Height of wedge-shaped guitar sidewall

Question

Imagine a guitar. Looking at its profile, its body tapers from full depth at the bottom of the body to nothing at the point where the body meets the neck. Meanwhile, looking from the top, the body is curved. In profile, you can see the sidewall, which is a continuous piece of wood bent to the contours of the body. Because of the wedge, the sidewall, when stretched out, is sloped. It starts at full height, at the bottom, and, like the profile, tapers to nothing.

My question is: what function describes the sidewall's taper, given that the body's curve is described by some function and the wedge shape has a constant slope?

I've taken a stab at it, and this is what I've got:

My axes are:

$x=0$ at the neck and $total$ at the bottom of the guitar

$y=0$ at the center line of the guitar, where the strings are, and increases going outward. I assume that the guitar is symmetrical and that the sidewall is therefore symmetrical.

$z=0$ at the neck and increases going down the depth of the guitar

$s(x)=$ arc length from 0 to x. $s(total)$ is the sidewall's entire length. (Or, half its length--this guitar is symmetrical.)

$h(x)=$ height of the sidewall at $x$.

My solution is very simple, but I'm not sure it takes everything into consideration, hence this question. I also have a follow-up question.

$h(x)=\frac{s(x)}{s(total)}\frac{dz}{dx}$

I approach this question assuming I would need to do much more complicated things, but my solution is very simple. Is this correct?

My follow-up question is this: how could I find $h$ on the basis of the point along the sidewall, not along the length of the guitar? I'm having trouble figuring out how to translate from x-coordinates to arc-length-coordinates, if you forgive my terminology.

Answer 1

Your approach is exactly correct, although I think you could think of the variable relationships a bit more clearly. Think of it like this:

• Parameterize the side wall by the distance along it, $s$. That is, you have functions $Y(s), Z(s)$, and what you ultimately want is the height there $h(s)$.
• To do the parameterization: given a desired shape of the sidewall, parameterize that by some variable $t$ or whatever; that could actually just be $z$ for a lot of designs, probably; then compute the arclength of $s$ in terms of $z$. Invert that to get $z$ in terms of $s$. Now write all your coordinates in terms of $s$.
• You want $h(s)$ to be proportional to $Z(s)$, so we're going to require that $$h(s) = \textrm{total}\cdot\frac{Z(s)}{\textrm{length}}$$ where here "length" is the distance in the $Z$ direction from neck to base.
• You have $Z(s)$, so scale by the above formula and you'll have $h(s)$ as well. "Plotting" $h(s)$ will give the curve you need to cut out to make the wood.