On a geometry of the flat violin boundary (Q1354181)

The violin is devided into three parts: the upper and lower bouts, called the A- and B-bouts, and the middle part, called the C-bout. The contours of the A- and B-bouts are modelled by a polar equation in the form \[ (r/a)^n= 1- 2\sum^{N-1}_{m=1} c_m\sin^2(m\theta/2)\tag{1} \] and the contour of the C-bout is represented by a rotated and translated ellipse. In order to determine the unknowns \(c_1,\dots,c_{N-1}\) in (1) a number, say \(K>0\), of measure points \((x_i,y_i)\), \(i=1,\dots,K\), on the contour of a given violin are taken and the linear system \[ 2\sum^{N-1}_{m=1} \sin^2(m\theta_i/2)c_m= 1-(r_i/a)^n,\quad i=1,\dots,K,\tag{2} \] is solved where \[ r^2_i= x^2_i+ y^2_i\quad\text{and} \quad \tan\theta_i= {y_i\over x_i}\quad\text{for }i=1,\dots,K. \] The parameters \(a\), \(n\), and \(N\) are also chosen and (2) is solved via the least squares method. This curve fitting method is applied to a 1720 Stradivarius.