Initial appearance: November 7, 2010
Last updated: July 31, 2020
A curtate cycloid curve is generated as shown in the following diagram, drawn by David Cohen of Cohen Musical Instruments and which appeared in his American Lutherie article "Curtate Cycloid Arching"1:
Violin makers and guitarmakers first became interested in these curves for purposes of arching profiles of carved plate instruments following publication of an article by Quentin Playfair2 in which he compared these curves to arching profiles of golden age violins. They don't really model the arching curves of violins well at all3 but they are nice looking curves and are readily rendered (curves that more realistically model those of Cremonese violin arching profiles can be found here).
The parametric equations for calculating locations of points on a curtate cycloid curve are:
x = aφ - b sin φ
y = a - b cos φ
a is the radius of the circle;
φ is the phase, 0 to π;
b is some displacement from the center of the circle;
As you can see, the equations yield coordinate values as functions of phase φ. For purposes of generating graphs or graphic curves, it is highly desirable to be able to generate points using a fixed x interval. And for purposes of generating a series of contour lines approximating an arched surface or for generating instructions to a CNC machine for cutting an arched plate, it would be convenient to have a series of points that used a fixed y interval (contour interval).
Here is how these parameters relate to a typical transverse arching profile:
The generated curve is shown in black in the above drawing. It should be obvious how to mirror a piece of the start of the curve to form the recurve area, and how to mirror the whole thing to get a profile for the right side of the plate too.
Note that the x or y interval is specified implicitly. When requested to generate a list of points for a half curtate cycloid curve with a fixed x interval, the calculator uses (the length - the x offset) divided by (the number of points - 1) as the x interval. When requested to generate a list of points for a half curtate cycloid curve with a fixed y interval, the calculator uses (the total height - the Y offset) divided by (the number of points - 1) as the y interval.
You can specify the format you want for the output. The calculator will generate a tab delineated .csv file for use by a spreadsheet, a .dxf file containing a polyline representing the specified curve in default units for use by CAD drawing software, or an .svg file containing a polyline representing the specified curve in default units for use by artistic drawing software. The calculator will also generate a coordinate list output in a popup window. If you specify this be sure to allow popups for this website. The coordinate list is output as a list suitable for copying to the clipboard and then being pasted into a spreadsheet or other program for further use.
Here's the calculator which generates a list of points for a half curtate cycloid curve:
Cohen, D. “Curtate Cycloid Arching”
American Lutherie #96, p. 26.
Playfair, Q. “Curtate Cycloid Arching in Golden Age Cremonese Violin Family Instruments” CAS Journal, Vol. 4 ( No. 7 (Series II)), pp. 48-58.
Mottola, R.M. “Comparison of Arching Profiles of Golden Age Cremonese Violins and Some Mathematically Generated Curves” Savart Journal, Vol. 1 ( No. 1)