Projectors

Depends on a lot of factors. What Ron said plus how much of a punch does your stage wash have.

Power is another factor.

Budget.

A blind suggestion would be a stack of 12ks with a wide angle lens to fill your cyc from wall to wall.
 
As tantalizing as projection is, you realize how dim they are in relation to theatrical lighting.
The first show I did with projection was Beauty and the Beast. The artwork was cheesy and projecting 30'tall x 45' wide those 1080p pixels were ENORMOUS!
 
There is no set answer when it comes to what you are asking. A projector is a light that you want to see as opposed to the object that it is illuminating. This will change how you need to specify what you need. On many sites (including this one), you will get some stock answers as to what will work. Ultimately, it depends on what is good enough for you.

You have a lot of factors that have been thrown at you, which only scratch the surface of what is necessary to make the best decision.

Brightness (which is a relative term to begin with) is only one factor that you require. Projector manufacturers will provide you with a lumen output. That is how much light is emitting from the projector (new, with a new lamp), not what the audience will see.

I found this great primer video that should help you to be ready for the answer that you want.
 
Also depends how well you can control spill from adjacent areas .. type of fixture, advantageous positions or compromise angles, etc. also the cyc material and how clean it is
 
Very true. 4' doesn't give you anything to work with. If you have the budget, and this will be expensive, I would recommend looking for a projector that can be oriented in many directions (laser projectors are an example) and then get one of the extreme short throw lenses (usually contain a mirror). This should get you the best coverage with minimum amount of keystone.

Epson makes a projector/lens combination that would work. I would recommend that you still try to stick with a single projector and not fill the entire cyc (only 23' wide with your 13' height at 16:9). To get the full width, you would either need to blend projectors, which is relatively complicated, and will complicate content. The other option is to overshoot to get the width, crop the image height, and lose a lot of brightness (the extra 7' will make a lot of difference).
 
Very true. 4' doesn't give you anything to work with. If you have the budget, and this will be expensive, I would recommend looking for a projector that can be oriented in many directions (laser projectors are an example) and then get one of the extreme short throw lenses (usually contain a mirror). This should get you the best coverage with minimum amount of keystone.

Epson makes a projector/lens combination that would work. I would recommend that you still try to stick with a single projector and not fill the entire cyc (only 23' wide with your 13' height at 16:9). To get the full width, you would either need to blend projectors, which is relatively complicated, and will complicate content. The other option is to overshoot to get the width, crop the image height, and lose a lot of brightness (the extra 7' will make a lot of difference).
Thanks. I like Epson. We also looked at an laser Eiki with 10,000 lumens and a short throw lens. But $25,000 for the combo!
 
Yeah, the lens itself is crazy expensive. I have been working with @gafftaper for a few years to try to figure out a decent projection option for him. It isn't easy or cheap. This is where you have to decide what is acceptable.

You have to decide how important the projection is to the production. Is it acceptable for the projection to be on the actors (shadow in the projection)? Is distortion of the image acceptable (extreme angle keystone)? Is a smaller image acceptable (image above performers, look at War Horse)?
 

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