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How did Kubrick shoot HAL's "eye" in 2001: A Space Odyssey? We see the reflection of the astronauts approaching HAL, but not the camera.

2001: A Space Odyssey

How was it done?

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You don't see a reflection. What you see is an actor coming towards the camera. The HAL lense and light reflexes are added from a different layer.

In Odyssey Kubrick used a technique called "Front Screen Projection"

A technology that uses projecting a pre-recorded movie on a very reflective material (Scotchlite by 3M in the case of 2001:SO) using a one way mirror set at 45 degrees toward the projector. Beetwen the mirror and Scotchlite the actor was placed (or in case of 2001:SO the whole scenography). The camera record that through that one way mirror.

Here's a picture of the setup they used (photo taken from Stanley Kubrick Exibition in National Museum in Krakow/Poland).

enter image description here

In the left part of the blue cube you can see HAL's eye. On the black wall a pre-recorded movie (or in the case of this exibition - visitors) were present.

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Nowadays CGI would be used for a scene like this (see Mirror shot without camera reflection in "Velvet Buzzsaw").

CGI Lens shot:

CGI Lens shot

But since that technique was most likely not available at the time and might also not have been a great fit for Kubricks artistic approach, there must be a practical solution at works here. Other movies have done similar things as discussed in How do they film the mirror scenes in movies?.

I would suggest that Kubrick used a combination of multiple techniques for the given effect. Similar to the scene in the linked SO, it might not a reflection but instead a lensed window into a room with the actor/astronaut and most of the lightning. The yellow light in the very center, might be a smart reflection from the cameras side using a slightly different technique of using an off-center object to be reflected towards the off-center camera, having them both appear centered (this is the go-to practical mirror-filming technique).

Off-center mirror shot:

Simple off-center mirror shot

The framing of the lens might also have been on the cameras side and been carefully lit to have the desired look. Afterall, Kubrick was a master of lightning as well.

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