Guitar Outline Equation The Nealon Equation
Arctan
The tangent of an angle, α, in a right triangle is the ratio of the length of the opposite side to the adjacent side.
In a Cartesian coordinate system, for an angle measured with respect to the x-axis, the opposite side is the y coordinate and the adjacent side is the x coordinate.
However, any straight line in this coordinate system has the slope
Δ
y/
Δ
x, so the slope is the tangent of the angle.
If the angle is
π
/2 (90
º
) then the slope is infinite, since
Δ
x is zero, regardless of the value of
Δ
y, so the tangent is undefined; in other words, the tangent "blows up" at 90
º
.
Calculations are in radians, not degrees.
The cotangent is defined as
Δ
x/
Δ
y. For this function, if α = 0, then
Δ
y is zero and the cotangent blows up; however, the cotangent is defined at 90
º.
The tangent function is periodic over
π
and goes to infinity at odd multiples of
π
/2; videlicet,
±
1
π
/2,
±
3
π
/2,
±
5
π
/2, . . . . Arctangent, sometimes called the inverse tangent, is the angle for a given tangent.
If the tangent of an angle is known (or if the slope of a straight line is known), what angle corresponds to that tangent? Arctangent.
This inverse of the tangent is found by rotating the tangent about an imaginary axis 45
º
with respect to the x-axis. The tangent is turned over on its side, so to speak, and it asymptotically approaches
π
/2 as x goes to infinity.
By writing (BL - x) as the argument, the arctan is offset - and inverted.
Notice how this arctan term is pinned to the x-axis at the body length, BL. Here I have set BL = 20 and plotted the graph only for 0 < x < BL.
Here is a graph of two arctan terms multiplied by a sin function. The "1+" inside the parentheses is just for the purpose of illustration. For the guitar curve fit there would be a series of sines. The equation for fitting the guitar outline also has a polynomial multiplying the arctans. The constant A is the amplitude, B is the frequency, and C is the phase of the sine. Notice that this equation has only three parameters plus the body length. This example has A = .2, B = 1, and C = .3 Now the curve is starting to look suspicious - but it's just a coincidence. To obtain a perfect fit to a guitar body, at least 19 coefficients are required! For this curve, A = .2, B = .3, and C = .4 webpages and Eos image copyright 2014 M Nealon |