Correct bugs in polygon_triangulation and _cleave_connected_region(

geometry.scad

@@ -38,14 +38,10 @@ function _is_point_on_line(point, line, bounded=false, eps=EPSILON) =
         t  = v0*v1/(v1*v1),
         bounded = force_list(bounded,2)
     ) 
-    abs(cross(v0,v1))<eps*norm(v1) 
+    abs(cross(v0,v1))<=eps*norm(v1) 
     && (!bounded[0] || t>=-eps) 
     && (!bounded[1] || t<1+eps) ;
 
-function xis_point_on_line(point, line, bounded=false, eps=EPSILON) =
-    assert( is_finite(eps) && (eps>=0), "The tolerance should be a non-negative value." )
-    point_line_distance(point, line, bounded)<eps;
-
 
 ///Internal - distance from point `d` to the line passing through the origin with unit direction n
 ///_dist2line works for any dimension
@@ -61,7 +57,67 @@ function _valid_line(line,dim,eps=EPSILON) =
 function _valid_plane(p, eps=EPSILON) = is_vector(p,4) && ! approx(norm(p),0,eps);
 
 
-/// Internal Function: point_left_of_line2d()
+/// Internal Function: _is_at_left()
+/// Usage:
+///   pt = point_left_of_line2d(point, line);
+/// Topics: Geometry, Points, Lines
+/// Description:
+///   Return true iff a 2d point is on or at left of the line defined by `line`.
+/// Arguments:
+///   pt = The 2d point to check position of.
+///   line  = Array of two 2d points forming the line segment to test against.
+///   eps = Tolerance in the geometrical tests.
+function _is_at_left(pt,line,eps=EPSILON) = _tri_class([pt,line[0],line[1]],eps) <= 0;
+
+
+/// Internal Function: _degenerate_tri()
+/// Usage:
+///   degen = _degenerate_tri(triangle);
+/// Topics: Geometry, Triangles
+/// Description:
+///   Return true for a specific kind of degeneracy: any two triangle vertices are equal
+/// Arguments:
+///   tri = A list of three 2d points
+///   eps = Tolerance in the geometrical tests.
+function _degenerate_tri(tri,eps) =
+    max(norm(tri[0]-tri[1]), norm(tri[1]-tri[2]), norm(tri[2]-tri[0])) < eps ;
+    
+
+/// Internal Function: _tri_class()
+/// Usage:
+///   class = _tri_class(triangle);
+/// Topics: Geometry, Triangles
+/// Description:
+///   Return  1 if the triangle `tri` is CW.
+///   Return  0 if the triangle `tri` has colinear vertices.
+///   Return -1 if the triangle `tri` is CCW.
+/// Arguments:
+///   tri = A list of the three 2d vertices of a triangle.
+///   eps = Tolerance in the geometrical tests.
+function _tri_class(tri, eps=EPSILON) =
+    let( crx = cross(tri[1]-tri[2],tri[0]-tri[2]) )
+    abs( crx ) <= eps*norm(tri[1]-tri[2])*norm(tri[0]-tri[2]) ? 0 : sign( crx );
+    
+    
+/// Internal Function: _pt_in_tri()
+/// Usage:
+///   class = _pt_in_tri(point, tri);
+/// Topics: Geometry, Points, Triangles
+/// Description:
+///   Return  1 if point is inside the triangle interion.
+///   Return =0 if point is on the triangle border.
+///   Return -1 if point is outside the triangle.
+/// Arguments:
+///   point = The point to check position of.
+///   tri  =  A list of the three 2d vertices of a triangle.
+///   eps = Tolerance in the geometrical tests.
+function _pt_in_tri(point, tri, eps=EPSILON) = 
+    min(   _tri_class([tri[0],tri[1],point],eps), 
+          _tri_class([tri[1],tri[2],point],eps), 
+          _tri_class([tri[2],tri[0],point],eps) );
+        
+
+/// Internal Function: _point_left_of_line2d()
 /// Usage:
 ///   pt = point_left_of_line2d(point, line);
 /// Topics: Geometry, Points, Lines
@@ -72,10 +128,11 @@ function _valid_plane(p, eps=EPSILON) = is_vector(p,4) && ! approx(norm(p),0,eps
 /// Arguments:
 ///   point = The point to check position of.
 ///   line  = Array of two points forming the line segment to test against.
-function _point_left_of_line2d(point, line) =
+function _point_left_of_line2d(point, line, eps=EPSILON) =
     assert( is_vector(point,2) && is_vector(line*point, 2), "Improper input." )
-    cross(line[0]-point, line[1]-line[0]);
+//    cross(line[0]-point, line[1]-line[0]);
-
+    _tri_class([point,line[1],line[0]],eps);
+    
 
 // Function: is_collinear()
 // Usage:
@@ -1648,18 +1705,19 @@ function point_in_polygon(point, poly, nonzero=false, eps=EPSILON) =
 //   .
 //   The function produce correct triangulations for some non-twisted non-simple polygons. 
 //   A polygon is non-twisted iff it is simple or there is a partition of it in
-//   simple polygons with the same winding. These polygons may have "touching" vertices 
+//   simple polygons with the same winding such that the intersection of any two partitions is
+//   made of full edges of both partitions. These polygons may have "touching" vertices 
 //   (two vertices having the same coordinates, but distinct adjacencies) and "contact" edges 
 //   (edges whose vertex pairs have the same pairwise coordinates but are in reversed order) but has 
-//   no self-crossing. See examples bellow. If all polygon edges are contact edges, 
+//   no self-crossing. See examples bellow. If all polygon edges are contact edges (polygons with 
-//   it returns an empty list for 2d polygons and issues an error for 3d polygons. 
+//   zero area), it returns an empty list for 2d polygons and issues an error for 3d polygons. 
 //   .
-//   Self-crossing polygons have no consistent winding and usually produce an error but 
+//   Twisted polygons have no consistent winding and when input to this function usually produce 
-//   when an error is not issued the outputs are not correct triangulations. The function
+//   an error but when an error is not issued the outputs are not correct triangulations. The function
 //   can work for 3d non-planar polygons if they are close enough to planar but may otherwise 
 //   issue an error for this case. 
 // Arguments:
-//   poly = Array of vertices for the polygon.
+//   poly = Array of the polygon vertices.
 //   ind = A list indexing the vertices of the polygon in `poly`.
 //   eps = A maximum tolerance in geometrical tests. Default: EPSILON
 // Example(2D,NoAxes):
@@ -1686,7 +1744,7 @@ function point_in_polygon(point, poly, nonzero=false, eps=EPSILON) =
 // Example(2D,NoAxes): a polygon with "contact" edges and no holes
 //   poly = [ [0,0], [10,0], [10,10], [0,10], [0,0], [3,3], [7,3], 
 //            [7,7], [7,3], [3,3] ];
-//   tris =  polygon_triangulate(poly); // see from the top
+//   tris =  polygon_triangulate(poly); // see from above
 //   color("lightblue") for(tri=tris) polygon(select(poly,tri));
 //   color("blue")    up(1) for(tri=tris) { stroke(select(poly,tri),.15,closed=true); }
 //   color("magenta") up(2) stroke(poly,.25,closed=true); 
@@ -1703,97 +1761,118 @@ function polygon_triangulate(poly, ind, eps=EPSILON) =
     assert(is_undef(ind) 
            || (is_vector(ind) && min(ind)>=0 && max(ind)<len(poly) ),
            "Improper or out of bounds list of indices")
+    (! is_undef(ind) ) && len(ind) == 0 ? [] :
     let( ind = is_undef(ind) ? count(len(poly)) : ind )
     len(ind) == 3 
-      ? _is_degenerate([poly[ind[0]], poly[ind[1]], poly[ind[2]]], eps) ? [] :
+      ? _degenerate_tri([poly[ind[0]], poly[ind[1]], poly[ind[2]]], eps) ? [] : 
         // non zero area
         assert( norm(scalar_vec3(cross(poly[ind[1]]-poly[ind[0]], poly[ind[2]]-poly[ind[0]]))) > 2*eps,
                 "The polygon vertices are collinear.") 
         [ind]
       : len(poly[ind[0]]) == 3 
-          ? // represents the polygon projection on its plane as a 2d polygon 
+          ? // find a representation of the polygon as a 2d polygon by projecting it on its own plane
             let( 
                 ind = deduplicate_indexed(poly, ind, eps) 
             )
             len(ind)<3 ? [] :
             let(
                 pts = select(poly,ind),
-                nrm = polygon_normal(pts)
+                nrm = -polygon_normal(pts)
             )
             assert( nrm!=undef, 
-                    "The polygon has self-intersections or its vertices are collinear or non coplanar.") 
+                    "The polygon has self-intersections or zero area or its vertices are collinear or non coplanar.") 
             let(
                 imax  = max_index([for(p=pts) norm(p-pts[0]) ]),
                 v1    = unit( pts[imax] - pts[0] ),
                 v2    = cross(v1,nrm),
-                prpts = pts*transpose([v1,v2])
+                prpts = pts*transpose([v1,v2]) // the 2d projection of pts on the polygon plane
             )
             [for(tri=_triangulate(prpts, count(len(ind)), eps)) select(ind,tri) ]
-          : let( cw = is_polygon_clockwise(select(poly, ind)) )
+          : is_polygon_clockwise(select(poly, ind)) 
-            cw 
+              ? _triangulate( poly, ind,  eps )
-              ? [for(tri=_triangulate( poly, reverse(ind), eps )) reverse(tri) ]
+              : [for(tri=_triangulate( poly, reverse(ind), eps )) reverse(tri) ];
-              : _triangulate( poly, ind, eps );
 
 
-function _triangulate(poly, ind, eps=EPSILON, tris=[]) =
+// poly is supposed to be a 2d cw polygon
+// implements a modified version of ear cut method for non-twisted polygons
+// the polygons accepted by this function are (tecnically) the ones whose interior
+// is homeomoph to the interior of a simple polygon
+function _triangulate(poly, ind,  eps=EPSILON, tris=[]) =
     len(ind)==3 
-    ?   _is_degenerate(select(poly,ind),eps)  
+    ?   _degenerate_tri(select(poly,ind),eps) 
-        ?   tris // last 3 pts perform a degenerate triangle, ignore it
+        ?   tris // if last 3 pts perform a degenerate triangle, ignore it
         :   concat(tris,[ind]) // otherwise, include it
     :   let( ear = _get_ear(poly,ind,eps) )
+/*
+let( x= [if(is_undef(ear)) echo(ind=ind) 0] ) 
+is_undef(ear) ? tris :
+*/
         assert( ear!=undef, 
-                "The polygon has self-intersections or its vertices are collinear or non coplanar.") 
+            "The polygon has twists or all its vertices are collinear or non coplanar.") 
-        is_list(ear) // degenerate ear
+        is_list(ear) // is it a degenerate ear ?
-        ?   _triangulate(poly, select(ind,ear[0]+2, ear[0]), eps, tris) // discard it
+        ?   len(ind) <= 4 ? tris :
+            _triangulate(poly, select(ind,ear[0]+3, ear[0]),  eps, tris) // discard it
         :   let(
                 ear_tri = select(ind,ear,ear+2),
-                indr    = select(ind,ear+2, ear) // remaining point indices
+                indr    = select(ind,ear+2, ear) //  indices of the remaining path
             )
-            _triangulate(poly, indr, eps, concat(tris,[ear_tri]));
+            _triangulate(poly, indr,  eps, concat(tris,[ear_tri]));
 
 
 // a returned ear will be:
-// 1. a CCW (non-degenerate) triangle, made of subsequent vertices, without other 
+// 1. a CW non-reflex triangle, made of subsequent poly vertices, without any other 
-//    points inside except possibly at its vertices
+//    poly points inside except possibly at its own vertices
 // 2. or a degenerate triangle where two vertices are coincident
 // the returned ear is specified by the index of `ind` of its first vertex
-function _get_ear(poly, ind, eps, _i=0) =
+function _get_ear(poly, ind,  eps, _i=0) =
-    _i>=len(ind) ? undef : // poly has no ears
+    let( lind = len(ind) )
+    lind==3 ? 0 :
     let( // the _i-th ear candidate
         p0 = poly[ind[_i]],
-        p1 = poly[ind[(_i+1)%len(ind)]],
+        p1 = poly[ind[(_i+1)%lind]],
-        p2 = poly[ind[(_i+2)%len(ind)]]
+        p2 = poly[ind[(_i+2)%lind]]
     )
-    // degenerate triangles are returned codified
+    // if vertex p1 is a convex candidate to be an ear,
-    _is_degenerate([p0,p1,p2],eps)  ? [_i] : 
+    // check if the triangle [p0,p1,p2] contains any other point
-    // if it is not a convex vertex, check the next one
+    // except possibly p0 and p2
-    _is_cw2(p0,p1,p2,eps) ? _get_ear(poly,ind,eps, _i=_i+1) : 
+    // exclude the ear candidate central vertex p1 from the verts to check 
-    let( // vertex p1 is convex
+    _tri_class([p0,p1,p2],eps) > 0  
-         // check if the triangle contains any other point
+    &&  _none_inside(select(ind,_i+2, _i),poly,p0,p1,p2,eps) ? _i : // found an ear
-         // except possibly its own vertices
+    // otherwise check the next ear candidate 
-        to_tst = select(ind,_i+3, _i-1),
+    _i<lind-1 ?  _get_ear(poly, ind,  eps, _i=_i+1) :
-        q      = [(p0-p2).y, (p2-p0).x],  // orthogonal to ray [p0,p2] pointing right
+    // poly has no ears, look for wiskers
-        r      = [(p2-p1).y, (p1-p2).x],  // orthogonal to ray [p2,p1] pointing right
+    let( 
-        s      = [(p1-p0).y, (p0-p1).x],  // orthogonal to ray [p1,p0] pointing right
+        wiskers = [for(j=idx(ind)) if(norm(poly[ind[j]]-poly[ind[(j+2)%lind]])<eps) j ]
-        inside = [for(p=select(poly,to_tst)) // for vertices other than p0, p1 and p2
-                      if( (p-p0)*q<=0 && (p-p2)*r<=0 && (p-p1)*s<=0  // p is on the triangle
-                          && norm(p-p0)>eps  // but not on any vertex of it
-                          && norm(p-p1)>eps  
-                          && norm(p-p2)>eps ) 
-                          p ]                       
     )
-    inside==[] ? _i : // found an ear
+    wiskers==[] ? undef : [wiskers[0]];
-    // check the next ear candidate
+    
-    _get_ear(poly, ind, eps, _i=_i+1);
+    
-
-
-// true for some specific kinds of degeneracy
-function _is_degenerate(tri,eps) =
-       norm(tri[0]-tri[1])<eps || norm(tri[1]-tri[2])<eps || norm(tri[2]-tri[0])<eps ;
-
-
-function _is_cw2(a,b,c,eps=EPSILON) = cross(a-c,b-c)<eps*norm(a-c)*norm(b-c);
 
+// returns false ASA it finds some reflex vertex of poly[idxs[.]] 
+// inside the triangle different from p0 and p2
+// note: to simplify the expressions it is assumed that the input polygon has no twists 
+function _none_inside(idxs,poly,p0,p1,p2,eps,i=0) =
+    i>=len(idxs) ? true :
+    let( 
+        vert      = poly[idxs[i]], 
+        prev_vert = poly[select(idxs,i-1)], 
+        next_vert = poly[select(idxs,i+1)]
+    )
+    // check if vert prevent [p0,p1,p2] to be an ear
+    // this conditions might have a simpler expression
+    _tri_class([prev_vert, vert, next_vert],eps) <= 0  // reflex condition
+    &&  (  // vert is a cw reflex poly vertex inside the triangle [p0,p1,p2]
+          ( _tri_class([p0,p1,vert],eps)>0 && 
+            _tri_class([p1,p2,vert],eps)>0 && 
+            _tri_class([p2,p0,vert],eps)>=0  )
+          // or it is equal to p1 and some of its adjacent edges cross the open segment (p0,p2)
+          ||  ( norm(vert-p1) < eps 
+                && (   _is_at_left(p0,[prev_vert,p1],eps) 
+                    && _is_at_left(p2,[p1,next_vert],eps) ) 
+              ) 
+        )
+    ?   false
+    :    _none_inside(idxs,poly,p0,p1,p2,eps,i=i+1);
 
 
 // Function: is_polygon_clockwise()

tests/test_all.scad

@@ -0,0 +1,32 @@
+include <test_affine.scad>
+include <test_attachments.scad>
+include <test_comparisons.scad>
+include <test_coords.scad>
+include <test_cubetruss.scad>
+include <test_distributors.scad>
+include <test_drawing.scad>
+include <test_edges.scad>
+include <test_fnliterals.scad>
+include <test_geometry.scad>
+include <test_hull.scad>
+include <test_linalg.scad>
+include <test_linear_bearings.scad>
+include <test_lists.scad>
+include <test_math.scad>
+include <test_mutators.scad>
+include <test_paths.scad>
+include <test_quaternions.scad>
+include <test_regions.scad>
+include <test_rounding.scad>
+include <test_screw_drive.scad>
+include <test_shapes2d.scad>
+include <test_shapes3d.scad>
+include <test_skin.scad>
+include <test_strings.scad>
+include <test_structs.scad>
+include <test_transforms.scad>
+include <test_trigonometry.scad>
+include <test_utility.scad>
+include <test_vectors.scad>
+include <test_version.scad>
+include <test_vnf.scad>

tests/test_geometry.scad

@@ -84,14 +84,14 @@ module test_polygon_triangulate() {
 		poly1 = [ [-10,0,-10], [10,0,10], [0,10,0], [-10,0,-10], [-4,4,-4], [4,4,4], [0,2,0], [-4,4,-4] ];
 		poly2 = [ [0,0], [5,5], [-5,5], [0,0], [-5,-5], [5,-5] ];
 		poly3 = [ [0,0], [10,0], [10,10], [10,13], [10,10], [0,10], [0,0], [3,3], [7,3], [7,7], [7,3], [3,3] ];
-		tris0 = sort(polygon_triangulate(poly0));
+		tris0 = (polygon_triangulate(poly0));
 		assert(approx(tris0, [[0, 1, 2]]));
 		tris1 = (polygon_triangulate(poly1));
-		assert(approx(tris1,( [[2, 3, 4], [6, 7, 0], [2, 4, 5], [6, 0, 1], [1, 2, 5], [5, 6, 1]])));
+		assert(approx(tris1,(  [[2, 3, 4], [6, 7, 0], [2, 4, 5], [6, 0, 1], [1, 2, 5], [5, 6, 1]])));
 		tris2 = (polygon_triangulate(poly2));
-		assert(approx(tris2,([[0, 1, 2], [3, 4, 5]])));
+		assert(approx(tris2,( [[3, 4, 5], [1, 2, 3]])));
 		tris3 = (polygon_triangulate(poly3));
-		assert(approx(tris3,( [[5, 6, 7], [7, 8, 9], [10, 11, 0], [5, 7, 9], [10, 0, 1], [4, 5, 9], [9, 10, 1], [1, 4, 9]])));
+		assert(approx(tris3,( [[5, 6, 7], [11, 0, 1], [5, 7, 8], [10, 11, 1], [5, 8, 9], [10, 1, 2], [4, 5, 9], [9, 10, 2]])));
 }
 
 module test__normalize_plane(){

vectors.scad

@@ -491,12 +491,11 @@ function _bt_tree(points, ind, leafsize=25) =
         bounds = pointlist_bounds(select(points,ind)),
         coord  = max_index(bounds[1]-bounds[0]), 
         projc  = [for(i=ind) points[i][coord] ],
-        pmc    = mean(projc), 
+        meanpr = mean(projc), 
-        pivot  = min_index([for(p=projc) abs(p-pmc)]),
+        pivot  = min_index([for(p=projc) abs(p-meanpr)]),
         radius = max([for(i=ind) norm(points[ind[pivot]]-points[i]) ]),
-        median = median(projc),
+        Lind   = [for(i=idx(ind)) if(projc[i]<=meanpr && i!=pivot) ind[i] ],
-        Lind   = [for(i=idx(ind)) if(projc[i]<=median && i!=pivot) ind[i] ],
+        Rind   = [for(i=idx(ind)) if(projc[i] >meanpr && i!=pivot) ind[i] ]
-        Rind   = [for(i=idx(ind)) if(projc[i] >median && i!=pivot) ind[i] ]
       )
     [ ind[pivot], radius, _bt_tree(points, Lind, leafsize), _bt_tree(points, Rind, leafsize) ];
 

vnf.scad

@@ -318,14 +318,13 @@ function vnf_merge(vnfs, cleanup=false, eps=EPSILON) =
     cleanup? _vnf_cleanup(verts,faces,eps) : [verts,faces];
 
 
-
 function _vnf_cleanup(verts,faces,eps) = 
     let(
         dedup  = vector_search(verts,eps,verts),                 // collect vertex duplicates
         map    = [for(i=idx(verts)) min(dedup[i]) ],             // remap duplic vertices
         offset = cumsum([for(i=idx(verts)) map[i]==i ? 0 : 1 ]), // remaping face vertex offsets 
         map2   = list(idx(verts))-offset,                        // map old vertex indices to new indices
-        nverts = [for(i=idx(verts)) if(map[i]==i) verts[i] ],    // eliminates all unreferenced vertices
+        nverts = [for(i=idx(verts)) if(map[i]==i) verts[i] ],    // this doesn't eliminate unreferenced vertices
         nfaces = 
             [ for(face=faces) 
                 let(
@@ -388,7 +387,7 @@ function _join_paths_at_vertices(path1,path2,v1,v2) =
 // 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) =
+function _old_cleave_connected_region(region) =
     len(region)==0? [] :
     len(region)<=1? clockwise_polygon(region[0]) :
     let(
@@ -411,8 +410,126 @@ function _cleave_connected_region(region) =
         ]
     )
     assert(len(orgn)<len(region))
-    _cleave_connected_region(orgn);
+    _old_cleave_connected_region(orgn);
 
+/// Internal Function: _cleave_connected_region(region, eps)
+/// Description:
+/// 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. 
+/// It expect that region[0] be a simple closed CW path and that each hole,
+/// region[i] for i>0, be a simple closed CCW path.
+/// The paths are also supposed to be disjoint except for common vertices and
+/// common edges but no crossing.
+/// This function implements an extension of the algorithm discussed in:  
+/// https://www.geometrictools.com/Documentation/TriangulationByEarClipping.pdf
+function _cleave_connected_region(region, eps=EPSILON) =
+    len(region)==1 ? region[0] :
+    let( 
+        outer   = deduplicate(clockwise_polygon(region[0])),  // 
+        holes   = [for(i=[1:1:len(region)-1])                 // possibly unneeded
+                      let(poly=region[i])                     //
+                      deduplicate( ccw_polygon(poly) ) ],     //
+        extridx = [for(li=holes) max_index(column(li,0)) ],
+        // the right extreme vertex for each hole sorted by decreasing x values
+        extremes = sort( [for(i=idx(holes)) [ i, extridx[i], -holes[i][extridx[i]].x] ], idx=2 )
+    ) 
+    _polyHoles(outer, holes, extremes, eps, 0);
+
+
+// connect the hole paths one at a time to the outer path.
+// 'extremes' is the list of the right extreme vertex of each hole sorted by decreasing abscissas
+// see   _cleave_connected_region(region, eps)
+function _polyHoles(outer, holes, extremes, eps=EPSILON, n=0) =
+    let( 
+        extr = extremes[n],    // 
+        hole = holes[extr[0]], // hole path to bridge to the outer path
+        ipt  = extr[1],        // index of the hole point with maximum abscissa
+        brdg = _bridge(hole[ipt], outer, eps)  // the index of a point in outer to bridge hole[ipt] to
+    )
+    assert(brdg!=undef, "Error: check input polygon restrictions")
+    let(
+        l  = len(outer),
+        lh = len(hole),
+        // the new outer polygon bridging the hole to the old outer
+        npoly =
+            approx(outer[brdg], hole[ipt], eps) 
+            ?   [ for(i=[brdg: 1: brdg+l])  outer[i%l] ,
+                  for(i=[ipt+1:1: ipt+lh-1])  hole[i%lh] ]
+            :   [ for(i=[brdg: 1: brdg+l])  outer[i%l] ,
+                  for(i=[ipt:1: ipt+lh])  hole[i%lh] ]
+    )
+    n==len(holes)-1 ?  npoly : 
+    _polyHoles(npoly, holes, extremes, eps, n+1);          
+          
+// find a point in outer to be connected to pt in the interior of outer 
+// by a segment that not cross or touch any non adjacente edge of outer.
+// return the index of a vertex in the outer path where the bridge should end
+// see _polyHoles(outer, holes, extremes, eps)
+function _bridge(pt, outer,eps) =
+    // find the intersection of a ray from pt to the right 
+    // with the boundary of the outer cycle
+    let(  
+        l    = len(outer),
+        crxs = 
+            [for( i=idx(outer) )
+                let( edge = select(outer,i,i+1) )
+                // consider just descending outer edges at right of pt crossing ordinate pt.y
+                if(    (edge[0].y> pt.y) 
+                    && (edge[1].y<=pt.y) 
+                    && ( norm(edge[1]-pt)<eps // accepts touching vertices
+                         || _tri_class([pt, edge[0], edge[1]], eps)>0 ) )
+                    [ i,
+                      // the point of edge with ordinate pt.y
+                      abs(pt.y-edge[1].y)<eps ? edge[1] :
+                      let( u = (pt-edge[1]).y / (edge[0]-edge[1]).y )
+                      (1-u)*edge[1] + u*edge[0]
+                    ]
+             ]
+    )
+    assert(crxs!=[], "Error: check input polygon restrictions") 
+    let( 
+        // the intersection point nearest to pt
+        minX    = min([for(p=crxs) p[1].x]),
+        crxcand = [for(crx=crxs) if(crx[1].x < minX+eps) crx ],
+        nearest = min_index([for(crx=crxcand) outer[crx[0]].y]),
+        proj    = crxcand[nearest],
+        vert0   = outer[proj[0]],    // the two vertices of the nearest crossing edge
+        vert1   = outer[(proj[0]+1)%l],
+        isect   = proj[1]            // the intersection point
+    )
+    // if pt touches the middle of an outer edge -> error
+    assert( ! approx(pt,isect,eps) || approx(pt,vert0,eps) || approx(pt,vert1,eps),
+            "There is a forbidden self_intersection" )
+    norm(pt-vert0) < eps ? proj[0] :  // if pt touches an outer vertex, return its index
+    norm(pt-vert1) < eps ? (proj[0]+1)%l :
+    let( 
+        // the edge [vert0, vert1] necessarily satisfies vert0.y > vert1.y
+        // indices of candidates to an outer bridge point
+        cand  = 
+            (vert0.x > pt.x) 
+            ?   [ proj[0], 
+                  // select reflex vertices inside of the triangle [pt, vert0, isect]
+                  for(i=idx(outer)) 
+                      if( _tri_class(select(outer,i-1,i+1),eps) <= 0 
+                          && _pt_in_tri(outer[i], [pt, vert0, isect], eps)>=0 )
+                        i 
+                ]
+            :   [ (proj[0]+1)%l,
+                  // select reflex vertices inside of the triangle [pt, isect, vert1] 
+                  for(i=idx(outer)) 
+                      if( _tri_class(select(outer,i-1,i+1),eps) <= 0 
+                          &&  _pt_in_tri(outer[i], [pt, isect, vert1], eps)>=0 )
+                        i 
+                ],
+        // choose the candidate outer[i] such that the line [pt, outer[i]] has minimum slope
+        // among those with minimum slope choose the nearest to pt
+        slopes  = [for(i=cand) 1-abs(outer[i].x-pt.x)/norm(outer[i]-pt) ],
+        min_slp = min(slopes),
+        cand2   = [for(i=idx(cand)) if(slopes[i]<=min_slp+eps) cand[i] ],
+        nearest = min_index([for(i=cand2) norm(pt-outer[i]) ]) 
+    )
+    cand2[nearest];
 
 
 // Function: vnf_from_region()