Span an area between 3 points

I have reduced my problem (see
Trouble-with-rotating-Text-to-correct-orientation
<http://forum.openscad.org/Trouble-with-rotating-Text-to-correct-orientation-td30280.html>
) to one rotating angel, I cannot find the correct calculation.

The task is to span an area between two points P1 and P2 in a direction
given by a direction vector D (which is identical to the problem in the
subject).

The image illustrates it:
<http://forum.openscad.org/file/t2988/span-area.jpg>

If you load the attached script ( span-area.scad
<http://forum.openscad.org/file/t2988/span-area.scad>  ) into OpenSCAD and
open the Customizer, you can move P1 and P2 anywhere and the grey fill area
is placed perfect. You can also change d.z from the direction vector d. But
if you change d.x or d.y, then the fill area goes out of the target plane.

I only miss one rotation angel (rx), I tried several combinations of
trigonometrics, but none of them were correct.

Any help would be thankfully appreciaded. If I have solved this problem, I'm
ready to share some - I believe - very helpful modules for OpenSCAD.

Regards




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Euler rotations always confuse me so I did it with Frenet-Seret style vector maths. Seems to work.

function unit(v) = let(n = norm(v)) n ? v / n : v; // Convert v to a unit vector

function orientate(p, r) = // Matrix to translate to p and rotate to r
    let(x = r[0], y = r[1], z = r[2])
        [[x.x, x.y, x.z, p.x],
         [y.x, y.y, y.z, p.y],
         [z.x, z.y, z.z, p.z],
         [  0,   0,   0, 1  ]];


//----------------------------------
// WorkInProgress
// fill area spanned by p1-p2 and d
//----------------------------------
module fill(p1,p2,d)
{
    // this is the vector
    v=p2-p1;

    // length of vector
    l=norm(v);

    // length of d
    h=norm(d);

    // angel between v and d is amout to shear
    sxy=angel(v,d);
   
    tangent = v;
    normal = d;
    binormal = cross(tangent, normal);
    x = unit(tangent);
    z = unit(binormal);
    y = -unit(cross(tangent, binormal));
    rot = [[x.x, y.x, z.x],
           [x.y, y.y, z.y],
           [x.z, y.z, z.z]];
         
  multmatrix(orientate(p1, rot) * sxy(sxy)) area(l,h,"p1","p2","p1+d","p2+d");

// DEBUG: show sheared area before translate/rotate
#multmatrix(sxy(sxy)) area(l,h,"p1","p2","p1+d","p2+d");

}

On Sat, 31 Oct 2020 at 08:38, jjvbhh <[hidden email]> wrote:

I have reduced my problem (see
Trouble-with-rotating-Text-to-correct-orientation
<http://forum.openscad.org/Trouble-with-rotating-Text-to-correct-orientation-td30280.html>
) to one rotating angel, I cannot find the correct calculation.

The task is to span an area between two points P1 and P2 in a direction
given by a direction vector D (which is identical to the problem in the
subject).

The image illustrates it:
<http://forum.openscad.org/file/t2988/span-area.jpg>

If you load the attached script ( span-area.scad
<http://forum.openscad.org/file/t2988/span-area.scad>  ) into OpenSCAD and
open the Customizer, you can move P1 and P2 anywhere and the grey fill area
is placed perfect. You can also change d.z from the direction vector d. But
if you change d.x or d.y, then the fill area goes out of the target plane.

I only miss one rotation angel (rx), I tried several combinations of
trigonometrics, but none of them were correct.

Any help would be thankfully appreciaded. If I have solved this problem, I'm
ready to share some - I believe - very helpful modules for OpenSCAD.

Regards




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nophead wrote
> Euler rotations always confuse me so I did it with Frenet-Seret style
> vector maths. Seems to work.

Yes, works fine! Your great, many thanks for your quick response.  This
really brings me a giant step forward my targets. Now I only have to
understand your magic code.

I will come back soon with some fine measure modules

Regards



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My code basically sets up a new coordinate system aligned to the vectors and then maps the original plane x, y, z to that new set of axes.

On Sat, 31 Oct 2020 at 12:36, jjvbhh <[hidden email]> wrote:

nophead wrote
> Euler rotations always confuse me so I did it with Frenet-Seret style
> vector maths. Seems to work.

Yes, works fine! Your great, many thanks for your quick response.  This
really brings me a giant step forward my targets. Now I only have to
understand your magic code.

I will come back soon with some fine measure modules

Regards



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On 2020-10-31 12:47, nop head wrote:
> Euler rotations always confuse me so I did it with Frenet-Seret style
> vector maths. Seems to work.

Why such complicated approach? What is wrong with simply adding the
vector to the two points to obtain the rectangle area?

Carsten Arnholm

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On 2020-10-31 13:51, nop head wrote:
> It isn't a rectangle, it is a parallelogram with sheared text on it.

My question still stands if it is a parallelogram

Carsten Arnholm

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On 2020-10-31 13:58, nop head wrote:
> So what is your solution to construct a transform to rotate and shear
> a plane to align with an arbitrary parallelogram?

The problem was presented with this illustration
http://forum.openscad.org/file/t2988/span-area.jpg

As far as I can tell you can do exactly what the figure indicates, add
the vector d to each point to arrive at the parallelogram. But maybe I
am missing something?

Carsten Arnholm

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On 2020-10-31 14:10, nop head wrote:
> Yes it would be easy to calculate the forth vector to complete the
> polygon but then how do you draw it with text, which are 2D operations
> operating in the XY plane?

I have not studied the details, but obviously you can compute the plane
equation from 3 points.  Or even simpler, create a vector from P1 to P2
and compute a couple of cross products (the first with vector d) to
arrive at the 3 first columns of the transformation matrix. The fourth
is simply choosing an origin. So I thing you can transform the text with
the resulting matrix.

Carsten Arnholm


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That is pretty much exactly what I did. 

On Sat, 31 Oct 2020 at 13:22, <[hidden email]> wrote:

On 2020-10-31 14:10, nop head wrote:
> Yes it would be easy to calculate the forth vector to complete the
> polygon but then how do you draw it with text, which are 2D operations
> operating in the XY plane?

I have not studied the details, but obviously you can compute the plane
equation from 3 points.  Or even simpler, create a vector from P1 to P2
and compute a couple of cross products (the first with vector d) to
arrive at the 3 first columns of the transformation matrix. The fourth
is simply choosing an origin. So I thing you can transform the text with
the resulting matrix.

Carsten Arnholm


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On 2020-11-01 10:15, nop head wrote:
> Here is a simpler version. Instead of computing the shear and applying
> it I simply set up the X and Y axes along the edges of the
> parallelogram and make Z orthogonal to both with a cross product.
>
> module fill(p1,p2,d)
> {
>     // this is the vector
>     v=p2-p1;
>
>     // length of vector
>     l=norm(v);
>
>     // length of d
>     h=norm(d);
>
>     x = unit(v);
>     y = unit(d);
>     z = unit(cross(v, d));
>     m = [[x.x, y.x, z.x, p1.x],
>          [x.y, y.y, z.y, p1.y],
>          [x.z, y.z, z.z, p1.z],
>          [0,   0,   0,   1   ]];
>
>     multmatrix(m) area(l,h,"p1","p2","p1+d","p2+d");
> }


That's pretty much it, except with this you are assuming it is a
rectangle, not a parallelogram. In the above, d is typically not
orthogonal to v, so the matrix m is then incorrect. That's why I said "a
couple of cross products" and not just one as in your code. If you cross
z with x you can arrive at a new y and thus arrive at an orthogonal m.

Carsten Arholm

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