# [OpenSCAD] Unusual hull() and minkowski modelling

Ronaldo rcmpersiano at gmail.com
Thu Jun 16 23:54:56 EDT 2016

```I have found from a discussion in another thread a whole new set of
possibilities of modelling with hull() and minkowski(). I will present here
some of them.

To start consider the following module definitions:

> module square3(a,b) scale([a,b,0]) cube(1,center=true);
> module circle3(r=1) scale([1,1,0]) cylinder(r=r);
> module point3(p)    translate(p)   scale(0) cube();

Essentially, the first and second are a cube and a cylinder smashed onto the
xy plane, and the third a  cube shrinked to the origin. They don't have
volume and the last one does not have even an area. They are 2D and 0D
"shapes" but they are 3D objects in some sense (the reason I added a 3 in
their names).

If you try to preview any of them you will see nothing. But they are not
empty sets. For instance, you can hull() the point3() with a sphere and get
a drop like model:

> hull() { sphere(5); point3([0,0,12]); }

or start the model of a T-joint:

> hull() { translate([0,-10,0]) rotate([90,0,0]) circle3(5,10);
> cylinder(r=5,h=20,center=true); }

It is possible to do intersections of those shapes with real 3D forms before
the hull:

> hull() {
>     intersection(){ cube(20); circle3(7); }
>     translate([0,0,20]) sphere(10);}

Or even make the hull() of just those strange forms to make a cone:

> translate([0,-20,0]) hull() { circle3(5); point3([0,0,10]); }

Let us add one 1D object to the arsenal:

> module segment3(p,p0=[0,0,0]) {
>     q = p-p0;
>     l = norm(q);
>     b = acos(q/l);
>     c = atan2(q,q);
>     translate(p0)
>         rotate([0, b, c])
>             scale([0,0,l])
>                 cube(1);
> }

Now the cube is smashed to an edge then rotate and scaled in such a way to
bring the edge (a line segment) to lay between the points p and p0. Again
you can't see any preview of segment3(). But it appears when you do the hull
of a set of them:

> module tetrahedron(h){
>     hull() {
>         segment3([0,0,h]); // a segment from [0,0,h] to [0,0,0]
>         segment3([h,0,0]);
>         segment3([0,h,0]);
>     }
> }

Now minkowski. This operator is more restricted. To do a minkowski you need
at least one real 3D object. So when I tried change the hull() in the
tetrahedron() to a minkowski I got a system crash. In other trials, I got
one CGAL error message or nothing at all.

But there is a lot of interesting thing to do with the operator. For
instance, a solid linear extrusion:

> module solid_linear_extrude(h=1){
>     minkowski(){
>         children();
>         segment3([0,0,h]);
>     }
> }
> solid_linear_extrude(20) difference(){ sphere(10); cylinder(r=5,
> h=30,center=true); }

This code took a longer time to preview but it worked. It is easy and faster
do it with hull().

But hull() does not solve the sweep operation. Minkowski does with the
module:

> module sweep_solid(line) {
>     for(i=[0:len(line)-2])
>         minkowski(){
>             children();
>             segment3(line[i],line[i+1]);
>         }
> }

We cannot sweep a 2D shape with sweep_solid() but it is possible to sweep a
sphere(), a cube or, if you have plenty of time, any 3D model. Note that
this is a translational sweep.

Finally, an interesting approach to morphing using minkowski:

> module morphing(t) {
>     minkowski() {
>         scale((1-t)) children(0);
>         scale(t) children(1);
>     }
> }

that is applied here to a sphere and a cylinder in an animation:

> morphing(1/2-cos(360*\$t)/2) { A(); B(); }
> module A() rotate([90,0,0]) cylinder(r=3,h=15,center=true);
> module B() translate([20,0,0]) sphere(5);

For any given 0<t<1,  morphing(t) is a blend of the two children.

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