BACKSTORY: I have a flat tortoise. I need to figure out its original dimensions. I'm a paleontologist, and the site I'm working at has produced a [Hespertestudo crassiscutata], a giant tortoise that lived in the southern US through Mexico and into Central America until about 11Ka BP. As part of my dissertation I'd like to mention our tortoise's size in terms of carapace length and mass. The problem is that our tortoise has been completely flattened by the weight of the sediments above it. It is in many, many pieces, and I don't have the time or facilities to reassemble it at present.
REQUEST: If I know our fossil's original proportions and current dimensions, can I figure out the animal's original size? Given the dimensions of the flattened shape (major axis X millimeters, minor axis Y millimeters), determine the dimensions of the original half triaxial ellipsoid. Original shell height = a, width = 2b, and length = 2c, where a, b, and c are the semi-principal axes of the half ellipsoid. In most tortoises the proportion (a : 2b : 2c) is (1 : 1.49 : 1.98).
DESIRED RESULT: An equation I can use to solve for a, 2b, and 2c given X and Y and whatever other information you might need. Alternately, I can give you X and Y and you can just give me the answer.
OTHER AVAILABLE DATA: Do you need the thickness of the shell to do this? I can provide it upon request. If you can arrive at a general answer, that's perfect. If the answer depends on how the flattening happened, I can help: DIAGRAM. The carapace flattened by breaking along the suture lines -- in this diagram, the suture lines are the light ones.
PS: Thanks for the earlier help with Sr sampling!
For a mathematician who is naive about the subject matter, the logical approach would be to attempt to fit the observed partial proportions to the theoretical proportions.
The first thing that you'd want to do is try to reconstruct one or more of the actual dimensions that the tortoise had. If the fossil does not appear to be distorted (i.e., there has not been any skewing of the imprint created by the creature), then the size of the imprint could give you an indication of such dimensions, but it is not in the scope of this site to tell you how to make any adjustments to the measurements, without any further details provided.
You may wish to consult the extant literature regarding the estimation of organism sizes based on measurements of fossilized remnants, not just among tortoises but perhaps among creatures with exoskeletons, shells, or other hard exteriors.
Now, for the actual math, if we somehow could say that the turtle had a width of $2b = 30$ centimeters (hypothetically), then the naive approach would be to say that $$\frac{2c}{2b} = \frac{1.98}{1.49},$$ or $$2c = \frac{1.98}{1.49} \cdot 30 = 39.9$$ centimeters, and similarly, $$\frac{a}{2b} = \frac{1}{1.49},$$ so $$a = \frac{1}{1.49}\cdot 30 = 20.1$$ centimeters. But this is based on an idealized tortoise, whereas if you had more specific data (perhaps from other fossils) about the dimensions of tortoises within the same species or broader classification, then you could improve your estimate.
The other thing you can do is to look at the distribution of tortoise proportions across a variety of species, to get a margin of error on the estimated proportion of the tortoise from your fossil.
Finally, you can do more if you could measure multiple dimensions. Suppose you found $2b = 30$ cm but you were also able to determine that $2c = 35$ cm. Note that this doesn't match up with your theoretical proportions of $1.49 : 1.98$. Then a simple way to estimate $a$ is to take the average of the values you would get if you used $2b$ as the basis, and $2c$ as the basis: i.e., you would compute $$a = \frac{1}{2}\left(\frac{1}{1.49} \cdot 30 + \frac{1}{1.98} \cdot 35\right) = 18.9$$ centimeters. There isn't a real way to get the "true" value of $a$ for this turtle. All you can do is estimate, make a guess, and explain how you arrived at it.