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tutorial update, doc fixes, bugfixes

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Adrian Mariano 2024-04-29 21:18:06 -04:00
parent 3fa4f967d4
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2 changed files with 153 additions and 81 deletions
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attachments.scad
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@ -602,14 +602,13 @@ module orient(anchor, spin) {
// align = optional alignment direction or directions for aligning the children. Default: CENTER // align = optional alignment direction or directions for aligning the children. Default: CENTER
// --- // ---
// inside = if true, place object inside the parent instead of outside. Default: false // inside = if true, place object inside the parent instead of outside. Default: false
// inset = a value to shift the child inward, away from the alignent location. Default: 0 // inset = shift the child away from the alignment edge/corner by this amount. Default: 0
// shiftout = A value to shift an inside object outward so that it overlaps all the aligned faces. Default: 0 // shiftout = Shift an inside object outward so that it overlaps all the aligned faces. Default: 0
// overlap = Amount to sink the child into the parent. Defaults to `$overlap` which is zero by default. // overlap = Amount to sink the child into the parent. Defaults to `$overlap` which is zero by default.
// Side Effects: // Side Effects:
// `$anchor` set to the anchor value used for the child. // `$anchor` set to the anchor value used for the child.
// `$align` set to the align value used for the child. // `$align` set to the align value used for the child.
// `$idx` set to a unique index for each child, increasing by alignment first. // `$idx` set to a unique index for each child, increasing by alignment first.
// `$pos` position where child was placed.
// `$attach_anchor` for each anchor given, this is set to the `[ANCHOR, POSITION, ORIENT, SPIN]` information for that anchor. // `$attach_anchor` for each anchor given, this is set to the `[ANCHOR, POSITION, ORIENT, SPIN]` information for that anchor.
// if inside is true then set default tag to "remove" // if inside is true then set default tag to "remove"
// Example: Cuboid positioned on the right of its parent. Note that it is in its native orientation. // Example: Cuboid positioned on the right of its parent. Note that it is in its native orientation.
@ -687,9 +686,7 @@ module align(anchor,align=CENTER,inside=false,inset=0,shiftout=0,overlap)
anchor = is_vector(anchor) ? [anchor] : anchor; anchor = is_vector(anchor) ? [anchor] : anchor;
align = is_vector(align) ? [align] : align; align = is_vector(align) ? [align] : align;
two_d = _attach_geom_2d($parent_geom); two_d = _attach_geom_2d($parent_geom);
factor = inside?-1:1; factor = inside?-1:1;
for (i = idx(anchor)) { for (i = idx(anchor)) {
face = anchor[i]; face = anchor[i];
$anchor=face; $anchor=face;
@ -731,7 +728,7 @@ function _quant_anch(x) = approx(x,0) ? 0 : sign(x);
// Make arbitrary anchor legal for a given geometry // Make arbitrary anchor legal for a given geometry
function _make_anchor_legal(anchor,geom) = function _make_anchor_legal(anchor,geom) =
in_list(geom[0], ["prismoid","trapzeoid"]) ? [for(v=anchor) _quant_anch(v)] in_list(geom[0], ["prismoid","trapezoid"]) ? [for(v=anchor) _quant_anch(v)]
: in_list(geom[0], ["conoid", "extrusion_extent"]) ? [anchor.x,anchor.y, _quant_anch(anchor.z)] : in_list(geom[0], ["conoid", "extrusion_extent"]) ? [anchor.x,anchor.y, _quant_anch(anchor.z)]
: anchor; : anchor;
@ -743,7 +740,7 @@ function _make_anchor_legal(anchor,geom) =
// Topics: Attachments // Topics: Attachments
// See Also: attachable(), position(), align(), face_profile(), edge_profile(), corner_profile() // See Also: attachable(), position(), align(), face_profile(), edge_profile(), corner_profile()
// Usage: // Usage:
// PARENT() attach(parent, child, [align=], [spin=], [overlap=], [inside=]) CHILDREN; // PARENT() attach(parent, child, [align=], [spin=], [overlap=], [inside=], [inset=], [shiftout=]) CHILDREN;
// PARENT() attach(parent, [overlap=], [spin=]) CHILDREN; // PARENT() attach(parent, [overlap=], [spin=]) CHILDREN;
// Description: // Description:
// Attaches children to a parent object at an anchor point or points, oriented in the anchor direction. // Attaches children to a parent object at an anchor point or points, oriented in the anchor direction.
@ -762,23 +759,29 @@ function _make_anchor_legal(anchor,geom) =
// When an object is attached to one of the other anchors its FRONT will be pointed DOWN and its // When an object is attached to one of the other anchors its FRONT will be pointed DOWN and its
// BACK pointed UP. You can change this using the `spin=` argument to attach(). Note that this spin // BACK pointed UP. You can change this using the `spin=` argument to attach(). Note that this spin
// rotates around the attachment vector and is not the same as the spin argument to the child, which // rotates around the attachment vector and is not the same as the spin argument to the child, which
// will usually rotate around some other direction that may be hard to predict. // will usually rotate around some other direction that may be hard to predict. For 2D objects you cannot
// give spin because it is not possible to spin around the attachment vector; spinning the object around the Z axis
// would change the child orientation so that the anchors are no longer parallel. Furthermore, any spin
// parameter you give to the child will be ignored so that the attachment condition of parallel anchors is preserved.
// . // .
// As with {{align()}} you can use the `align=` parameter to align the child to an edge or corner of the // As with {{align()}} you can use the `align=` parameter to align the child to an edge or corner of the
// face where that child is attached. For example `attach(TOP,BOT,align=RIGHT)` would stand the child // face where that child is attached. For example `attach(TOP,BOT,align=RIGHT)` would stand the child
// up on the top while aligning it with the right edge of the top, and `attach(RIGHT,BOT,align=TOP)` which // up on the top while aligning it with the right edge of the top face, and `attach(RIGHT,BOT,align=TOP)` which
// stand the object on the right face while aligning with the top edge. If you apply spin using the // stand the object on the right face while aligning with the top edge. If you apply spin using the
// argument to `attach()` then it will be taken into account for the alignment. If you apply spin with // argument to `attach()` then it will be taken into account for the alignment. If you apply spin with
// a parameter to the child it will NOT be taken into account. // a parameter to the child it will NOT be taken into account. Note that spin is not permitted for
// . // 2D objects because it would change the child orientation so that the anchors are no longer parallel.
// Because the attachment process forces an orientation and anchor point for the child, it overrides // When you use `align=` you can also adjust the position using `inset=`, which shifts the child
// any such specifications you give to the child: both `anchor=` and `orient=` given to the child are // away from the edge or corner it is aligned to.
// ignored with the **double argument** version of `attach()`. As noted above, you can give `spin=` to the
// child but using the `spin=` parameter to `attach()` is more likely to be useful.
// . // .
// If you give `inside=true` then the anchor arrows are lined up so they are pointing the same direction and // If you give `inside=true` then the anchor arrows are lined up so they are pointing the same direction and
// the child object will be located inside the parent. In this case a default "remove" tag is applied to // the child object will be located inside the parent. In this case a default "remove" tag is applied to
// the children. // the children.
// .
// Because the attachment process forces an orientation and anchor point for the child, it overrides
// any such specifications you give to the child: **both `anchor=` and `orient=` given to the child are
// ignored** with the **double argument** version of `attach()`. As noted above, you can give `spin=` to the
// child but using the `spin=` parameter to `attach()` is more likely to be useful.
// . // .
// For the single parameter version of `attach()` you give only the `parent` anchor. The `align` direction // For the single parameter version of `attach()` you give only the `parent` anchor. The `align` direction
// is not permitted. In this case the child is placed at the specified parent anchor point // is not permitted. In this case the child is placed at the specified parent anchor point
@ -787,7 +790,7 @@ function _make_anchor_legal(anchor,geom) =
// from the parent. If you want the cube sitting on the parent you need to anchor the cube to its bottom: // from the parent. If you want the cube sitting on the parent you need to anchor the cube to its bottom:
// `attach(TOP) cuboid(2,anchor=BOT);`. // `attach(TOP) cuboid(2,anchor=BOT);`.
// . // .
// The **single argument** version of `attach()` respects `anchor=` and `orient=` given to the child. // The **single argument** version of `attach()` **respects `anchor=` and `orient=` given to the child.**
// These options will probably be necessary, in fact, to get the child correctly positioned. Note that // These options will probably be necessary, in fact, to get the child correctly positioned. Note that
// giving `spin=` to `attach()` in this case is the same as applying `zrot()` to the child. // giving `spin=` to `attach()` in this case is the same as applying `zrot()` to the child.
// . // .
@ -802,18 +805,68 @@ function _make_anchor_legal(anchor,geom) =
// parent = The parent anchor point to attach to or a list of parent anchor points. // parent = The parent anchor point to attach to or a list of parent anchor points.
// child = Optional child anchor point. If given, orients the child to connect this anchor point to the parent anchor. // child = Optional child anchor point. If given, orients the child to connect this anchor point to the parent anchor.
// --- // ---
// align = If `child` is given you can specify alignment to shift the child to an edge or corner of the parent. // align = If `child` is given you can specify alignment or list of alistnments to shift the child to an edge or corner of the parent.
// inset = Shift aligned children away from their alignment edge/corner by this amount. Default: 0
// shiftout = Shift an inside object outward so that it overlaps all the aligned faces. Default: 0
// inside = If `child` is given you can set `inside=true` to attach the child to the inside of the parent for diff() operations. Default: false
// overlap = Amount to sink child into the parent. Equivalent to `down(X)` after the attach. This defaults to the value in `$overlap`, which is `0` by default. // overlap = Amount to sink child into the parent. Equivalent to `down(X)` after the attach. This defaults to the value in `$overlap`, which is `0` by default.
// spin = Amount to rotate the parent around the axis of the parent anchor. // spin = Amount to rotate the parent around the axis of the parent anchor. (Only permitted in 3D.)
// Side Effects: // Side Effects:
// `$idx` is set to the index number of each anchor if a list of anchors is given. Otherwise is set to `0`. // `$anchor` set to the parent anchor value used for the child.
// `$attach_anchor` for each `from=` anchor given, this is set to the `[ANCHOR, POSITION, ORIENT, SPIN]` information for that anchor. // `$align` set to the align value used for the child.
// `$attach_to` is set to the value of the `to=` argument, if given. Otherwise, `undef` // `$idx` set to a unique index for each child, increasing by alignment first.
// Example: // `$attach_anchor` for each anchor given, this is set to the `[ANCHOR, POSITION, ORIENT, SPIN]` information for that anchor.
// spheroid(d=20) { // if inside is true then set default tag to "remove"
// attach(TOP) down(1.5) cyl(l=11.5, d1=10, d2=5, anchor=BOTTOM); // `$attach_to` is set to the value of the `child` argument, if given. Otherwise, `undef`
// attach(RIGHT, BOTTOM) down(1.5) cyl(l=11.5, d1=10, d2=5); // Example: Cylinder placed on top of cube:
// attach(FRONT, BOTTOM, overlap=1.5) cyl(l=11.5, d1=10, d2=5); // cuboid(50)
// attach(TOP,BOT) cylinder(d1=30,d2=15,h=25);
// Example: Cylinder on right and front side of cube:
// cuboid(50)
// attach([RIGHT,FRONT],BOT) cylinder(d1=30,d2=15,h=25);
// Example: Using `align` can align child object(s) with edges
// prismoid(50,25,25) color("green"){
// attach(TOP,BOT,align=[BACK,FWD]) cuboid(4);
// attach(RIGHT,BOT,align=[TOP,BOT]) cuboid(4);
// }
// Example: One aligned to the corner upside down (light blue) and one inset fromt the corner (pink), one aligned on a side (orange) and one rotated and aligned (green).
// cuboid(30) {
// attach(TOP,TOP,align=FRONT+RIGHT) color("lightblue") prismoid(5,3,3);
// attach(TOP,BOT,inset=3,align=FRONT+LEFT) color("pink") prismoid(5,3,3);
// attach(FRONT,RIGHT,align=TOP) color("orange") prismoid(5,3,3);
// attach(FRONT,RIGHT,align=RIGHT,spin=90) color("lightgreen") prismoid(5,3,3);
// }
// Example: Rotation not a multiple of 90 degrees with alignment. The children are aligned on a corner.
// cuboid(30)
// attach(FRONT,BOT,spin=33,align=[RIGHT,LEFT,TOP,BOT,RIGHT+TOP])
// color("lightblue")cuboid(4);
// Example: Anchoring the cone onto the sphere gives a single point of contact.
// spheroid(d=20)
// attach([1,1.5,1], BOTTOM) cyl(l=11.5, d1=10, d2=5);
// Example: Using the `overlap` option can help:
// spheroid(d=20)
// attach([1,1.5,1], BOTTOM, overlap=1.5) cyl(l=11.5, d1=10, d2=5);
// Example: Alignment works for cylinders but you can only align with either the top or bototm face:
// cyl(h=30,d=10)
// attach([LEFT,[1,1.3]], BOT,align=TOP) cuboid(6);
// Example: Attaching to edges. The light blue and orange objects are attached to edges. The purple object is attached to an edge and aligned.
// prismoid([20,10],[10,10],7){
// attach(RIGHT+TOP,BOT,align=FRONT) color("pink")cuboid(2);
// attach(BACK+TOP, BOT) color("lightblue")cuboid(2);
// attach(RIGHT+BOT, RIGHT,spin=90) color("orange")cyl(h=8,d=1);
// }
// Example: Attaching inside the parent. For inside attachment the anchors are lined up pointing the same direction, so the most natural way to anchor the child is using its TOP anchor.
// back_half()
// diff()
// cuboid(20)
// attach(TOP,TOP,inside=true,shiftout=0.01) cyl(d1=10,d2=5,h=10);
// Example: Attaching inside the parent with alignment
// diff()
// cuboid(20){
// attach(TOP,TOP,inside=true,align=RIGHT,shiftout=.01) cuboid([8,7,3]);
// attach(TOP,TOP,inside=true,align=LEFT+FRONT,shiftout=0.01) cuboid([3,4,5]);
// attach(RIGHT+FRONT, TOP, inside=true) cuboid([10,3,5]);
// attach(RIGHT+FRONT, TOP, inside=true, align=TOP,shiftout=.01) cuboid([5,1,2]);
// } // }
module attach(parent, child, overlap, align, spin=0, norot, inset=0, shiftout=0, inside=false, from, to) module attach(parent, child, overlap, align, spin=0, norot, inset=0, shiftout=0, inside=false, from, to)
@ -830,62 +883,72 @@ module attach(parent, child, overlap, align, spin=0, norot, inset=0, shiftout=0,
req_children($children); req_children($children);
dummy=assert($parent_geom != undef, "No object to attach to!") dummy=assert($parent_geom != undef, "No object to attach to!")
assert(is_undef(align) || (is_vector(align) && (len(align)==2 || len(align)==3)), "align must be a 2-vector or 3-vector")
assert(is_undef(child) || is_string(child) || (is_vector(child) && (len(child)==2 || len(child)==3)), "child must be a named anchor (a string) or a 2-vector or 3-vector") assert(is_undef(child) || is_string(child) || (is_vector(child) && (len(child)==2 || len(child)==3)), "child must be a named anchor (a string) or a 2-vector or 3-vector")
assert(is_undef(align) || !is_string(child), "child is a named anchor. Named anchors are not supported with align="); assert(is_undef(align) || !is_string(child), "child is a named anchor. Named anchors are not supported with align=");
overlap = (overlap!=undef)? overlap : $overlap;
anchors = (is_vector(parent)||is_string(parent))? [parent] : parent;
two_d = _attach_geom_2d($parent_geom); two_d = _attach_geom_2d($parent_geom);
overlap = (overlap!=undef)? overlap : $overlap;
parent = first_defined([parent,from]); parent = first_defined([parent,from]);
dummy4 = assert(is_string(parent) || is_list(parent), "Invalid parent anchor or anchor list"); anchors = is_vector(parent) || is_string(parent) ? [parent] : parent;
align_list = is_undef(align) ? [undef]
: is_vector(align) || is_string(align) ? [align] : align;
dummy4 = assert(is_string(parent) || is_list(parent), "Invalid parent anchor or anchor list")
assert(spin==0 || (!two_d || is_undef(child)), "spin is not allowed for 2d objects when 'child' is given");
child_temp = first_defined([child,to]); child_temp = first_defined([child,to]);
child = two_d ? _force_anchor_2d(child_temp) : child_temp; child = two_d ? _force_anchor_2d(child_temp) : child_temp;
align = is_undef(align) ? undef dummy2=assert(align_list==[undef] || is_def(child), "Cannot use 'align' without 'child'")
: two_d ? _force_anchor_2d(align) : point3d(align);
dummy2=assert(is_undef(align) || is_def(child), "Cannot use 'align' without 'child'")
assert(!inside || is_def(child), "Cannot use 'inside' without 'child'") assert(!inside || is_def(child), "Cannot use 'inside' without 'child'")
assert(inset==0 || is_def(child), "Cannot specify 'inset' without 'child'") assert(inset==0 || is_def(child), "Cannot specify 'inset' without 'child'")
assert(shiftout==0 || is_def(child), "Cannot specify 'shiftout' without 'child'"); assert(shiftout==0 || is_def(child), "Cannot specify 'shiftout' without 'child'");
for ($idx = idx(anchors)) { factor = inside?-1:1;
dummy2= $attach_to = u_mul(factor,child);
assert(is_string(anchors[$idx]) || (is_vector(anchors[$idx]) && (len(anchors[$idx])==2 || len(anchors[$idx])==3)), for (anch_ind = idx(anchors)) {
str("parent[",$idx,"] is ",anchors[$idx]," but it must be a named anchor (string) or a 2-vector or 3-vector")) dummy=assert(is_string(anchors[anch_ind]) || (is_vector(anchors[anch_ind]) && (len(anchors[anch_ind])==2 || len(anchors[anch_ind])==3)),
assert(is_undef(align) || !is_string(anchors[$idx]), str("parent[",anch_ind,"] is ",anchors[anch_ind]," but it must be a named anchor (string) or a 2-vector or 3-vector"))
str("parent[",$idx,"] is a named anchor (",anchors[$idx],"), but named anchors are not wupported with align=")); assert(align_list==[undef] || !is_string(anchors[anch_ind]),
anchr = is_string(anchors[$idx])? anchors[$idx] str("parent[",anch_ind,"] is a named anchor (",anchors[anch_ind],"), but named anchors are not supported with align="));
: two_d?_force_anchor_2d(anchors[$idx]) anchor = is_string(anchors[anch_ind])? anchors[anch_ind]
:anchors[$idx]; : two_d?_force_anchor_2d(anchors[anch_ind])
dummy=assert(is_undef(align) || all_zero(v_mul(anchr,align)), : point3d(anchors[anch_ind]);
str("align (",align,") cannot include component parallel to parent anchor (",anchr,")")); anchor_data = _find_anchor(anchor, $parent_geom);
anch = _find_anchor(anchr, $parent_geom); $anchor=anchor;
pos = is_undef(align) ? anch[1] : _find_anchor(anchr+align, $parent_geom)[1]; for(align_ind = idx(align_list)){
factor = inside?-1:1; align = is_undef(align_list[align_ind]) ? undef
$attach_to = is_vector(child) ? factor*child : child; : assert(is_vector(align_list[align_ind],2) || is_vector(align_list[align_ind],3), "align direction must be a 2-vector or 3-vector")
$attach_anchor = list_set(anch, 1, pos); /// two_d ? _force_anchor_2d(align_list[align_ind])
startdir = anchr==UP || anchr==DOWN ? BACK : UP; : point3d(align_list[align_ind]);
enddir = is_undef(child) || child.z==0 ? UP : BACK; $idx = align_ind+len(align_list)*anch_ind;
anchor_adjustment = is_undef(align)? CTR $align=align;
: two_d ? zrot(spin, rot(to=factor*child,from=-anchr,p=align)) dummy=assert(is_undef(align) || all_zero(v_mul(anchor,align)),
: apply( frame_map(x=factor*child, z=enddir) str("Invalid alignment: align value (",align,") includes component parallel to parent anchor (",anchor,")"));
*frame_map(x=-anchr, z=startdir, reverse=true) pos = is_undef(align) ? anchor_data[1] : _find_anchor(anchor+align, $parent_geom)[1];
*rot(v=parent,-spin), align); $attach_anchor = list_set(anchor_data, 1, pos); ///
$anchor_override=all_zero(anchor_adjustment)? inside?child:undef startdir = anchor==UP || anchor==DOWN ? BACK : UP - (anchor*UP)*anchor/(anchor*anchor);
:child+anchor_adjustment; enddir = is_undef(child) || child.z==0 ? UP : BACK;
reference = two_d? BACK : UP; child_adjustment = is_undef(align)? CTR
offsetdir = is_undef(align) ? CTR : two_d ? rot(to=factor*child,from=-anchor,p=align)
: apply(zrot(-spin)*frame_map(x=reference, z=BACK)*frame_map(x=anchr, z=startdir, reverse=true), : apply( frame_map(x=factor*child, z=enddir)
align); *frame_map(x=-anchor, z=startdir, reverse=true)
spinaxis = two_d? UP : anch[2]; *rot(v=anchor,-spin), align);
olap = - overlap * reference - inset*offsetdir - shiftout * (-offsetdir - reference); $anchor_override = all_zero(child_adjustment)? inside?child:undef
if (norot || (approx(anch[2],reference) && anch[3]==0)) { : two_d ? zrot(-spin, child+child_adjustment)
translate(pos) rot(v=spinaxis,a=spin) translate(olap) default_tag("remove",inside) children(); : child+child_adjustment;
} else { reference = two_d? BACK : UP;
translate(pos) inset_dir = is_undef(align) ? CTR
rot(v=spinaxis,a=spin) : two_d ? zrot(-spin, rot(to=reference, from=anchor,p=align))
rot(anch[3],from=reference,to=anch[2]){ : apply(zrot(-spin)*frame_map(x=reference, z=BACK)*frame_map(x=anchor, z=startdir, reverse=true),
translate(olap) align);
default_tag("remove",inside) children();}} spinaxis = two_d? UP : anchor_data[2];
olap = - overlap * reference - inset*inset_dir - shiftout * (-inset_dir - reference);
if (norot || (approx(anchor_data[2],reference) && anchor_data[3]==0)) {
translate(pos) rot(v=spinaxis,a=spin) translate(olap) default_tag("remove",inside) children();
} else {
translate(pos)
rot(v=spinaxis,a=spin)
rot(anchor_data[3],from=reference,to=anchor_data[2]){
translate(olap)
default_tag("remove",inside) children();}}
}
} }
} }
@ -3399,8 +3462,8 @@ function _attach_transform(anchor, spin, orient, geom, p) =
) )
two_d? two_d?
assert(is_num(spin)) assert(is_num(spin))
affine3d_zrot(spin) /*affine3d_zrot(spin) * */
* rot(to=FWD, from=point3d(anch[2])) rot(to=FWD, from=point3d(anch[2]))
* affine3d_translate(point3d(-pos)) * affine3d_translate(point3d(-pos))
: :
assert(is_num(spin) || is_vector(spin,3)) assert(is_num(spin) || is_vector(spin,3))

21
tutorials/Attachments.md
View file

@ -1172,25 +1172,34 @@ cube(50, center=true)
## Attaching 2D Children ## Attaching 2D Children
You can use attachments in 2D as well. As usual for the 2D case you You can use attachments in 2D as well. As usual for the 2D case you
can use TOP and BOTTOM as alternative to BACK and FORWARD. can use TOP and BOTTOM as alternative to BACK and FORWARD. With
parent-child anchor attachment you cannot use the spin parameter to
`attach()` nor can you specify spin to the child. Spinning the child
on the Z axis would rotate the anchor arrows out of alignment.
```openscad-2D ```openscad-2D
include <BOSL2/std.scad> include <BOSL2/std.scad>
square(50,center=true) rect(50){
attach(RIGHT,FRONT) attach(RIGHT,FRONT)
trapezoid(w1=30,w2=0,h=30); color("red")trapezoid(w1=30,w2=0,h=30);
attach(LEFT,FRONT,align=[FRONT,BACK],inset=3)
color("green") trapezoid(w1=25, w2=0,h=30);
}
``` ```
```openscad-2D ```openscad-2D
include <BOSL2/std.scad> include <BOSL2/std.scad>
circle(d=50) diff()
circle(d=50){
attach(TOP,BOT,overlap=5) attach(TOP,BOT,overlap=5)
trapezoid(w1=30,w2=0,h=30); trapezoid(w1=30,w2=0,h=30);
attach(BOT,BOT,inside=true)
tag("remove")
trapezoid(w1=30,w2=0,h=30);
}
``` ```
## Tagged Operations ## Tagged Operations
BOSL2 introduces the concept of tags. Tags are names that can be given to attachables, so that BOSL2 introduces the concept of tags. Tags are names that can be given to attachables, so that
you can refer to them when performing `diff()`, `intersect()`, and `conv_hull()` operations. you can refer to them when performing `diff()`, `intersect()`, and `conv_hull()` operations.