On 10/22/2020 7:48 PM, Jordan Brown wrote:

For anybody who is curious about the mistake that I made back then...

I had assumed that a module yielded an object based on its parameters,
period.  That object could then be transformed, after it was created. 
Because it could be transformed after it was created, it couldn't
possibly know the relationship between its local coordinate system and
the world coordinate system, because at the time it was created that
relationship wasn't known.

There are perfectly reasonable graphics systems that use this model.

But that's not how OpenSCAD objects are created.  They are always
created in the context of their ancestral tree of transformations.  That
tree exists before the object is constructed.  In addition, that tree
can pass information down into the object in $ variables.

As a result, it is possible for an OpenSCAD object to know how its
local coordinate system relates to the world coordinate system.

It sounds similar to the Relativity library:
https://github.com/davidson16807/relativity.scad

On Thu, Oct 22, 2020, at 10:48 PM, Jordan Brown wrote:

$mm = [
[1,0,0,0],
[0,1,0,0],
[0,0,1,0],
[0,0,0,1]
];

// Given a point in local coordinates, return its location in world coordinates.
function whereis(p3) =
let(r4 = $mm * [p3.x, p3.y, p3.z, 1])
[ r4.x, r4.y, r4.z];

// Translate, but tracking the transformation.
module translate(delta) {
m = [
[1,0,0,delta.x],
[0,1,0,delta.y],
[0,0,1,delta.z],
[0,0,0,1]
];
$mm = $mm * m;
multmatrix(m) children();
}

// Rotate, tracking transformations.
module rotate(a) {
a = is_num(a) ? [ 0,0,a ] : a;
echo(a);
rx = [
[1, 0,        0,        0],
[0, cos(a.x), -sin(a.x), 0],
[0, sin(a.x), cos(a.x),  0],
[0, 0,        0,        1]
];
ry = [
[ cos(a.y),  0, sin(a.y), 0],
[ 0,        1, 0,        0],
[ -sin(a.y), 0, cos(a.y), 0],
[ 0,        0, 0,        1]
];
rz = [
[ cos(a.z), -sin(a.z), 0, 0],
[ sin(a.z), cos(a.z),  0, 0],
[ 0,        0,        1, 0],
[ 0,        0,        0, 1]
];
m = rxryrz;
$mm = $mm * m;
multmatrix(m) children();
}

// Scale, tracking transformations
module scale(s) {
s = is_num(s) ? [ s,s,s ] : s;
m = [
[ s.x, 0,  0,  0 ],
[ 0,  s.y, 0,  0 ],
[ 0,  0,  s.z, 0 ],
[ 0,  0,  0,  1 ]
];
$mm = $mm * m;
multmatrix(m) children();
}

scale(2) rotate(45) translate([0,5,0]) translate([10,0,0]) {
cube();
echo(whereis([0,0,0]));
}

// Translate, but tracking the transformation and reverse transformation.
module translate(delta) {
m = [
[1,0,0,delta.x],
[0,1,0,delta.y],
[0,0,1,delta.z],
[0,0,0,1]
];
$mm = $mm * m;
rm = [
[1,0,0,-delta.x],
[0,1,0,-delta.y],
[0,0,1,-delta.z],
[0,0,0,1]
];
$rm = $rm * rm;
multmatrix(m) children();
}

multmatrix($rm) children();

}

cube();
origin() cube();

}

On 10/22/2020 8:27 PM, Doug Moen wrote:

Not really.  Relativity (from a brief look) is about constructing
objects connected to one another.

This concept is about managing the relationship between local
coordinates and world coordinates.

The two are orthogonal to one another.

This is assuming there is some reverse logic provided at the time the forward logic of an openscad design is conceived. Unfortunately for people who ask these questions about openscad the original designers of the scripts (let's not call it a programming language until it can do X=X+1) only looked forward to what it could produce (ie parts for 3D printers).  Not how it it could be introspective, and analyse itself. Its current powers are a noble and awesome achievement for sure, and should be applauded, but in this race to simply produce "things" lies its main restriction.

Those of you who desire to work your way back through the 3-D machinations of your design to reliable coordinates in a defined universe will be ultimately disappointed. I am not, because I don't aspire to these heights of intellectual achievement. Design it, print it for me.

But I do understand what you desire, however only a total rewrite of openscad could achieve that. There are too many hours invested by the people who have built Openscad in the way it is useful for them now, to invest the intellectual effort you improve it beyond what will make most of their previous designs unworkable.

I have sympathies on both sides.
Cheers, RobW

On 23 October 2020 3:35:06 pm AEDT, Jordan Brown openscad@jordan.maileater.net wrote:

Actually if you want to go to the effort of overriding all the transformation built-ins (translate, rotate, scale) with user modules calling multmatrix(), and provide an alternate call for multmatrix() itself, it should become fairly trivial to keep track of the world coordinates compared to local.

-Revar

On 2020-10-23 04:48, Jordan Brown wrote:

If you want to be a purist you might say that an object and its
transformation(s) are separate, the transformation defines the context
and an object can exist in several contexts at the same time. I come
from the tradition of Finite Element Analysis and super-element
techniques where this is important. A super-element can thus not know
its transformation, because it exists in multiple contexts on multiple
levels, and the same super-element might be "re-tracked" in several
contexts.

However, in Constructive Solid Geometry (CSG) this way of looking at
things is not so central, and unlike for Finite Element super-elements a
CSG object is typically copied with each transformation, so each copy
can thus know its transformation. This is exactly what happens in
AngelCAD. Also, since 3d objects are named variables there, they can be
used to return some information. Any 2d or 3d object inherits from
"shape" ( https://arnholm.github.io/angelcad-docs/docs/classshape.html )
and a shape can return its transformation in the form of a tmatrix
object which is essentially just a 4x4 homogenous transformation matrix
like in your code examples

General NxN matrix inversion isn't trivial, but the special case of 4x4
inversion is pretty much trivial and can be coded symbolically.

Carsten Arnholm

This seems to work:

$local_matrix = undef;

function _wc_transpose(m) = [
    for (c=[0:3]) [
        for (r=[0:3]) m[r][c]
    ]
];

function _wc_minor(m,r,c) = [
    for (rr=[0:1:len(m)-1])
    if (rr != r) [
        for (cc=[0:1:len(m[rr])-1])
        if (cc != c) 
        m[rr][cc]
    ]
];

function _wc_sum(v,_total=0,_i=0) =
    _i>=len(v) ? _total :
    _wc_sum(v, _total+v[_i], _i+1);

function _wc_determinant(m) =
    len(m)==2? m[0][0] * m[1][1] - m[0][1] * m[1][0] :
    _wc_sum([
        for (c = [0:1:len(m[0])-1])
        ((c%2==0)? 1 : -1) * m[0][c] *
        _wc_determinant(_wc_minor(m,0,c))
    ]);

function _wc_inverse(m) =
    let(
        det = _wc_determinant(m),
        inv = _wc_transpose([
            for (r = [0:3]) [
                for (c = [0:3])
                ((r+c) % 2 == 0? 1 : -1) *
                _wc_determinant(_wc_minor(m,r,c))
            ]
        ]) / det
    ) inv;


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

function _wc_translate(v) = [
    [1, 0, 0, v.x],
    [0, 1, 0, v.y],
    [0, 0, 1, v.z],
    [0 ,0, 0,   1]
];

function _wc_scale(v) = [
    [v.x,   0,   0, 0],
    [  0, v.y,   0, 0],
    [  0,   0, v.z, 0],
    [  0,   0,   0, 1]
];

function _wc_rotx(ang) = [
    [1,        0,         0,   0],
    [0, cos(ang), -sin(ang),   0],
    [0, sin(ang),  cos(ang),   0],
    [0,        0,         0,   1]
];

function _wc_roty(ang) = [
    [ cos(ang), 0, sin(ang),   0],
    [        0, 1,        0,   0],
    [-sin(ang), 0, cos(ang),   0],
    [        0, 0,        0,   1]
];

function _wc_rotz(ang) = [
    [cos(ang), -sin(ang), 0, 0],
    [sin(ang),  cos(ang), 0, 0],
    [       0,         0, 1, 0],
    [       0,         0, 0, 1]
];

function _wc_rot_by_axis(u, ang) =
    ang==0? _wc_ident() :
    let(
        u = u/norm(u),
        c = cos(ang),
        c2 = 1-c,
        s = sin(ang)
    ) [
        [u.x*u.x*c2+c    , u.x*u.y*c2-u.z*s, u.x*u.z*c2+u.y*s, 0],
        [u.y*u.x*c2+u.z*s, u.y*u.y*c2+c    , u.y*u.z*c2-u.x*s, 0],
        [u.z*u.x*c2-u.y*s, u.z*u.y*c2+u.x*s, u.z*u.z*c2+c    , 0],
        [               0,                0,                0, 1]
    ];

function _wc_vec_angle(v1,v2) =
    let(
        n0 = norm(v1),
        n1 = norm(v2)
    )
    assert(n0>0 && n1>0, "Zero length vector.")
    let (
        c1 = (v1*v2)/(n0*n1),
        c2 = min(max(c1,-1),1) // Correct for FP rounding errors.
    ) acos(c2);
 
function _wc_to3d(v) = [for (i=[0:2]) i<len(v)? v[i] : 0];

function _wc_vec_axis(v1,v2) =
        let(
          eps = 1e-6,
          w1 = _wc_to3d(v1/norm(v1)),
          w2 = _wc_to3d(v2/norm(v2)),
          w3 = (norm(w1-w2) > eps && norm(w1+w2) > eps) ? w2 
               : (norm([abs(w2.x),abs(w2.y),abs(w2.z)]-[0,0,1]) > eps)? [0,0,1] 
               : [1,0,0],
          x = cross(w1,w3)
        ) x/norm(x);


module translate(v) {
    multmat(_wc_translate(v)) children();
}


module scale(v) {
    multmat(_wc_scale(v)) children();
}


module rotate(a=0, v) {
    mat = is_undef(v)? (
        is_list(a)? (
            _wc_rotx(len(a)>=1? assert(is_num(a.x)) a.x : 0) *
            _wc_roty(len(a)>=2? assert(is_num(a.y)) a.y : 0) *
            _wc_rotz(len(a)>=3? assert(is_num(a.z)) a.z : 0)
        ) : (
            assert(is_num(a)) _wc_rotz(a)
        )
    ) : (
        assert(is_num(a))
        assert(is_list(v))
        let( v = [ for (i=[0:2]) v[i] ] )
        _wc_rot_by_axis(v, a)
    );
    multmat(mat) children();
}


module multmat(mat) {
    $local_matrix = is_undef($local_matrix)? mat : ($local_matrix * mat);
    multmatrix(mat) children();
}


// Returns the local transformation matrix.
function local_matrix() =
    is_undef($local_matrix)? _wc_ident() :
    $local_matrix;


// Returns the local translation vector [X,Y,Z]
function local_translation() =
    is_undef($local_matrix)? [0,0,0] :
    let(
        vec = [0, 0, 0, 1],
        lvec = $local_matrix * vec
    ) _wc_to3d(lvec);


// If mat is undef, resets to the world reference frame.
// If given a matrix in mat, resets to that reference frame.
module reference_frame(mat) {
    if (is_undef(mat)) {
        multmat(_wc_inverse($local_matrix)) children();
    } else {
        multmat(_wc_inverse(mat)) children();
    }
}

translate([30,40,50]) {
    scale([2,3,4]) {
        rotate(30,v=[1,1,0]) {
            cylinder(d1=5,d2=0,h=5);
            reference_frame() {
                cylinder(d1=5,d2=0,h=5);  // This gets rendered in the world reference frame
            }
        }
    }
}

You should be able to also store local reference frames in the middle there, in $special_vars and pass those to reference_frame() to access arbitrary reference frames.  You just have to make sure that your code never calls multmatrix(), but multmat() instead.

• Revar

On 10/23/2020 1:43 AM, arnholm@arnholm.org wrote:

See my separate message on the mistake that I made in March 2019.  That
was how I was thinking, that an object was a black box to which a
transformation was applied.

But that's not what OpenSCAD does.  I wouldn't say that it "copies" the
object, but rather that each invocation creates a new instance of the
object, in the context of its transformation.

Neither scheme is wrong... they're just different.

On 10/23/2020 1:00 AM, Rob Ward wrote:

But, for almost all transformations, there is.

When you say

translate([10,0,0]) ...

the inverse is

translate([-10,0,0]) ...

and similarly for any other transformation, except scale(0) variants.

The trick is in keeping track of those transformations and their
inverses as you work your way through the stack of transformations.

Note that I'm not attempting to "undo" Boolean operations... just
transformations.