Measure the velocity factor (at or near the operating frequency)! If
you are using solid polyethylene dielectric cable, then the v.f. really
should be close to 0.66. For a flat pattern, you want the
center-to-center spacing of the elements to be 1/2 wave, considering
the net velocity factor. I say net because at the junctions between
elements, it's possible that the effective v.f. is slightly higher than
in the coax itself. You'd cut the elements long by enough to let you
solder them together and end up with that 1/2 wave center to center.
(It's easier and equivalent to measure from top of one element to the
top of the next element, etc.)

Since the _pattern_ depends mainly on the phasing of the feed system,
and the phasing is established mainly by the propagation velocity in
the line, and not by the surrounding environment, putting it next to
something won't affect the pattern, except that if there are conductors
in that display you want to mount it to, they may very well screw up
the pattern by allowing current where you don't want it. Especially if
there is some piece that's resonant near the operating frequency it
would be like putting an unwanted parasitic element in a Yagi: not a
good idea for optimum performance.

OK, so the phasing is designed to get you the right pattern. But the
phasing is independent of the feedpoint impedance. You need to match
to that, and you also need to decouple the antenna from nearby
conductors. Any conductors in the display fall into that category, but
the feedline from the xcvr also does. A choke balun, and other
feedline choking, is very highly recommended.

I used to have a web page with quite a bit of text about the why and
the how of coaxial collinears; it wasn't a construction article, but it
armed you with enough info that you could go out and make one that
would work well on pretty much any frequency you might want. It
included things like why the center-center half wave spacing, and why
it's slightly better to use foam dielectric cable. If I got enough
requests, I'd make a revised version of that available, maybe even with
some pictures this time. Every once in a while I get a request to
re-print it in a club newsletter, so I know there are some of those
floating around out there.

Cheers,
Tom