Been considering doing this for quite a while, I've always wanted a long range spotlight like those HID throwers.
The only issue is that the LED surface area is much larger than what a short arc would be, and far less lumens too.
That leads to a large spread and less intense beam, unless you use a really large reflector.
If I do this I will be watercooling the LED, because the distance of the copper "arm" is too long to provide good cooling for a high power LED running at 12 amps or more.
There are some good reflectors from edmund options for $50-100 that are 12-24" diameter and have a short focal length. Problem is that edmund optics import costs are extremely high.
I've ordered a lens from them before and it the COD was more expensive than the lens itself. Can't imagine what it would be like for a 12" or 24" huge box....
http://www.edmundoptics.com/optics/optical-mirrors/focusing-concave-mirrors/large-parabolic-reflectors/1364/
Another option is this guy that makes acrylic parabolic mirrors for sun collection, not sure how accurate the parabola is but the 24" and 37" deep parabolics seem pretty good quality, and shipping within the US is cheap.
http://greenpowerscience.com/SHOPARABLOICHOME.html
The larger the diameter, the tighter the beam will be. This is really important if you plan to use a large LED like XHP70, SST90, SBT140, or something like that which has lots of lumens.
I'm not sure how large you're planning to make yours, based on the image I would guess 6-12". I am interested if anyone can find a good cheap parabolic reflector below 12" with an accurate parabola.
Remember that the smaller the mirror is, the more accurate the parabola needs to be, and the smaller the LED die needs to be. XP-L HI or dedomed XP-G3 would be optimal IMO.
Best surface brightness (lumens/mm^2) I have found so far.