One reason to what to know the global coordinates would be for
labeling.  I posted questions about this recently and got good feedback.

For example, if I want to produce a printable schematic or design
document for translating the on-screen object into a real-world object
through traditional machining techniques, I need to know real world
coordinates.    So if a particular edge or point is the result of
multiple transformations, it is difficult to determine that location
for labeling.

It was a thought experiment, not a real-world example.  The point was
that not only is it impossible to retrieve the absolute position from
inside a series of transformations, but it is also impossible to
specify an absolute position for whatever reason.

The easiest way to to it would be to add them to my model, and during
compilation of the model have it echo the marker-id and location to
the console. Having x,y,z would be ok for me, but having all other
details as rotations and scale wouldn't hurt.

How practical would it be to have a function that "rendered"
an [x,y,z] triple and returned an [x,y,z] triple in the
top-level coordinate system?

At the time that functions and echo( ) are executed, has
OpenSCAD even constructed all of the "parent" transformations?

Would it be possible to walk up the work-in-progress tree
transforming a point, at execution time?

Not quite as general, but would it be possible to have an
"object" that didn't actually generate any geometry, but
that got transformed like geometry and, at the time that
one would otherwise output geometry, would print a value
to the console?

The question is whether OpenSCAD has equivalent data already available to
it.

Given a set of functions identity(), rotate([xa,ya,za]),
translate([x,y,z]), et cetera, which return transformation matrices, one
could imagine:

$m = identity();
module xrot(a) {
$m = $m * rotate(a);
rotate(a) children();
}
module xtrans(off) {
$m = $m * translate(off);
translate(off) children();
}
function prender(p)  = $m * p;

... and if you used xrot() and xtrans() instead of rotate() and
translate(), you could at any point
echo(prender([0,0,0]));
to print the "global" coordinates of the local origin.

Does that make sense?  (See a more complete example at the bottom.)

Other notes:

- At any point you could set $m to the identity matrix to "reset" it
for the children, so that they would report coordinates relative to *that*
coordinate system.
- Given $m, you can invert it to get a global-to-local transformation,
allowing you to then do things like add labels that are described in terms
of the global coordinate system.

So it seems pretty clear that one could construct such a mechanism on top
of current OpenSCAD, entirely in userspace.  The question is whether
OpenSCAD has equivalent data already available to it.

To prove to myself that the $m scheme above was actually viable, I
implemented some of it.  As long as I've written it...

ECHO: "marker", [21.2132, 21.2132, 10, 1]

Note that the unit cube emitted by whoami() has been transformed - it's
rotated and elongated - but the label is aligned with the axes and is
undistorted.

Here's the source:

$m = identity();

xrot([0,0,45]) xscale([3,1,1]) xtrans([10,0,10]) whereami("marker");

function identity() = [
[1,0,0,0],
[0,1,0,0],
[0,0,1,0],
[0,0,0,1]
];

function translate(p) = [
[1,0,0,p.x],
[0,1,0,p.y],
[0,0,1,p.z],
[0,0,0,  1]
];

function scale(s) = [
[s.x,  0,  0, 0],
[  0, s.y,  0, 0],
[  0,  0, s.z, 0],
[  0,  0,  0, 1]
];

function rotatex(a) = [
[ 1,      0,      0, 0 ],
[ 0, cos(a), -sin(a), 0 ],
[ 0, sin(a),  cos(a), 0 ],
[ 0,      0,      0, 1 ]
];
function rotatey(a) = [
[  cos(a), 0, sin(a), 0 ],
[      0, 1,      0, 0 ],
[ -sin(a), 0, cos(a), 0 ],
[      0, 0,      0, 1 ]
];
function rotatez(a) = [
[ cos(a), -sin(a), 0, 0 ],
[ sin(a),  cos(a), 0, 0 ],
[      0,      0, 1, 0 ],
[      0,      0, 0, 1 ]
];
function rotate(xyz) =
rotatex(xyz[0])
* rotatey(xyz[1])
* rotatez(xyz[2]);

module xrot(xyz) {
$m = $m * rotate(xyz);
rotate(xyz) children();
}

module xtrans(off) {
$m = $m * translate(off);
translate(off) children();
}
module xscale(s) {
$m = $m * scale(s);
scale(s) children();
}

// Place a unit cube (in local coordinates) at the origin.
// Place a label (with global-default parameters) at that point.
// Echo the label and the global coordinates.
module whereami(label) {
p = $m * [0,0,0,1];
echo(label,p);
cube(1, center=true);
multmatrix(matrix_invert($m)) translate([p.x,p.y,p.z]) text(label);
}

// Matrix inversion, from
//https://github.com/openscad/scad-utils/pull/1/files
function remove(list, i) = [for (i_=[0:1:len(list)-2]) list[i_ < i ? i_ : i_ + 1]];
function det(m) = let(r=[for(i=[0:1:len(m)-1]) i]) det_help(m, 0, r);
function det_help(m, i, r) = len(r) == 0 ? 1 :
m[len(m)-len(r)][r[i]]*det_help(m,0,remove(r,i)) - (i+1<len(r)? det_help(m, i+1, r) : 0);

function matrix_invert(m) = let(r=[for(i=[0:len(m)-1]) i]) [for(i=r) [for(j=r)
((i+j)%2==0 ? 1:-1) * matrix_minor(m,0,remove(r,j),remove(r,i))]] / det(m);
function matrix_minor(m,k,ri, rj) = let(len_r=len(ri)) len_r == 0 ? 1 :
m[ri[0]][rj[k]]*matrix_minor(m,0,remove(ri,0),remove(rj,k)) - (k+1<len_r?matrix_minor(m,k+1,ri,rj) : 0);

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