Date Published: February 6, 2017
Publisher: Public Library of Science
Author(s): Steven Piantadosi, Suzannah Rutherford.
This paper presents an approach to describing the three dimensional shape of a violin plate in mathematical form. The shape description begins with standard contour lines and ends with an equation for a surface in three dimensional space. The traditional specification of cross sectional arching is unnecessary. Advantages of this approach are that it employs simple and universal description of plate geometry and yields a complete, smoothed, precise mathematical equation of the plate that is suitable for modern three dimensional production. It is quite general and suitable for both exterior and interior plate surfaces, yielding the ability to control thicknesses along with shape. This method can produce mathematical descriptions with tolerances easily less than 0.001 millimeters suitable for modern computerized numerical control carving and hand finishing.
This paper describes an approach for constructing three dimensional mathematical models for the shape of violin plates that has not previously been used in the violin community. The method consists of two modeling steps beginning with ordinary plate contour lines. First, each contour is individually modeled with a general flexible equation independent of its elevation. Second, the coefficients from contour equations are quantitatively related to elevation using a second set of simple models. These steps jointly smooth and synthesize contour lines into a complete surface. The result also allows any number of additional contours to be drawn consistent with the originals and resulting surface. This represents essentially a full mathematical description of the plate surface shape that can be applied to both exterior and interior surfaces.
The goal of this work has been to provide a precise mathematical characterization of the surface of a violin plate. My motivations are oriented toward labor saving technologies, CNC carving in particular. Having control over details of the process, even when leaving significant hand finishing for aesthetics and acoustics is essential. Beginning such an effort from common simple empirical characterizations of shape such as contour curves is highly desirable, as opposed to assuming a specific mathematical form a priori. I have avoided some specific formal mathematical prescriptions because they seem not to be correct, but more importantly because they are unnecessary.