To develop the curves in the various brackets–here the support for the back fence on the lid of a correspondence desk– I followed the ancient practice of melding arcs of a circle along a straight lineSweet

I begin by making a few concept sketches to get an intuitive feel for the curve I would like to see transition the horizontal lid surface to the vertical back fence. I’m going to go with the shape below.

Sweet 5

From the sketch, it reveals that the overall form suits that of a 1:2 rectangle. (An octave, by the way–but that’s another story).  Next, I divide the horizontal length into four equal segments. The first of these segments defines the flat at the top of the curve. I then draw a baseline for the sine curve from this segment point to the lower right hand corner then divide that baseline into three equal segments (see drawings below).

Sweet 2Sweet 4

To find the focal point of the arcs–which will each be one-sixth of a circle’s circumference–I set the dividers to the length of the segment (which is the chord of the arc) and swing out intersections to locate the focal point of the arc. . Next, without changing the span of the dividers (because the chord equals the radius for sixth sector arcs as you may remember from Mr. Hammersmacker’s seventh grade geometry class) I swing the arc from the focal point to each segment point. The transition between the two arcs is seamless–proven to be so because a line connecting the two focal points will pass through the arc’s transition point.

Sweet 1

….and you’re done. You can use the drawing to produce a thick paper template to trace onto any number of brackets.Sweet 3