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I'd like to build an installation which projects a star sky on the ceiling. The picture will explain the concept and the problem I've encountered:

enter image description here

As you can see, there is plenty factors that I would need to take into account which I don't know. I'm looking for suggestions on how to approach this problem and would be grateful for some buzzwords on what to google for similar projects (light projecting and similar didn't yield results).

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When you say "LED lamp" do you mean some random thing sold in stores as a replacement for incandescent bulbs? Those are terrible "point light sources", because they go to extremes to diffuse the light like the old frosted-bulb incandescent. Precisely because one of the things people hate about LEDs is their intense single-point light.

The technology is not a problem, rather it's finding it in a screw-in off-the-shelf consumer product. You might want to look at bulbs intended for track lighting, flashlight or automotive replacement - specifically ones that have a single multi-watt LED emitter. Explore SuperBrightLEDs.com ... you'll know it when you see it.

It is likely to be 12 volts, which means getting a 12 volt DC power supply also (those are readily available, and if you kept the old power bricks off old Internet routers, you may even own a few.)

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Focusing the light requires some tinkering. The size of the hole is a factor. The distance between the light source and the apeture is another factor. Also, if you are using multiple LEDs then there will be multiple stars per hole; each LED will make a star.

Some words to google might be diffraction, interferance, and apeture. Also you could perhaps pick up some info from making an eclipse viewer.

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This is also known as a GOBO light. That might be able to help you find more resources online, like this: http://www.instructables.com/id/Gobo-Light-Projector

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To get sharp shadow projection you need either something close to a single-point light source, or lenses

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