Calculating the side height of a guitar with a radiused back
If the guitar you plan to build has a flat soundboard rim and a single radius domed back, it is possible to calculate the side height at various positions.
This will allow you to cut the sides to near finished size before bending, saving lots of planing and sanding later in the build.
Of course once you have built one you can copy the shape onto a bit of paper and use that
This will allow you to cut the sides to near finished size before bending, saving lots of planing and sanding later in the build.
Of course once you have built one you can copy the shape onto a bit of paper and use that
Basic method
The basic method is to calculate
 the drop, due to the curve in the back, where the back meets the side, relative to the centre point of the guitar body,
 the height of the basic taper of the side as it climbs from the heel to the tail.
AssumptionsWe are after calculating side height "s", at a combination of lateral and longitudinal positions around the guitar.
These we will give coordinates for "x" and "y", starting at the centre in both directions. Basic build technique The guitar back is going to be made using a radius dish, so the lateral and longitudinal radii will be the same. To put it another way, the radius has to be the same heel to tail as side to side. (It is possible to have different radii one to end and side to side but it's just different maths) Assuming
Sure things

Maths for the drop of the curve  in one direction
This picture shows how to calculate the drop "h" as distance "x" increases for the known radius of the dish "r"
When "x" = 0 "h" = 0, and as "x" increases "h" increases. So the further way from the centre point the bigger the drop. This is what accounts for the sides getting lower at the bouts but higher again at the waist. Since we are looking at guitars in reality the dish radius "r" is very big number, typically 7620mm (25') compared to the "x" which is up to 250mm approx heel to tail. So "h" never gets very big This picture should show how the drop is seen in reality. 
Maths for drop of the curve  in both directions
So now we know how the sides height drops as we move across the guitar "x". What about up and down "y" and a combination of both (x and y).
The drop as we move up and down is just the same with “y” instead of “x”, but a combination of the two is only slightly more tricky. Since the dish is the inside of a sphere it is simply a matter of working out how far the contact point is from the centre. So with known “x” and “y” we can figure that out using Pythagoras. Then we can use “d” as an intermediate step to get from a combination of “x” and “y” to “h” 
Maths for the basic side taper
The basic taper height of the side, due to the difference in the heel and tail heights, needs to be worked out, so it can be added later to the drop caused by to the back radius.
This height varies linearly with changes in “x”.
We need to calculate the difference in height, tail to heel, and add this proportionally with “x”, to the basic heel height. This gives the gentle taper down form one height to the other.
The only tricky bit is that the zero value of “x” is in the centre, so “x” goes from negative to positive as we move up the guitar back. This leads to some slightly over complicated maths to compensate.
Use "m" as the whole length, which is actually 2 times the maximum "x"
This height varies linearly with changes in “x”.
We need to calculate the difference in height, tail to heel, and add this proportionally with “x”, to the basic heel height. This gives the gentle taper down form one height to the other.
The only tricky bit is that the zero value of “x” is in the centre, so “x” goes from negative to positive as we move up the guitar back. This leads to some slightly over complicated maths to compensate.
Use "m" as the whole length, which is actually 2 times the maximum "x"
Adding up the curve and basic taper
How to do this is reality
Coming soon with photo's of bits of paper !!!
Notes
Centring
The dome being centred in the middle sounds odd as the high point is normally seen to be nearer the tail.
The dome has to centred really as it is a spherical shape and the high point going back is due to the higher tail block not an off centre dome.
The dome being centred in the middle sounds odd as the high point is normally seen to be nearer the tail.
The dome has to centred really as it is a spherical shape and the high point going back is due to the higher tail block not an off centre dome.
Heel to Tail diffrence
If the heel and tail height are massively different it starts to affect the angle between the two values calculated, the sagitta and the basic side profile taper. (red and green the diagram)
This means they don't add up to well, but for small differences it has been assumed they just sum up. This should be accommodated for when cutting the sides just a bit generously to allow finishing.
So watch out if making something really tapered.
If the heel and tail height are massively different it starts to affect the angle between the two values calculated, the sagitta and the basic side profile taper. (red and green the diagram)
This means they don't add up to well, but for small differences it has been assumed they just sum up. This should be accommodated for when cutting the sides just a bit generously to allow finishing.
So watch out if making something really tapered.