You know the opening is 1200 wide and you want the curve to rise 200 in the middle. This gives you the radius, where to put the trammel point, and — if the radius is too big for a compass — a table of offsets to plot the curve point by point.
Notes
Where the formula comes from
With chord C and rise H, the radius is (C squared divided by 8H) plus H over 2. It falls out of the intersecting chords theorem, and it is exact, not an approximation.
When the radius is bigger than your shop
A 1200 opening rising 100 needs a radius of 1850 — fine. Rising 20 needs 9010, and no trammel reaches that. That is what the offsets table is for: measure along the chord, mark the offset up from it, and bend a batten through the points.
Marking with a batten
Drive a nail at each plotted point, spring a thin strip of straight-grained stock against them, and pencil along it. A batten naturally takes a fair curve, which is more forgiving to the eye than a slightly wrong true arc.
What this does not do
It handles circular arcs only. Ellipses, ogees and free curves are a different problem, and a batten sprung through three points is usually the better answer for those anyway.