaled photos where imaging sensors were with points in front of them. Now we can draw light rays of different intensity – where these rays cross, we sum their intensity and get cat’s 3D model. Then we simulate how the light beam should be refracted to get that 3D image (observed by putting the glass between source and viewer).

1. LordNothing says:

then there is the difficulty of performing photogrammetry on a cat.

1. KVG says:

Easy. Just put a keyboard down in the center of the photogrammetry rig and pretend to work on it.

People in the future will wonder why we made cats sit on blocky plastic trays, but what do you do?

2. poiuyt says:

Spectacular!

1. damntech says:

Once you’ve made one of those windows, could you create copies (without the need for sanding and polishing) by casting ?

1. CRJEEA says:

If you can cast microstructure holograms on to chocolate, you can cast this.
What I’m wondering is if this could be combined with a liquid crystal to redirect the incoming light, perhaps producing animations or even using a sandwich of materials with different refractive indexs to show different images at different wavelengths and reproduce colour images from white light.

Intentional chromatic aberration ?

4. Dave says:

Amazing.

5. daveboltman says:

Very cool!

The title made me wonder why those speaking American English say “math” with no ‘s’, but ‘optics’, with an ‘s’? If it’s ‘math’, shouldn’t it be ‘optic’, with no ‘s’ as well then? On the other hand, those speaking English English say ‘maths’ and ‘optics’.

I always thought ‘maths’ was short for ‘mathematics’, so I guess students in the United States get taught ‘mathematic’, with no ‘s’? Maybe someone cleverer than I could explain it?

1. Dave says:

I’m just baffled by “a pair of pants.”

1. Bob Magana says:

Pants used to come in two (pant) halves that were connected in the middle.

1. paulvdh says:

Just like scissors?

2. Hirudinea says:

Well, first you put one leg in, then the other leg, and pull them up over your pelvis. Don’t worry, you’ll get it one of these days. :)

1. Dave says:

I just spit my coffee. Thanks for the tutorial!

2. Quin says:

Occasionally you have to take a leg out to do the hokey pokey. Turning around also helps.

But what if that really IS what it’s all about ?

2. Jace says:

Correct, ‘math’ is short for ‘mathematics’. it’s not ‘mathsematic’. :-)

3. ThisGuy says:

US and England. Two countries divided by a “common” language.

Problem is that English isn’t a single language. It’s more like 3 languages in a trench-coat pretending to be a single entity. One with a nasty habit of following other languages down dark alleys, hitting them over the head and rifling through their pocket for loose grammar and vocabulary.

A lot of the division between US English and UK English is that they’ve been influenced by different cultures and different language reforms. And when it comes to language reforms it seems the universal constant across languages that those language reforms never actually make a language more uniform or logical.

4. Charles says:

I can only offer an unsatisfying explanation: the fundamental assumption that language (English in particular) follows a consistent set of rules is flawed.

We have already inherited plenty of words, phrases, and usages that make no sense. Add to that the fact that language changes over time, and it is easy for it to become even more of a mess.

5. Misterlaneous says:

I’ve always assumed optics has an S because there are generally several parts to a lens.

I say shrimps too, but that’s just to annoy my environmental scientist gf.

1. Ren says:

My B-I-L is Italian-American, he combines “scampi” and “shrimp” into “scrimps”.
I don’t mind, whatever he serves us tastes great!

6. Hirudinea says:

“Maybe someone cleverer than I could explain it?” If someone could explain the “logic” of language they’re not just clever, they’re a friggin’ genius!

7. Steven Gann says:

As an American English speaker it makes a lot more sense. Math is a singular concept. There are different maths (plural) such as arithmetic, calculus, algebra, statistics, etc., but they’re all facets of math. It is similar to science. The concept of science is singular, but there are many sciences which are facets of science.

Optics also makes sense to me because any optical system involves multiple elements, hence optics are a plurality.

This intuition breaks down when you toss in physics, or other concepts. It sort of highlights that around the 18th Century English was pretty fluid but the proliferation of public literacy on both sides of the Atlantic resulted in rigid school books that made arbitrary choices, which typically manifests in UK English having extra vowels and US English having fewer, because both forms were in common use prior to arbitrary rules being made up.

Language is fluid. Right and wrong are silly concepts to ascribe to a system designed for communicating ideas. As long as both parties grok your jive, it’s all good.

6. johnrpm says:

An excellent example of defining then resolving a project, this looks like (to me at least) to be original work, impressive.

1. sebastian says:

It is not. He even quotes the paper that he was inspired by.

“Poisson-Based Continuous Surface Generation for Goal-Based Caustics” by Yue et al. from 2014.

Another paper that does the same thing (and much more) with a different approach is “Mitsuba 2: A Retargetable Forward and Inverse Renderer”. https://rgl.epfl.ch/publications/NimierDavidVicini2019Mitsuba2

7. Alice Lalita Heald says:

Maybe a built in sun clock into a window?

1. Dave says:

Nice idea!

2. Hirudinea says:

Actually this technique would be great for making “stained glass” windows.

3. π π says:

No you need a hologram to do that, and it has been done, may even be a “product”.

8. BT says:

Excellent write-up, and don’t miss [Matt Ferraro]’s equally excellent write-up about homemade CNC router choices also on his site.

I want one !
I would love to see my wife’s dog being mad looking at that cat on the wall ^^ (it is stupid enough to bark to “that dog” behind the mirror …)

10. xdlyes says:

A famous swiss entreprise called Rayform has develloped and used the same technology. Even sells services, like customized reflective jewels. Thumbs up for Matt to have sorted out all of that by himself !

11. Andrew says:

The internet was invented to distribute cat videos. This is the next logical step.

12. Roberto Garcia says:

What software was used to create the caustic image?

1. Paul says:

13. nikescar says:

I wonder if the same thing could be accomplished by carving the acrylic but then instead of polishing it, pour a layer of clear resin over it. The difference in refractive index, transmission, etc might give the same effect.

1. MacAttack says:

Or even modifying a FDM printer to dispense UV resin and curing it. Do multiple passes to get the layers needed to implement the heightmap directly. With a fine enough head and Z resolution you could probably dispense with most, if not all, of the polishing.

IIRC wasn’t such a resin printer recently mentioned here at HaD ?

1. MacAttack says:

Obviously a good resin printer would also work.

14. π π says:

I remember discussing the possibility of doing this back in the mid 1990’s on the POVRay mailing list. We knew it was possible as you can break the problem down in an array of lenses but it is great to see such a good writeup of the maths out in the public domain after all of this time.

15. Edward Bear says:

I always thought optics were the liquor dispensers in pubs. :)

That, and the study of the physics of light (especially it’s transmission through flavoured alcohol)

But what it absolutely isn’t, is the PR appearance of a subject. Or am I the only one who finds that irritating ?

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