Copyright © 2018 by Wayne Stegall
Created July 24, 2018. See Document History at end for
details.
Guitar Shape
Mathematical
modeling
of
guitar
shape leads to modeling program
Introduction
Some time ago I wondered if the shape of a classical guitar was
mathematical and further if a mathematically modeled shape would be
better in some way. I imagined this to be a polar plot.
Manual Attempts
When I began to explore these ideas, I first tried for a peanut (or
hourglass) shape with the following equation and succeeded. This
resulted in upper and lower bouts of same width with the expected waist
in between.
()

r = a_{0} + a_{2}cos2θ

I already had in mind that if I added a
_{1}cosθ to
the equation the lower bout could be larger than the upper.
()

r = a_{0} + a_{1}cosθ
+
a_{2}cos2θ 
Now the width of the outline was unavoidably narrow relative to the
length without some further change. The required change was to
stretch the y axis of the polar plot.
()

x = r·cosθ 
()

y = y_{mult}·r·sinθ 
Now that could get the right bout widths, the outline seemed too
roundish. Adding a sixth harmonic then allowed the familiar shape
with considerable tinkering with the coefficients.
()

r = a_{0} + a_{1}cosθ
+
a_{2}cos2θ + a_{6}cos6θ 
Computer solution
When I came to want a program to calculate the coefficients and plot
the curve, I was stuck on the idea of and empirical method of refining
the coefficients but failed to imagine a final algorithm. Then I
thought of the min/max algorithm used to calculate digital filters and
thought it right.
Min/Max algorithm process
 User specifies length, lower bout, waist, and upper bout
dimensions and a_{4} and a_{6} coefficients
 Choose reasonable θ values for minimum and maximum points
 Calculate coefficients a_{0}, a_{1}, and a_{2}
from current min/max θ values by solving simultaneous linear equations.
 Find actual min/max points for next calculation.
 Repeat 3 and 4 until convergence.
Equations on which simultaneous linear equations are based
r = a
_{0} + a
_{1}cosθ + a
_{2}cos2θ + a
_{4}cos4θ
+
a
_{6}cos6θ
x = r·cosθ
y = r·sinθ/a
_{y}
r = a
_{y}·y
/sinθ
a
_{0} + a
_{1}cosθ + a
_{2}cos2θ + a
_{4}cos4θ
+
a
_{6}cos6θ  a
_{y}·y/sinθ
=
0
Length = 2(a
_{0} + a
_{2} + a
_{4}
+ a
_{6})
Matrix input to Gaussian elimination
a_{0} 

a_{1} 

a_{2} 

a_{y} 











2


0


2


0


Length – 2(a_{4}
+ a_{6}) 
1 

cosθ_{0} 

cos2θ_{0} 

–Lower/sinθ_{0} 

–(a_{4}cos4θ_{0}
+ a_{6}cos6θ_{0}) 
1 

cosθ_{1} 

cos2θ_{1} 

–Waist/sinθ_{1} 

–(a_{4}cos4θ_{1}
+ a_{6}cos6θ_{1}) 
1 

cosθ_{2} 

cos2θ_{2} 

–Upper/sinθ_{2} 

–(a_{4}cos4θ_{2}
+ a_{6}cos6θ_{2}) 
_{
}
Program and its operation
Program name is
guio.exe
Download
program version
1.0.0.
Run program and a default program shape appears.
Menu>Edit>Input Shape
brings up dialog to define your own shape loaded with default
parameters.
Figure
1:
Input
Shape
dialog.


Enter all required values experimenting with small values of a
_{4}
and a
_{6}.
in>cm and
cm>in buttons convert units.
Press
Calc to calculate and
draw shape.
Press
OK to do the same and
exit dialog.
Figure
2:
Guitar
outline
in
horizontal orientation


Menu>View>Vertical
toggles between vertical and horizontal modes with vertical mode
indicated by check mark.
Figure
3:
Guitar
outline
in
vertical orientation


Help in adjusting coefficients a_{4} and a_{6}.
A
_{4} and a
_{6} should only have small experimental
values. Figure 4 below shows positive and negative values of a
_{4}
and a
_{6} added to circles to help you to understand their
effects. Starting from the inside working out they are:
 + a_{4}.
 – a_{4}.
 + a_{6}.
 – a_{6}.
Figure
4:
Graph
of
the
effects of coefficients a_{4} and
a_{6}.


Lutherie use
For those who want life sized drawings of guitar shapes the number
displayed in the upper left hand corner of the plots indicates the
number of pixels per unit measure. Then follow this procedure.
 AltPrtSc with program selected to get its image.
 Paste into a graphics program.
 Scale the image to the indicated dots per unit.
In GIMP this is done as follows:
Menu>Image>Scale Image
Change X resolution and Y
resolution to <your value> pixels/in (or pixels/cm)
 Use graphics program to create subimages that when printed can
be taped or glued together to create a complete blueprint.
(guitar bodies are larger than standard letter paper.)
Dreadnoughts too
Although I originally sought to model the Spanish guitar the following
shows an attempt at a dreadnought.
Figure
5: Input and results of attempt to model a dreadnought.



Document History
July 24, 2018 Created.