BOSL2/regions.scad

//////////////////////////////////////////////////////////////////////
// LibFile: regions.scad
//   This file provides 2D boolean geometry operations on paths, where you can
//   compute the intersection or union of the shape defined by point lists, producing
//   a new point list.  Of course, boolean operations may produce shapes with multiple
//   components.  To handle that, we use "regions" which are defined by sets of
//   multiple paths.  
// Includes:
//   include <BOSL2/std.scad>
//////////////////////////////////////////////////////////////////////


// CommonCode:
//   include <BOSL2/rounding.scad>


// Section: Regions
//   A region is a list of non-crossing simple polygons.  Simple polygons are those without self intersections,
//   and the polygons of a region can touch at corners, but their segments should not
//   cross each other.  The actual geometry of the region is defined by XORing together
//   all of the polygons on the list.  This may sound obscure, but it simply means that nested
//   boundaries make rings in the obvious fashion, and non-nested shapes simply union together.
//   Checking that the polygons on a list are simple and non-crossing can be a time consuming test,
//   so it is not done automatically.  It is your responsibility to ensure that your regions are
//   compliant.  You can construct regions by making a list of polygons, or by using
//   boolean function operations such as union() or difference().  And if you must you
//   can clean up an ill-formed region using sanitize_region().  


// Function: is_region()
// Usage:
//   is_region(x);
// Description:
//   Returns true if the given item looks like a region.  A region is defined as a list of zero or more paths.
function is_region(x) = is_list(x) && is_path(x.x);



// Function: check_and_fix_path()
// Usage:
//   check_and_fix_path(path, [valid_dim], [closed], [name])
// Description:
//   Checks that the input is a path.  If it is a region with one component, converts it to a path.
//   Note that arbitrary paths must have at least two points, but closed paths need at least 3 points.  
//   valid_dim specfies the allowed dimension of the points in the path.
//   If the path is closed, removes duplicate endpoint if present.
// Arguments:
//   path = path to process
//   valid_dim = list of allowed dimensions for the points in the path, e.g. [2,3] to require 2 or 3 dimensional input.  If left undefined do not perform this check.  Default: undef
//   closed = set to true if the path is closed, which enables a check for endpoint duplication
//   name = parameter name to use for reporting errors.  Default: "path"
function check_and_fix_path(path, valid_dim=undef, closed=false, name="path") =
    let(
        path =
          is_region(path)? 
               assert(len(path)==1,str("Region ",name," supplied as path does not have exactly one component"))
               path[0]
          :
               assert(is_path(path), str("Input ",name," is not a path"))
               path
    )
    assert(len(path)>(closed?2:1),closed?str("Closed path ",name," must have at least 3 points")
                                        :str("Path ",name," must have at least 2 points"))
    let(valid=is_undef(valid_dim) || in_list(len(path[0]),force_list(valid_dim)))
    assert(
        valid, str(
            "Input ",name," must has dimension ", len(path[0])," but dimension must be ",
            is_list(valid_dim) ? str("one of ",valid_dim) : valid_dim
        )
    )
    closed && approx(path[0], last(path))? list_head(path) : path;



// Function: sanitize_region()
// Usage:
//   r_fixed = sanitize_region(r, [nonzero], [eps]);
// Description:
//   Takes a malformed input region that contains self-intersecting polygons or polygons
//   that cross each other and converts it into a properly defined region without
//   these defects.
// Arguments:
//   r = region to sanitize
//   nonzero = set to true to use nonzero rule for polygon membership.  Default: false
//   eps = Epsilon for geometric comparisons.  Default: `EPSILON` (1e-9)
// Examples:
//   
function sanitize_region(r,nonzero=false,eps=EPSILON) =
     assert(is_region(r))
     exclusive_or(
                  [for(poly=r) each polygon_parts(poly,nonzero,eps)],
                  eps=eps);


// Module: region()
// Usage:
//   region(r);
// Description:
//   Creates 2D polygons for the given region.  The region given is a list of closed 2D paths.
//   Each path will be effectively exclusive-ORed from all other paths in the region, so if a
//   path is inside another path, it will be effectively subtracted from it.
// Example(2D):
//   region([circle(d=50), square(25,center=true)]);
// Example(2D):
//   rgn = concat(
//       [for (d=[50:-10:10]) circle(d=d-5)],
//       [square([60,10], center=true)]
//   );
//   region(rgn);
module region(r)
{
    no_children($children);
    r = is_path(r) ? [r] : r;
    points = flatten(r);
    lengths = [for(path=r) len(path)];
    starts = [0,each cumsum(lengths)];
    paths = [for(i=idx(r)) count(s=starts[i], n=lengths[i])];
    polygon(points=points, paths=paths);
}



// Function: point_in_region()
// Usage:
//   check = point_in_region(point, region, [eps]);
// Description:
//   Tests if a point is inside, outside, or on the border of a region.
//   Returns -1 if the point is outside the region.
//   Returns 0 if the point is on the boundary.
//   Returns 1 if the point lies inside the region.
// Arguments:
//   point = The point to test.
//   region = The region to test against.  Given as a list of polygon paths.
//   eps = Acceptable variance.  Default: `EPSILON` (1e-9)
function point_in_region(point, region, eps=EPSILON, _i=0, _cnt=0) =
    _i >= len(region) ? ((_cnt%2==1)? 1 : -1)
   : let(
           pip = point_in_polygon(point, region[_i], eps=eps)
     )
     pip==0? 0
   : point_in_region(point, region, eps=eps, _i=_i+1, _cnt = _cnt + (pip>0? 1 : 0));



// Function: is_region_simple()
// Usage:
//   bool = is_region_simple(region, [eps]);
// Description:
//   Returns true if the region is entirely non-self-intersecting, meaning that it is
//   formed from a list of simple polygons that do not intersect each other.
// Arguments:
//   region = region to check
//   eps = tolerance for geometric omparisons.  Default: `EPSILON` = 1e-9
function is_region_simple(region, eps=EPSILON) =
   [for(p=region) if (!is_path_simple(p,closed=true,eps)) 1] == [];

/*   &&
   [for(i=[0:1:len(region)-2])
       if (_path_region_intersections(region[i], list_tail(region,i+1), eps=eps) != []) 1
   ] ==[];
*/

// Function: are_regions_equal()
// Usage:
//    b = are_regions_equal(region1, region2, [eps])
// Description:
//    Returns true if the components of region1 and region2 are the same polygons (in any order)
//    within given epsilon tolerance.
// Arguments:
//    region1 = first region
//    region2 = second region
//    eps = tolerance for comparison
function are_regions_equal(region1, region2) =
    assert(is_region(region1) && is_region(region2))
    len(region1) != len(region2)? false :
    __are_regions_equal(region1, region2, 0);

function __are_regions_equal(region1, region2, i) =
    i >= len(region1)? true :
    !is_polygon_in_list(region1[i], region2)? false :
    __are_regions_equal(region1, region2, i+1);


/// Internal Function: _path_region_intersections()
/// Usage:
///   _path_region_intersections(path, region, [closed], [eps]);
/// Description:
///   Returns a sorted list of [SEGMENT, U] that describe where a given path intersects the region
//    in a single point.  (Note that intersections of collinear segments, where the intersection is another segment, are
//    ignored.)
/// Arguments:
///   path = The path to find crossings on.
///   region = Region to test for crossings of.
///   closed = If true, treat path as a closed polygon.  Default: true
///   eps = Acceptable variance.  Default: `EPSILON` (1e-9)
function old_path_region_intersections(path, region, closed=true, eps=EPSILON) =
    let(
        pathclosed = closed && !is_closed_path(path),
        pathlen = len(path),
        regionsegs = [for(poly=region) each pair(poly, is_closed_path(poly)?false:true)]
    )
    sort(
         [for(si = [0:1:len(path)-(pathclosed?1:2)])
              let(
                  a1 = path[si],
                  a2 = path[(si+1)%pathlen],
                  maxax = max(a1.x,a2.x),
                  minax = min(a1.x,a2.x),
                  maxay = max(a1.y,a2.y),
                  minay = min(a1.y,a2.y)
              )
              for(rseg=regionsegs)
                  let(
                      b1 = rseg[0],
                      b2 = rseg[1],
                      isect =
                              maxax < b1.x && maxax < b2.x  ||
                              minax > b1.x && minax > b2.x  ||
                              maxay < b1.y && maxay < b2.y  ||
                              minay > b1.y && minay > b2.y
                            ? undef
                            : _general_line_intersection([a1,a2],rseg,eps)
                  )
                  if (isect && isect[1]>=-eps && isect[1]<=1+eps
                            && isect[2]>=-eps && isect[2]<=1+eps)
                      [si,isect[1]]
         ]
    );


// find the intersection points of a path and the polygons of a
// region; only crossing intersections are caught, no collinear
// intersection is returned.
function _path_region_intersections(path, region, closed=true, eps=EPSILON, extra=[]) =
    let( path = closed ? close_path(path,eps=eps) : path )
    _sort_vectors(
         [ each extra,
           for(si = [0:1:len(path)-2]) let(
            a1 = path[si],
            a2 = path[si+1],
            nrm = norm(a1-a2)
          )
          if( nrm>eps ) let( // ignore zero-length path edges
            seg_normal = [-(a2-a1).y, (a2-a1).x]/nrm,
            ref = a1*seg_normal
          )
          // `signs[j]` is the sign of the signed distance from
          // poly vertex j to the line [a1,a2] where near zero
          // distances are snapped to zero;  poly edges 
          //  with equal signs at its vertices cannot intersect
          // the path edge [a1,a2] or they are collinear and 
          // further tests can be discarded.
          for(poly=region) let(
              poly  = close_path(poly),
              signs = [for(v=poly*seg_normal) v-ref> eps ? 1 : v-ref<-eps ? -1 : 0]
          ) 
          if(max(signs)>=0 && min(signs)<=0 ) // some edge edge intersects line [a1,a2]
          for(j=[0:1:len(poly)-2]) 
          if( signs[j]!=signs[j+1] ) let( // exclude non-crossing and collinear segments
              b1 = poly[j],
              b2 = poly[j+1],
              isect = _general_line_intersection([a1,a2],[b1,b2],eps=eps) 
          )
          if (  isect 
//                && isect[1]> (si==0 && !closed? -eps: 0)  
                && isect[1]>= -eps 
                && isect[1]<= 1+eps 
//                && isect[2]>  0 
                && isect[2]>= -eps
                && isect[2]<= 1+eps )
              [si,isect[1]]
    ]);


// Function: split_path_at_region_crossings()
// Usage:
//   paths = split_path_at_region_crossings(path, region, [eps]);
// Description:
//   Splits a path into sub-paths wherever the path crosses the perimeter of a region.
//   Splits may occur mid-segment, so new vertices will be created at the intersection points.
// Arguments:
//   path = The path to split up.
//   region = The region to check for perimeter crossings of.
//   closed = If true, treat path as a closed polygon.  Default: true
//   eps = Acceptable variance.  Default: `EPSILON` (1e-9)
// Example(2D):
//   path = square(50,center=false);
//   region = [circle(d=80), circle(d=40)];
//   paths = split_path_at_region_crossings(path, region);
//   color("#aaa") region(region);
//   rainbow(paths) stroke($item, closed=false, width=2);
function split_path_at_region_crossings(path, region, closed=true, eps=EPSILON, extra=[]) =
    let(
        path = deduplicate(path, eps=eps),
        region = [for (path=region) deduplicate(path, eps=eps)],
        xings = _path_region_intersections(path, region, closed=closed, eps=eps, extra=extra),
        crossings = deduplicate(
            concat([[0,0]], xings, [[len(path)-1,1]]),
            eps=eps
        ),
        subpaths = [
            for (p = pair(crossings))
                deduplicate(
                    _path_select(path, p[0][0], p[0][1], p[1][0], p[1][1], closed=closed),
                    eps=eps
                )
        ]
    )
    [for(s=subpaths) if (len(s)>1) s];


// Function: region_parts()
// Usage:
//   rgns = region_parts(region);
// Description:
//   Divides a region into a list of connected regions.  Each connected region has exactly one outside boundary
//   and zero or more outlines defining internal holes.  Note that behavior is undefined on invalid regions whose
//   components intersect each other.
// Example(2D,NoAxes):
//   R = [for(i=[1:7]) square(i,center=true)];
//   region_list = region_parts(R);
//   rainbow(region_list) region($item);
// Example(2D,NoAxes):
//   R = [back(7,square(3,center=true)),
//        square([20,10],center=true),
//        left(5,square(8,center=true)),
//        for(i=[4:2:8])
//          right(5,square(i,center=true))];
//   region_list = region_parts(R);
//   rainbow(region_list) region($item);


function old_region_parts(region) =
    let(
        paths = sort(idx=0,
           [
            for(i = idx(region))
              let(
                pt = mean([region[i][0],region[i][1]]),
                cnt = sum([for (j = idx(region))
                              if (i!=j && point_in_polygon(pt, region[j]) >=0) 1])
              )
              [cnt, region[i]]
        ]),
        
        outs = [
            for (candout = paths)
               let(
                  lev = candout[0],
                  parent = candout[1]
               )
               if (lev % 2 == 0)
                  [
                    clockwise_polygon(parent),
                    for (path = paths)
                        if (
                            path[0] == lev+1
                            && point_in_polygon(
                                                lerp(path[1][0], path[1][1], 0.5)
                                                ,parent
                                               ) >= 0
                        )
                        ccw_polygon(path[1])
                  ]
        ]
    )
    outs;


function inside(region, ins=[]) =
  let(
      i = len(ins)
  )
  i==len(region) ? ins
  :
  let(
      pt=mean([region[i][0],region[i][1]])
  )
  i==0 ? inside(region,
                       [[0,
                         for(j=[1:1:len(region)-1]) point_in_polygon(pt,region[j])>=0 ? 1 : 0]])
  : let(
        prev = [for(j=[0:i-1]) point_in_polygon(pt,region[j])>=0 ? 1 : 0],
        check = sum(bselect(ins,prev),repeat(0,len(region))),
        next = [for(j=[i+1:1:len(region)-1]) check[j]>0 ? 1 : point_in_polygon(pt,region[j])>=0 ? 1 : 0]
    )
    inside(region, [
                    each ins,
                    [each prev, 0, each next]
                   ]);


function region_parts(region) =
   let(
       inside = [for(i=idx(region))
                    let(pt = mean([region[i][0], region[i][1]]))
                    [for(j=idx(region))  i==j ? 0
                                       : point_in_polygon(pt,region[j]) >=0 ? 1 : 0]
                ],
       level = inside*repeat(1,len(region))
   )
   [ for(i=idx(region))
      if(level[i]%2==0)
         let(
             possible_children = search([level[i]+1],level,0)[0],
             keep=search([1], select(inside,possible_children), 0, i)[0]
         )
         [
           clockwise_polygon(region[i]),
           for(good=keep)
              ccw_polygon(region[possible_children[good]])
         ]
    ];


// Section: Region Extrusion and VNFs

function _path_path_closest_vertices(path1,path2) =
    let(
        dists = [for (i=idx(path1)) let(j=closest_point(path1[i],path2)) [j,norm(path2[j]-path1[i])]],
        i1 = min_index(subindex(dists,1)),
        i2 = dists[i1][0]
    ) [dists[i1][1], i1, i2];


function _join_paths_at_vertices(path1,path2,v1,v2) =
    let(
        repeat_start = !approx(path1[v1],path2[v2]),
        path1 = clockwise_polygon(polygon_shift(path1,v1)),
        path2 = ccw_polygon(polygon_shift(path2,v2))
    )
    [
        each path1,
        if (repeat_start) path1[0],
        each path2,
        if (repeat_start) path2[0],
    ];                      

        
// Given a region that is connected and has its outer border in region[0],
// produces a polygon with the same points that has overlapping connected paths
// to join internal holes to the outer border.  Output is a single path.  
function _cleave_connected_region(region) =
    len(region)==0? [] :
    len(region)<=1? clockwise_polygon(region[0]) :
    let(
        dists = [
            for (i=[1:1:len(region)-1])
            _path_path_closest_vertices(region[0],region[i])
        ],
        idxi = min_index(subindex(dists,0)),
        newoline = _join_paths_at_vertices(
            region[0], region[idxi+1],
            dists[idxi][1], dists[idxi][2]
        )
    ) len(region)==2? clockwise_polygon(newoline) :
    let(
        orgn = [
            newoline,
            for (i=idx(region))
                if (i>0 && i!=idxi+1)
                    region[i]
        ]
    )
    assert(len(orgn)<len(region))
    _cleave_connected_region(orgn);



// Function: region_faces()
// Usage:
//   vnf = region_faces(region, [transform], [reverse], [vnf]);
// Description:
//   Given a region, applies the given transformation matrix to it and makes a VNF of
//   faces for that region, reversed if necessary.
// Arguments:
//   region = The region to make faces for.
//   transform = If given, a transformation matrix to apply to the faces generated from the region.  Default: No transformation applied.
//   reverse = If true, reverse the normals of the faces generated from the region.  An untransformed region will have face normals pointing `UP`.  Default: false
//   vnf = If given, the faces are added to this VNF.  Default: `EMPTY_VNF`
function region_faces(region, transform, reverse=false, vnf=EMPTY_VNF) =
    let (
        regions = region_parts(region),
        vnfs = [
            if (vnf != EMPTY_VNF) vnf,
            for (rgn = regions) let(
                cleaved = path3d(_cleave_connected_region(rgn)),
                face = is_undef(transform)? cleaved : apply(transform,cleaved),
                faceidxs = reverse? [for (i=[len(face)-1:-1:0]) i] : [for (i=[0:1:len(face)-1]) i]
            ) [face, [faceidxs]]
        ],
        outvnf = vnf_merge(vnfs)
    ) outvnf;


// Function&Module: linear_sweep()
// Usage:
//   linear_sweep(region, height, [center], [slices], [twist], [scale], [style], [convexity]);
// Description:
//   If called as a module, creates a polyhedron that is the linear extrusion of the given 2D region or path.
//   If called as a function, returns a VNF that can be used to generate a polyhedron of the linear extrusion
//   of the given 2D region or path.  The benefit of using this, over using `linear_extrude region(rgn)` is
//   that you can use `anchor`, `spin`, `orient` and attachments with it.  Also, you can make more refined
//   twisted extrusions by using `maxseg` to subsample flat faces.
// Arguments:
//   region = The 2D [Region](regions.scad) or path that is to be extruded.
//   height = The height to extrude the region.  Default: 1
//   center = If true, the created polyhedron will be vertically centered.  If false, it will be extruded upwards from the origin.  Default: `false`
//   slices = The number of slices to divide the shape into along the Z axis, to allow refinement of detail, especially when working with a twist.  Default: `twist/5`
//   maxseg = If given, then any long segments of the region will be subdivided to be shorter than this length.  This can refine twisting flat faces a lot.  Default: `undef` (no subsampling)
//   twist = The number of degrees to rotate the shape clockwise around the Z axis, as it rises from bottom to top.  Default: 0
//   scale = The amount to scale the shape, from bottom to top.  Default: 1
//   style = The style to use when triangulating the surface of the object.  Valid values are `"default"`, `"alt"`, or `"quincunx"`.
//   convexity = Max number of surfaces any single ray could pass through.  Module use only.
//   anchor_isect = If true, anchoring it performed by finding where the anchor vector intersects the swept shape.  Default: false
//   anchor = Translate so anchor point is at origin (0,0,0).  See [anchor](attachments.scad#anchor).  Default: `CENTER`
//   spin = Rotate this many degrees around the Z axis after anchor.  See [spin](attachments.scad#spin).  Default: `0`
//   orient = Vector to rotate top towards, after spin.  See [orient](attachments.scad#orient).  Default: `UP`
// Example: Extruding a Compound Region.
//   rgn1 = [for (d=[10:10:60]) circle(d=d,$fn=8)];
//   rgn2 = [square(30,center=false)];
//   rgn3 = [for (size=[10:10:20]) move([15,15],p=square(size=size, center=true))];
//   mrgn = union(rgn1,rgn2);
//   orgn = difference(mrgn,rgn3);
//   linear_sweep(orgn,height=20,convexity=16);
// Example: With Twist, Scale, Slices and Maxseg.
//   rgn1 = [for (d=[10:10:60]) circle(d=d,$fn=8)];
//   rgn2 = [square(30,center=false)];
//   rgn3 = [for (size=[10:10:20]) move([15,15],p=square(size=size, center=true))];
//   mrgn = union(rgn1,rgn2);
//   orgn = difference(mrgn,rgn3);
//   linear_sweep(orgn,height=50,maxseg=2,slices=40,twist=180,scale=0.5,convexity=16);
// Example: Anchors on an Extruded Region
//   rgn1 = [for (d=[10:10:60]) circle(d=d,$fn=8)];
//   rgn2 = [square(30,center=false)];
//   rgn3 = [for (size=[10:10:20]) move([15,15],p=square(size=size, center=true))];
//   mrgn = union(rgn1,rgn2);
//   orgn = difference(mrgn,rgn3);
//   linear_sweep(orgn,height=20,convexity=16) show_anchors();
module linear_sweep(region, height=1, center, twist=0, scale=1, slices, maxseg, style="default", convexity, anchor_isect=false, anchor, spin=0, orient=UP) {
    region = is_path(region)? [region] : region;
    cp = mean(pointlist_bounds(flatten(region)));
    anchor = get_anchor(anchor, center, "origin", "origin");
    vnf = linear_sweep(
        region, height=height,
        twist=twist, scale=scale,
        slices=slices, maxseg=maxseg,
        style=style
    );
    attachable(anchor,spin,orient, cp=cp, vnf=vnf, extent=!anchor_isect) {
        vnf_polyhedron(vnf, convexity=convexity);
        children();
    }
}


function linear_sweep(region, height=1, center, twist=0, scale=1, slices,
                      maxseg, style="default", anchor_isect=false, anchor, spin=0, orient=UP) =
    let(
        anchor = get_anchor(anchor,center,BOT,BOT),
        region = is_path(region)? [region] : region,
        cp = mean(pointlist_bounds(flatten(region))),
        regions = region_parts(region),
        slices = default(slices, floor(twist/5+1)),
        step = twist/slices,
        hstep = height/slices,
        trgns = [
            for (rgn=regions) [
                for (path=rgn) let(
                    p = cleanup_path(path),
                    path = is_undef(maxseg)? p : [
                        for (seg=pair(p,true)) each
                        let(steps=ceil(norm(seg.y-seg.x)/maxseg))
                        lerpn(seg.x, seg.y, steps, false)
                    ]
                )
                rot(twist, p=scale([scale,scale],p=path))
            ]
        ],
        vnf = vnf_merge([
            for (rgn = regions)
            for (pathnum = idx(rgn)) let(
                p = cleanup_path(rgn[pathnum]),
                path = is_undef(maxseg)? p : [
                    for (seg=pair(p,true)) each
                    let(steps=ceil(norm(seg.y-seg.x)/maxseg))
                    lerpn(seg.x, seg.y, steps, false)
                ],
                verts = [
                    for (i=[0:1:slices]) let(
                        sc = lerp(1, scale, i/slices),
                        ang = i * step,
                        h = i * hstep - height/2
                    ) scale([sc,sc,1], p=rot(ang, p=path3d(path,h)))
                ]
            ) vnf_vertex_array(verts, caps=false, col_wrap=true, style=style),
            for (rgn = regions) region_faces(rgn, move([0,0,-height/2]), reverse=true),
            for (rgn = trgns) region_faces(rgn, move([0,0, height/2]), reverse=false)
        ])
    ) reorient(anchor,spin,orient, cp=cp, vnf=vnf, extent=!anchor_isect, p=vnf);



// Section: Offsets and Boolean 2D Geometry


function _offset_chamfer(center, points, delta) =
    let(
        dist = sign(delta)*norm(center-line_intersection(select(points,[0,2]), [center, points[1]])),
        endline = _shift_segment(select(points,[0,2]), delta-dist)
    ) [
        line_intersection(endline, select(points,[0,1])),
        line_intersection(endline, select(points,[1,2]))
    ];


function _shift_segment(segment, d) =
    assert(!approx(segment[0],segment[1]),"Path has repeated points")
    move(d*line_normal(segment),segment);


// Extend to segments to their intersection point.  First check if the segments already have a point in common,
// which can happen if two colinear segments are input to the path variant of `offset()`
function _segment_extension(s1,s2) =
    norm(s1[1]-s2[0])<1e-6 ? s1[1] : line_intersection(s1,s2,LINE,LINE);


function _makefaces(direction, startind, good, pointcount, closed) =
    let(
        lenlist = list_bset(good, pointcount),
        numfirst = len(lenlist),
        numsecond = sum(lenlist),
        prelim_faces = _makefaces_recurse(startind, startind+len(lenlist), numfirst, numsecond, lenlist, closed)
    )
    direction? [for(entry=prelim_faces) reverse(entry)] : prelim_faces;


function _makefaces_recurse(startind1, startind2, numfirst, numsecond, lenlist, closed, firstind=0, secondind=0, faces=[]) =
    // We are done if *both* firstind and secondind reach their max value, which is the last point if !closed or one past
    // the last point if closed (wrapping around).  If you don't check both you can leave a triangular gap in the output.
    ((firstind == numfirst - (closed?0:1)) && (secondind == numsecond - (closed?0:1)))? faces :
    _makefaces_recurse(
        startind1, startind2, numfirst, numsecond, lenlist, closed, firstind+1, secondind+lenlist[firstind],
        lenlist[firstind]==0? (
            // point in original path has been deleted in offset path, so it has no match.  We therefore
            // make a triangular face using the current point from the offset (second) path
            // (The current point in the second path can be equal to numsecond if firstind is the last point)
            concat(faces,[[secondind%numsecond+startind2, firstind+startind1, (firstind+1)%numfirst+startind1]])
            // in this case a point or points exist in the offset path corresponding to the original path
        ) : (
            concat(faces,
                // First generate triangular faces for all of the extra points (if there are any---loop may be empty)
                [for(i=[0:1:lenlist[firstind]-2]) [firstind+startind1, secondind+i+1+startind2, secondind+i+startind2]],
                // Finish (unconditionally) with a quadrilateral face
                [
                    [
                        firstind+startind1,
                        (firstind+1)%numfirst+startind1,
                        (secondind+lenlist[firstind])%numsecond+startind2,
                        (secondind+lenlist[firstind]-1)%numsecond+startind2
                    ]
                ]
            )
        )
    );


// Determine which of the shifted segments are good
function _good_segments(path, d, shiftsegs, closed, quality) =
    let(
        maxind = len(path)-(closed ? 1 : 2),
        pathseg = [for(i=[0:maxind]) select(path,i+1)-path[i]],
        pathseg_len =  [for(seg=pathseg) norm(seg)],
        pathseg_unit = [for(i=[0:maxind]) pathseg[i]/pathseg_len[i]],
        // Order matters because as soon as a valid point is found, the test stops
        // This order works better for circular paths because they succeed in the center
        alpha = concat([for(i=[1:1:quality]) i/(quality+1)],[0,1])
    ) [
        for (i=[0:len(shiftsegs)-1])
            (i>maxind)? true :
            _segment_good(path,pathseg_unit,pathseg_len, d - 1e-7, shiftsegs[i], alpha)
    ];


// Determine if a segment is good (approximately)
// Input is the path, the path segments normalized to unit length, the length of each path segment
// the distance threshold, the segment to test, and the locations on the segment to test (normalized to [0,1])
// The last parameter, index, gives the current alpha index.
//
// A segment is good if any part of it is farther than distance d from the path.  The test is expensive, so
// we want to quit as soon as we find a point with distance > d, hence the recursive code structure.
//
// This test is approximate because it only samples the points listed in alpha.  Listing more points
// will make the test more accurate, but slower.
function _segment_good(path,pathseg_unit,pathseg_len, d, seg,alpha ,index=0) =
    index == len(alpha) ? false :
    _point_dist(path,pathseg_unit,pathseg_len, alpha[index]*seg[0]+(1-alpha[index])*seg[1]) > d ? true :
    _segment_good(path,pathseg_unit,pathseg_len,d,seg,alpha,index+1);


// Input is the path, the path segments normalized to unit length, the length of each path segment
// and a test point.  Computes the (minimum) distance from the path to the point, taking into
// account that the minimal distance may be anywhere along a path segment, not just at the ends.
function _point_dist(path,pathseg_unit,pathseg_len,pt) =
    min([
        for(i=[0:len(pathseg_unit)-1]) let(
            v = pt-path[i],
            projection = v*pathseg_unit[i],
            segdist = projection < 0? norm(pt-path[i]) :
                projection > pathseg_len[i]? norm(pt-select(path,i+1)) :
                norm(v-projection*pathseg_unit[i])
        ) segdist
    ]);


function _offset_region(region, r, delta, chamfer, check_valid, quality,closed,return_faces,firstface_index,flip_faces) =
    let(
//dg=        echo(region=region),
         reglist = [for(R=region_parts(region)) is_path(R) ? [R] : R],
//fdsa=         echo(reglist),
         ofsregs = [for(R=reglist)
             [for(i=idx(R)) offset(R[i], r=u_mul(i>0?-1:1,r), delta=u_mul(i>0?-1:1,delta), chamfer=chamfer, check_valid=check_valid,
                                    quality=quality,closed=true)]]
    )
    union(ofsregs);

function d_offset_region(
    paths, r, delta, chamfer, closed,
    check_valid, quality,
    return_faces, firstface_index,
    flip_faces, _acc=[], _i=0
) =
    _i>=len(paths)? _acc :
    _offset_region(
        paths, _i=_i+1,
        _acc = (paths[_i].x % 2 == 0)? (
            union(_acc, [
                offset(
                    paths[_i].y,
                    r=r, delta=delta, chamfer=chamfer, closed=closed,
                    check_valid=check_valid, quality=quality,
                    return_faces=return_faces, firstface_index=firstface_index,
                    flip_faces=flip_faces
                )
            ])
        ) : (
            difference(_acc, [
                offset(
                    paths[_i].y,
                    r=u_mul(-1,r), delta=u_mul(-1,delta), chamfer=chamfer, closed=closed,
                    check_valid=check_valid, quality=quality,
                    return_faces=return_faces, firstface_index=firstface_index,
                    flip_faces=flip_faces
                )
            ])
        ),
        r=r, delta=delta, chamfer=chamfer, closed=closed,
        check_valid=check_valid, quality=quality,
        return_faces=return_faces, firstface_index=firstface_index, flip_faces=flip_faces
    );


// Function: offset()
// Usage:
//   offsetpath = offset(path, [r|delta], [chamfer], [closed], [check_valid], [quality])
//   path_faces = offset(path, return_faces=true, [r|delta], [chamfer], [closed], [check_valid], [quality], [firstface_index], [flip_faces])
// Description:
//   Takes an input path and returns a path offset by the specified amount.  As with the built-in
//   offset() module, you can use `r` to specify rounded offset and `delta` to specify offset with
//   corners.  If you used `delta` you can set `chamfer` to true to get chamfers.
//   Positive offsets shift the path to the left (relative to the direction of the path).
//   .
//   When offsets shrink the path, segments cross and become invalid.  By default `offset()` checks
//   for this situation.  To test validity the code checks that segments have distance larger than (r
//   or delta) from the input path.  This check takes O(N^2) time and may mistakenly eliminate
//   segments you wanted included in various situations, so you can disable it if you wish by setting
//   check_valid=false.  Another situation is that the test is not sufficiently thorough and some
//   segments persist that should be eliminated.  In this case, increase `quality` to 2 or 3.  (This
//   increases the number of samples on the segment that are checked.)  Run time will increase.  In
//   some situations you may be able to decrease run time by setting quality to 0, which causes only
//   segment ends to be checked.
//   .
//   For construction of polyhedra `offset()` can also return face lists.  These list faces between
//   the original path and the offset path where the vertices are ordered with the original path
//   first, starting at `firstface_index` and the offset path vertices appearing afterwords.  The
//   direction of the faces can be flipped using `flip_faces`.  When you request faces the return
//   value is a list: [offset_path, face_list].
// Arguments:
//   path = the path to process.  A list of 2d points.
//   ---
//   r = offset radius.  Distance to offset.  Will round over corners.
//   delta = offset distance.  Distance to offset with pointed corners.
//   chamfer = chamfer corners when you specify `delta`.  Default: false
//   closed = path is a closed curve. Default: False.
//   check_valid = perform segment validity check.  Default: True.
//   quality = validity check quality parameter, a small integer.  Default: 1.
//   return_faces = return face list.  Default: False.
//   firstface_index = starting index for face list.  Default: 0.
//   flip_faces = flip face direction.  Default: false
// Example(2D):
//   star = star(5, r=100, ir=30);
//   #stroke(closed=true, star);
//   stroke(closed=true, offset(star, delta=10, closed=true));
// Example(2D):
//   star = star(5, r=100, ir=30);
//   #stroke(closed=true, star);
//   stroke(closed=true, offset(star, delta=10, chamfer=true, closed=true));
// Example(2D):
//   star = star(5, r=100, ir=30);
//   #stroke(closed=true, star);
//   stroke(closed=true, offset(star, r=10, closed=true));
// Example(2D):
//   star = star(5, r=100, ir=30);
//   #stroke(closed=true, star);
//   stroke(closed=true, offset(star, delta=-10, closed=true));
// Example(2D):
//   star = star(5, r=100, ir=30);
//   #stroke(closed=true, star);
//   stroke(closed=true, offset(star, delta=-10, chamfer=true, closed=true));
// Example(2D):
//   star = star(5, r=100, ir=30);
//   #stroke(closed=true, star);
//   stroke(closed=true, offset(star, r=-10, closed=true, $fn=20));
// Example(2D):  This case needs `quality=2` for success
//   test = [[0,0],[10,0],[10,7],[0,7], [-1,-3]];
//   polygon(offset(test,r=-1.9, closed=true, quality=2));
//   //polygon(offset(test,r=-1.9, closed=true, quality=1));  // Fails with erroneous 180 deg path error
//   %down(.1)polygon(test);
// Example(2D): This case fails if `check_valid=true` when delta is large enough because segments are too close to the opposite side of the curve.  
//   star = star(5, r=22, ir=13);
//   stroke(star,width=.2,closed=true);                                                           
//   color("green")
//     stroke(offset(star, delta=-9, closed=true),width=.2,closed=true); // Works with check_valid=true (the default)
//   color("red")
//     stroke(offset(star, delta=-10, closed=true, check_valid=false),   // Fails if check_valid=true 
//            width=.2,closed=true); 
// Example(2D): But if you use rounding with offset then you need `check_valid=true` when `r` is big enough.  It works without the validity check as long as the offset shape retains a some of the straight edges at the star tip, but once the shape shrinks smaller than that, it fails.  There is no simple way to get a correct result for the case with `r=10`, because as in the previous example, it will fail if you turn on validity checks.  
//   star = star(5, r=22, ir=13);
//   color("green")
//     stroke(offset(star, r=-8, closed=true,check_valid=false), width=.1, closed=true);
//   color("red")
//     stroke(offset(star, r=-10, closed=true,check_valid=false), width=.1, closed=true);
// Example(2D): The extra triangles in this example show that the validity check cannot be skipped
//   ellipse = scale([20,4], p=circle(r=1,$fn=64));
//   stroke(ellipse, closed=true, width=0.3);
//   stroke(offset(ellipse, r=-3, check_valid=false, closed=true), width=0.3, closed=true);
// Example(2D): The triangles are removed by the validity check
//   ellipse = scale([20,4], p=circle(r=1,$fn=64));
//   stroke(ellipse, closed=true, width=0.3);
//   stroke(offset(ellipse, r=-3, check_valid=true, closed=true), width=0.3, closed=true);
// Example(2D): Open path.  The path moves from left to right and the positive offset shifts to the left of the initial red path.
//   sinpath = 2*[for(theta=[-180:5:180]) [theta/4,45*sin(theta)]];
//   #stroke(sinpath);
//   stroke(offset(sinpath, r=17.5));
// Example(2D): Region
//   rgn = difference(circle(d=100), union(square([20,40], center=true), square([40,20], center=true)));
//   #linear_extrude(height=1.1) for (p=rgn) stroke(closed=true, width=0.5, p);
//   region(offset(rgn, r=-5));
function offset(
    path, r=undef, delta=undef, chamfer=false,
    closed=false, check_valid=true,
    quality=1, return_faces=false, firstface_index=0,
    flip_faces=false
) = 
    is_region(path)? _offset_region(path,r=r,delta=delta,chamfer=chamfer,quality=quality,check_valid=check_valid)
  
/*
        (
        assert(!return_faces, "return_faces not supported for regions.")
        let(
            path = [for (p=path) clockwise_polygon(p)],
            rgn = exclusive_or([for (p = path) [p]]),
            pathlist = sort(idx=0,[
                for (i=[0:1:len(rgn)-1]) [
                    sum(concat([0],[
                        for (j=[0:1:len(rgn)-1]) if (i!=j)
                            point_in_polygon(rgn[i][0],rgn[j])>=0? 1 : 0
                    ])),
                    rgn[i]
                ]
            ])
        ) _offset_region(
            pathlist, r=r, delta=delta, chamfer=chamfer, closed=true,
            check_valid=check_valid, quality=quality,
            return_faces=return_faces, firstface_index=firstface_index,
            flip_faces=flip_faces
        )
    )*/
    : let(rcount = num_defined([r,delta]))
    assert(rcount==1,"Must define exactly one of 'delta' and 'r'")
    let(
        chamfer = is_def(r) ? false : chamfer,
        quality = max(0,round(quality)),
        flip_dir = closed && !is_polygon_clockwise(path)? -1 : 1,
        d = flip_dir * (is_def(r) ? r : delta),
//        shiftsegs = [for(i=[0:len(path)-1]) _shift_segment(select(path,i,i+1), d)],
        shiftsegs = [for(i=[0:len(path)-2]) _shift_segment([path[i],path[i+1]], d),
                     if (closed) _shift_segment([last(path),path[0]],d)
                     else [path[0],path[1]]  // dummy segment, not used
                    ],
        // good segments are ones where no point on the segment is less than distance d from any point on the path
        good = check_valid ? _good_segments(path, abs(d), shiftsegs, closed, quality) : repeat(true,len(shiftsegs)),
        goodsegs = bselect(shiftsegs, good),
        goodpath = bselect(path,good)
    )
    assert(len(goodsegs)>0,"Offset of path is degenerate")
    let(
        // Extend the shifted segments to their intersection points
        sharpcorners = [for(i=[0:len(goodsegs)-1]) _segment_extension(select(goodsegs,i-1), select(goodsegs,i))],
        // If some segments are parallel then the extended segments are undefined.  This case is not handled
        // Note if !closed the last corner doesn't matter, so exclude it
        parallelcheck =
            (len(sharpcorners)==2 && !closed) ||
            all_defined(closed? sharpcorners : select(sharpcorners, 1,-2))
    )
    assert(parallelcheck, "Path contains sequential parallel segments (either 180 deg turn or 0 deg turn")
    let(
        // This is a boolean array that indicates whether a corner is an outside or inside corner
        // For outside corners, the newcorner is an extension (angle 0), for inside corners, it turns backward
        // If either side turns back it is an inside corner---must check both.
        // Outside corners can get rounded (if r is specified and there is space to round them)
        outsidecorner = len(sharpcorners)==2 ? [false,false]
           :
            [for(i=[0:len(goodsegs)-1])
                let(prevseg=select(goodsegs,i-1))
                (i==0 || i==len(goodsegs)-1) && !closed ? false  // In open case first entry is bogus
               :  
                (goodsegs[i][1]-goodsegs[i][0]) * (goodsegs[i][0]-sharpcorners[i]) > 0
                 && (prevseg[1]-prevseg[0]) * (sharpcorners[i]-prevseg[1]) > 0
            ],
        steps = is_def(delta) ? [] : [
            for(i=[0:len(goodsegs)-1])
                r==0 ? 0
                // floor is important here to ensure we don't generate extra segments when nearly straight paths expand outward
              : 1+floor(segs(r)*vector_angle(   
                                             select(goodsegs,i-1)[1]-goodpath[i],
                                             goodsegs[i][0]-goodpath[i])
                        /360)
        ],
        // If rounding is true then newcorners replaces sharpcorners with rounded arcs where needed
        // Otherwise it's the same as sharpcorners
        // If rounding is on then newcorners[i] will be the point list that replaces goodpath[i] and newcorners later
        // gets flattened.  If rounding is off then we set it to [sharpcorners] so we can later flatten it and get
        // plain sharpcorners back.
        newcorners = is_def(delta) && !chamfer ? [sharpcorners]
            : [for(i=[0:len(goodsegs)-1])
                  (!chamfer && steps[i] <=1)  // Don't round if steps is smaller than 2
                  || !outsidecorner[i]        // Don't round inside corners
                  || (!closed && (i==0 || i==len(goodsegs)-1))  // Don't round ends of an open path
                ? [sharpcorners[i]]
                : chamfer ? _offset_chamfer(
                                  goodpath[i], [
                                      select(goodsegs,i-1)[1],
                                      sharpcorners[i],
                                      goodsegs[i][0]
                                  ], d
                              )     
                : // rounded case
                  arc(cp=goodpath[i],
                      points=[
                          select(goodsegs,i-1)[1],
                          goodsegs[i][0]
                      ],
                      N=steps[i])
              ],
        pointcount = (is_def(delta) && !chamfer)?
            repeat(1,len(sharpcorners)) :
            [for(i=[0:len(goodsegs)-1]) len(newcorners[i])],
        start = [goodsegs[0][0]],
        end = [goodsegs[len(goodsegs)-2][1]],
        edges =  closed?
            flatten(newcorners) :
            concat(start,slice(flatten(newcorners),1,-2),end),
        faces = !return_faces? [] :
            _makefaces(
                flip_faces, firstface_index, good,
                pointcount, closed
            )
    ) return_faces? [edges,faces] : edges;

/// Internal Function: _tag_subpaths()
/// splits the polygon (path) into subpaths by region crossing and then tags each subpath:
///    "O" - the subpath is outside the region
///    "I" - the subpath is inside the region's interior
///    "S" - the subpath is on the region's border and the polygon and region are on the same side of the subpath
///    "U" - the subpath is on the region's border and the polygon and region meet at the subpath (from opposite sides)
/// The return has the form of a list with entries [TAG, SUBPATH]
function _tag_subpaths(region1, region2, eps=EPSILON, SUtags=true) =
    // We have to compute common vertices between paths in the region because
    // they can be places where the path must be cut, even though they aren't
    // found my the split_path function.  
    let(
        points = flatten(region1),
        tree =  len(points)>0 ? vector_search_tree(points): undef
    )
    [for(path=region1)
        let(
            self_int = is_undef(tree)?[]:[for(i=idx(path)) if (len(vector_search(path[i], eps, tree))>1) [i,0]],
            subpaths = split_path_at_region_crossings(path, region2, eps=eps, extra=self_int)
        )
        for (subpath = subpaths)
            let(
                midpt = mean([subpath[0], subpath[1]]),
                rel = point_in_region(midpt,region2,eps=eps)
            )
            if(rel<0) ["O", subpath]
            else if (rel>0) ["I", subpath]
            else if (SUtags)
                let(
                    sidept = midpt + 0.01*line_normal(subpath[0],subpath[1]),
                    rel1 = point_in_region(sidept,region1,eps=eps)>0,
                    rel2 = point_in_region(sidept,region2,eps=eps)>0
                 )
                 rel1==rel2? ["S", subpath] : ["U", subpath]
    ];


function _tagged_region(region1,region2,keep1,keep2,eps=EPSILON) =
    let(
        tagged1 = _tag_subpaths(region1, region2, eps=eps),
        tagged2 = _tag_subpaths(region2, region1, eps=eps),
        tagged = [
                  for (tagpath = tagged1) if (in_list(tagpath[0], keep1)) tagpath[1],
                  for (tagpath = tagged2) if (in_list(tagpath[0], keep2)) tagpath[1]
                 ]
    )
    _assemble_path_fragments(tagged, eps=eps);



// Function&Module: union()
// Usage:
//   union() {...}
//   region = union(regions);
//   region = union(REGION1,REGION2);
//   region = union(REGION1,REGION2,REGION3);
// Description:
//   When called as a function and given a list of regions, where each region is a list of closed
//   2D paths, returns the boolean union of all given regions.  Result is a single region.
//   When called as the built-in module, makes the boolean union of the given children.
// Arguments:
//   regions = List of regions to union.  Each region is a list of closed paths.
// Example(2D):
//   shape1 = move([-8,-8,0], p=circle(d=50));
//   shape2 = move([ 8, 8,0], p=circle(d=50));
//   for (shape = [shape1,shape2]) color("red") stroke(shape, width=0.5, closed=true);
//   color("green") region(union(shape1,shape2));
function union(regions=[],b=undef,c=undef,eps=EPSILON) =
    b!=undef? union(concat([regions],[b],c==undef?[]:[c]), eps=eps) :
    len(regions)==0? [] :
    len(regions)==1? regions[0] :
    let(regions=[for (r=regions) quant(is_path(r)? [r] : r, 1/65536)])
    union([
           _tagged_region(regions[0],regions[1],["O","S"],["O"], eps=eps),
            for (i=[2:1:len(regions)-1]) regions[i]
          ],
          eps=eps
    );


// Function&Module: difference()
// Usage:
//   difference() {...}
//   region = difference(regions);
//   region = difference(REGION1,REGION2);
//   region = difference(REGION1,REGION2,REGION3);
// Description:
//   When called as a function, and given a list of regions, where each region is a list of closed
//   2D paths, takes the first region and differences away all other regions from it.  The resulting
//   region is returned.
//   When called as the built-in module, makes the boolean difference of the given children.
// Arguments:
//   regions = List of regions to difference.  Each region is a list of closed paths.
// Example(2D):
//   shape1 = move([-8,-8,0], p=circle(d=50));
//   shape2 = move([ 8, 8,0], p=circle(d=50));
//   for (shape = [shape1,shape2]) color("red") stroke(shape, width=0.5, closed=true);
//   color("green") region(difference(shape1,shape2));
function difference(regions=[],b=undef,c=undef,eps=EPSILON) =
    b!=undef? difference(concat([regions],[b],c==undef?[]:[c]), eps=eps) :
    len(regions)==0? [] : 
    len(regions)==1? regions[0] :
    regions[0]==[] ? [] : 
    let(regions=[for (r=regions) quant(is_path(r)? [r] : r, 1/65536)])
    difference([
                _tagged_region(regions[0],regions[1],["O","U"],["I"], eps=eps),
                for (i=[2:1:len(regions)-1]) regions[i]
               ],
               eps=eps
    );


// Function&Module: intersection()
// Usage:
//   intersection() {...}
//   region = intersection(regions);
//   region = intersection(REGION1,REGION2);
//   region = intersection(REGION1,REGION2,REGION3);
// Description:
//   When called as a function, and given a list of regions, where each region is a list of closed
//   2D paths, returns the boolean intersection of all given regions.  Result is a single region.
//   When called as the built-in module, makes the boolean intersection of all the given children.
// Arguments:
//   regions = List of regions to intersection.  Each region is a list of closed paths.
// Example(2D):
//   shape1 = move([-8,-8,0], p=circle(d=50));
//   shape2 = move([ 8, 8,0], p=circle(d=50));
//   for (shape = [shape1,shape2]) color("red") stroke(shape, width=0.5, closed=true);
//   color("green") region(intersection(shape1,shape2));
function intersection(regions=[],b=undef,c=undef,eps=EPSILON) =
//    echo(regions=regions)
     b!=undef? intersection(concat([regions],[b],c==undef?[]:[c]),eps=eps)
   : len(regions)==0 ? []
   : len(regions)==1? regions[0]
   : regions[0]==[] || regions[1]==[] ? []   
   : let(regions=[for (r=regions) quant(is_path(r)? [r] : r, 1/65536)])
         intersection([
                       _tagged_region(regions[0],regions[1],["I","S"],["I"],eps=eps),
                       for (i=[2:1:len(regions)-1]) regions[i]
                      ],
                      eps=eps
         );



// Function&Module: exclusive_or()
// Usage:
//   exclusive_or() {...}
//   region = exclusive_or(regions);
//   region = exclusive_or(REGION1,REGION2);
//   region = exclusive_or(REGION1,REGION2,REGION3);
// Description:
//   When called as a function and given a list of regions, where each region is a list of closed
//   2D paths, returns the boolean exclusive_or of all given regions.  Result is a single region.
//   When called as a module, performs a boolean exclusive-or of up to 10 children.  Note that the
//   xor operator tends to produce shapes that meet at corners, which do not render in CGAL.  
// Arguments:
//   regions = List of regions to exclusive_or.  Each region is a list of closed paths.
// Example(2D): As Function.  A linear_sweep of this shape fails to render in CGAL.  
//   shape1 = move([-8,-8,0], p=circle(d=50));
//   shape2 = move([ 8, 8,0], p=circle(d=50));
//   for (shape = [shape1,shape2])
//       color("red") stroke(shape, width=0.5, closed=true);
//   color("green") region(exclusive_or(shape1,shape2));
// Example(2D): As Module.  A linear_extrude() of the resulting geometry fails to render in CGAL.  
//   exclusive_or() {
//       square(40,center=false);
//       circle(d=40);
//   }
function exclusive_or(regions=[],b=undef,c=undef,eps=EPSILON) =
    b!=undef? exclusive_or([regions, b, if(is_def(c)) c],eps=eps) :
    len(regions)==0? [] :
    len(regions)==1? regions[0] :
    let(regions=[for (r=regions) is_path(r)? [r] : r])
    exclusive_or([
                  _tagged_region(regions[0],regions[1],["I","O"],["I","O"],eps=eps),
                  for (i=[2:1:len(regions)-1]) regions[i]
                 ],
                 eps=eps
    );


module exclusive_or() {
    if ($children==1) {
        children();
    } else if ($children==2) {
        difference() {
            children(0);
            children(1);
        }
        difference() {
            children(1);
            children(0);
        }
    } else if ($children==3) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
            }
            children(2);
        }
    } else if ($children==4) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
            }
            exclusive_or() {
                children(2);
                children(3);
            }
        }
    } else if ($children==5) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
                children(2);
                children(3);
            }
            children(4);
        }
    } else if ($children==6) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
                children(2);
                children(3);
            }
            children(4);
            children(5);
        }
    } else if ($children==7) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
                children(2);
                children(3);
            }
            children(4);
            children(5);
            children(6);
        }
    } else if ($children==8) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
                children(2);
                children(3);
            }
            exclusive_or() {
                children(4);
                children(5);
                children(6);
                children(7);
            }
        }
    } else if ($children==9) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
                children(2);
                children(3);
            }
            exclusive_or() {
                children(4);
                children(5);
                children(6);
                children(7);
            }
            children(8);
        }
    } else if ($children==10) {
        exclusive_or() {
            exclusive_or() {
                children(0);
                children(1);
                children(2);
                children(3);
            }
            exclusive_or() {
                children(4);
                children(5);
                children(6);
                children(7);
            }
            children(8);
            children(9);
        }
    } else {
        assert($children<=10, "exclusive_or() can only handle up to 10 children.");
    }
}


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