Kevin,thank you,and yes,that is about what happens in this process.
Your diagram is indeed how this type of construction works,and it's how we think about it.
In practice,however,some things happen unexpectedly;some due to the lack of careful planning,others purely random(particularly when using essentially Junk materials).
Originally,long time ago,tools rarely were designed from scratch.
Apprenticeship lasted 6-7 years got you familiar with the tools and process in general,then as a Journeyman you traveled around different shops becoming familiar with different ways that other people did things.
But eventually a smith settled to a pre-existing method.Tools,materials,fuel,proportions of stock,style of the product et c. were all developed over a very long period of time by a succession of people,evolving very slowly.
At present we're missing So many parts of the puzzle that it's not even funny...Some things we may never learn,as in Why they were done that way.
Here's a funky fact that has an "elephant in the room" kind of effect,always present,but nothing can be done about it,so ignoring it becomes a habit:
Forging,by definition,as in Plastic Deformation,requires an N-th Minimal Effort.
Meaning that the energy applied to a chunk of steel must Penetrate all throughout the piece,with sufficient force.
ONLY then it qualifies as a Forging blow.If it is less than N,it is then only a Riveting blow,the distortion is limited to whatever depth from surface,not affecting the entire mass.
So on average the amount of steel that a person can forge by hand is somewhere around 5/8" square stock...Only that much can be truly Forged,in a controlled manner.
That force requirement increases not gradually but in a mathematical( i think)proportion,rapidly exceeding the ability of a human to truly forge a chunk of steel properly.
An axe of any size is already beyond what any human can handle(a scrawny old worn-out human like me especially).
To some extent at least this is helped by Welding,which is a combination of fusion and deformation.
And other assorted trickery.
All that makes the planning even more complicated.But yes,computations are great,and very necessary,and Mensuration was long a most important part of forging science.
BUT-it can only work properly if one had an unlimited force at one's disposal.
(let's say a self-contained air-hammer in,oh,maybe 1 ton capacity,i'd call that unlimited for an average axe).
So yes,it Is useful,and very helpful in a way,but very far from the Whole story...
