As I've been reading books on linguistics and phonetics, I've noticed the labelling of the regions of the tongue are quite vague. (ie apical, dorsal, laminal, etc.) Upon thinking about this problem, I thought of the following:

The regions of the tongue can be defined, relative to the length of the speakers tongue, at the angle-forming points of the tongue. The tongue's shape along the vocal tract creates distinct angles.

The distinct angles are as follows: the first is formed by the tongue section that is parallel to the lower jaw and the posterior section which connects the jaw parallel part to the pharynx parallel part. I will call this the dorsal angle. The second is the angle formed by the radical region and the pharynx parallel region. I will call this the radical angle. The third angle is formed by movement of the laminal region. This region, when bent upwards or downwards as in a laminal-alveolar fricative, forms an angle. I will call this the laminal angle.

My suggestion is to label the tongue's regions where these angles occur. The laminal region would be from the tip to the laminal angle, the dorsal from the laminal angle to the dorsal angle, and the radical from the dorsal angle to the radical angle.

Since nobody has the same length of each region, we can first note the total length of the tongue, the length of each of the three regions, and the degree of each angle of each speaker. By use of fractions, we could then describe accurately which region of the tongue we are referring to among a wide range of speakers. (ie. Speakers were found to produce such and such a fricative with the 1/3 dorsal region.) In the previous example the 1/3 dorsal would be the point of the tongue that marks 1/3 the length of the dorsal region with 0/1 dorsal being nearest to the tip.

I know this is a haphazard description but I hope you get my idea. I'm wondering if such a model has been formulated before, (particularly the measuring of the tongue by use of angles) and if so, by who.

Words like apical, laminal etc. are used to describe how a sound is articulated using a specific part of the tongue, no? I have never understood these terms to be names of specific (read: anatomically well-defined) regions of the tongue, but explanations of how the tongue can be used to make different sounds in reference to the areas of the tongue used in the articulation. Apical is an adjective relating to the noun apex which means the tip. I think that's a fairly uncontroversial place of the tongue that we can all imagine. Lamina is the name typically associated with the blade and therefore its adjective laminal references the blade of the tongue (which I think is pretty uncontroversially located in humans). Then you have dorsal which is obviously derived from the Latin for back.

With the proviso that these terms are used mainly to explain the articulatory process, and not to compartmentalise the regions of the tongue in any fixed and highly-specific way, I do not see what the problem is in the current labelling. I think a solution to such a 'problem' might be driven out of misunderstanding the purpose for which these terms are employed in a phonetic description of a speech sound. When what you need to do is to use a description of tongue regions, separating out the highly salient regions, such as tip (apex->apical), blade (lamina->laminal) and back (dorsum->dorsal), seems pretty normal.

Can you conceive of a problem that could arise in using these terms in a real-life context, which could be solved by a newer, more specific and well-defined approach like the one you are suggesting? If so, I'd be happy to hear it and would be more open to analysing the issue of that problem in then deciding whether the current definitions suffice or not.

There would be no point in doing this to replace phonetic notation systems like IPA, because languages don't make contrasts based on anything more specific than can be represented that way.

On the other hand, there is already research that does this sort of thing (possibly not the exact mathematical representations you suggested) for experimental reasons, though the results are usually averaged and generalized rather than reporting the exact measurements. So, yes, studies of articulation have done this, but studies of phonetics/phonology in general do not need to.

One example of a study that has done something like this is in the field of dialectometry, measuring the distances between dialects or languages. So in those cases, measurements based on the physical dimensions of the mouth may be used to determine the distance between certain sounds.

Another case may be in speech synthesizers that are trying to create the same manipulations of the base frequencies from the vocal folds, and in some cases this is done very precisely based on actual articulatory information, though not always.

The problem is the vagueness itself.

But that's exactly it: there is no problem. The general region is in fact where these sounds are produced and the details don't matter too much for most descriptions and so forth.

Though this scheme does not need to replace the IPA, it may enhance it's description when necessary.

Sure. But when is it necessary?