What kind of reflector geometry would I need to focus an American tubular(ish) style 1000 watt sodium lamp into a flat wide beam? I know it would need to be mounted horizontally. But would it be a curved reflector or a triangular reflector? I want to essentially make an overpowered fog light. Was thinking of getting some steel sheeting. Polishing it to almost chrome levels of shine and bending/tuning it into the correct shape to cast a crisp or at least flat beam.

Maybe some kind of Fresnel lens, but it would have to be specialized.

I mean if I can just do it with a reflector I could likely form it out of highly polished steel or chromed steel sheeting. Ideally I'd want it to be relatively thin but I'm not familiar with what math equations I'd even need to use to calculate focal points and what not.

I'd like to make it as efficient as possible. But compact so I can have it be portable. I think I vaguely recall linear sodium lamps existing in a form factor similar to linear halogens but I'm not sure. Something like that would probably fit my application far better than a bulky screw base lamp.

A parabolic shape (a sheet metal bend along a parabole; not speaking about paraboloid cup), with the lamp arctube in the focal line...

This was actually a thing! Look for a Wide-Lite 11-degree fixture. Lots of fishing boats had them. They would, literally, throw lift a mile away. The orange glare cuts fog/rain quite well, unlike LED. Many vessels even had two or three of them.
I think the reflectors were somewhat parabolic, with square bevel glass, but it's been years since I've seen one.

 I had a set of fog lights that used a glass rod to create a thin flat beam : https://www.ebay.com/itm/226144474249  The lights have no internal reflector . Don't know if this method can be adapted for use with such a long lightsource or not .

@wide-lite 1000:
These reflectors won't work, they are designed for a small cylinder of the halogen filament.
The HPS is significantly larger and way-way longer, so the focal length would need to be made larger as well.
Dunno what diameter is the 1kW HPS burner, my guess would be something nearing 10mm or so.
For a 10deg high beam it would mean about 6cm distance from the back of the parabole, so at least 30cm high reflector.
I think for the 10deg beam, we are talking about size and accuracy in reach for DIY, assume you have at least some basic wood/metal (hand saw, drilling,...) home workshop...