Here are three (or four) field-expedient methods for drawing curves that I use frequently when building houses and furniture. Each method produces a different type of curve, so pick whichever suits your application.
These are demonstrated here in small scale, but I usually use them in the field to draw curves from 1' or 2' long up to 20' or 30' long.
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Step 1: True Arc
This method produces an arc, a segment of a circle. Mark the width of the arc and the height of the arc at the midpoint. Place a nail or other restraining implement at either end of the arc.
Place two straight strips of material so that they are touching the nails and intersecting at the midpoint. Fasten them securely at the midpoint and place your marking implement in the apex. Slide the contraption to one side and then the other, keeping the marker in the apex, and the "legs" in contact with the nails.
Step 2: Batten or Spline
This method is often used by boat builders because it can be used to make "fair" (smooth) complex curves. Here is a boat-specific example. It is the method I used to layout the curve for my trestle table. I'm also not sure what kind of curve it produces, mathematically speaking, but I'm hoping someone here at Instructables can tell me...
Mark the end points of your curve, and the height. Place to nails or other restraining devices at these points. I often use clamps. Use a long, thin, consistent strip of material (a "batten", PVC pipe works well for large curves) to "fair the curve". You may have to experiment with different batten materials and dimensions to get a good result.
For compound curves, mark several "waypoints" and use the batten to connect them.
Step 3: Catenary
This method produces a catenary, which is almost, but not quite, a parabola, but is usually close enough. This method doesn't work well if there is a breeze.
The surface to be marked should be vertical. Drape a limp piece of string or thin rope so that it hangs freely. The piece I used is not quite right. I use a marker and "dab" it over the string every so often and then connect the gaps in the dots later by hand.
Again, you'll have to experiment with material. Woven cord works better than twisted, which is why the string in the example photo isn't quite right.