BOSL2/vnf.scad

//////////////////////////////////////////////////////////////////////
// LibFile: vnf.scad
//   The Vertices'N'Faces structure (VNF) holds the data used by polyhedron() to construct objects: a vertex
//   list and a list of faces.  This library makes it easier to construct polyhedra by providing
//   functions to construct, merge, and modify VNF data, while avoiding common pitfalls such as
//   reversed faces.  
// Includes:
//   include <BOSL2/std.scad>
// FileGroup: Advanced Modeling
// FileSummary: Vertices 'n' Faces structure.  Makes polyhedron() easier to use.
// FileFootnotes: STD=Included in std.scad
//////////////////////////////////////////////////////////////////////


// Section: Creating Polyhedrons with VNF Structures
//   VNF stands for "Vertices'N'Faces".  VNF structures are 2-item lists, `[VERTICES,FACES]` where the
//   first item is a list of vertex points, and the second is a list of face indices into the vertex
//   list.  Each VNF is self contained, with face indices referring only to its own vertex list.
//   You can construct a `polyhedron()` in parts by describing each part in a self-contained VNF, then
//   merge the various VNFs to get the completed polyhedron vertex list and faces.

/// Constant: EMPTY_VNF
/// Description:
///   The empty VNF data structure.  Equal to `[[],[]]`.  
EMPTY_VNF = [[],[]];  // The standard empty VNF with no vertices or faces.


// Function: vnf_vertex_array()
// Usage:
//   vnf = vnf_vertex_array(points, [caps], [cap1], [cap2], [style], [reverse], [col_wrap], [row_wrap]);
// Description:
//   Creates a VNF structure from a rectangular vertex list, by dividing the vertices into columns and rows,
//   adding faces to tile the surface.  You can optionally have faces added to wrap the last column
//   back to the first column, or wrap the last row to the first.  Endcaps can be added to either
//   the first and/or last rows.  The style parameter determines how the quadrilaterals are divided into
//   triangles.  The default style is an arbitrary, systematic subdivision in the same direction.  The "alt" style
//   is the uniform subdivision in the other (alternate) direction.  The "min_edge" style picks the shorter edge to
//   subdivide for each quadrilateral, so the division may not be uniform across the shape.  The "quincunx" style
//   adds a vertex in the center of each quadrilateral and creates four triangles, and the "convex" and "concave" styles
//   chooses the locally convex/concave subdivision.  Degenerate faces
//   are not included in the output, but if this results in unused vertices they will still appear in the output.  
// Arguments:
//   points = A list of vertices to divide into columns and rows.
//   ---
//   caps = If true, add endcap faces to the first AND last rows.
//   cap1 = If true, add an endcap face to the first row.
//   cap2 = If true, add an endcap face to the last row.
//   col_wrap = If true, add faces to connect the last column to the first.
//   row_wrap = If true, add faces to connect the last row to the first.
//   reverse = If true, reverse all face normals.
//   style = The style of subdividing the quads into faces.  Valid options are "default", "alt", "min_edge", "quincunx", "convex" and "concave".
// Example(3D):
//   vnf = vnf_vertex_array(
//       points=[
//           for (h = [0:5:180-EPSILON]) [
//               for (t = [0:5:360-EPSILON])
//                   cylindrical_to_xyz(100 + 12 * cos((h/2 + t)*6), t, h)
//           ]
//       ],
//       col_wrap=true, caps=true, reverse=true, style="alt"
//   );
//   vnf_polyhedron(vnf);
// Example(3D): Both `col_wrap` and `row_wrap` are true to make a torus.
//   vnf = vnf_vertex_array(
//       points=[
//           for (a=[0:5:360-EPSILON])
//               apply(
//                   zrot(a) * right(30) * xrot(90),
//                   path3d(circle(d=20))
//               )
//       ],
//       col_wrap=true, row_wrap=true, reverse=true
//   );
//   vnf_polyhedron(vnf);
// Example(3D): Möbius Strip.  Note that `row_wrap` is not used, and the first and last profile copies are the same.
//   vnf = vnf_vertex_array(
//       points=[
//           for (a=[0:5:360]) apply(
//               zrot(a) * right(30) * xrot(90) * zrot(a/2+60),
//               path3d(square([1,10], center=true))
//           )
//       ],
//       col_wrap=true, reverse=true
//   );
//   vnf_polyhedron(vnf);
// Example(3D): Assembling a Polyhedron from Multiple Parts
//   wall_points = [
//       for (a = [-90:2:90]) apply(
//           up(a) * scale([1-0.1*cos(a*6),1-0.1*cos((a+90)*6),1]),
//           path3d(circle(d=100))
//       )
//   ];
//   cap = [
//       for (a = [0:0.01:1+EPSILON]) apply(
//           up(90-5*sin(a*360*2)) * scale([a,a,1]),
//           wall_points[0]
//       )
//   ];
//   cap1 = [for (p=cap) down(90, p=zscale(-1, p=p))];
//   cap2 = [for (p=cap) up(90, p=p)];
//   vnf1 = vnf_vertex_array(points=wall_points, col_wrap=true);
//   vnf2 = vnf_vertex_array(points=cap1, col_wrap=true);
//   vnf3 = vnf_vertex_array(points=cap2, col_wrap=true, reverse=true);
//   vnf_polyhedron([vnf1, vnf2, vnf3]);
function vnf_vertex_array(
    points,
    caps, cap1, cap2,
    col_wrap=false,
    row_wrap=false,
    reverse=false,
    style="default"
) = 
    assert(!(any([caps,cap1,cap2]) && !col_wrap), "col_wrap must be true if caps are requested")
    assert(!(any([caps,cap1,cap2]) && row_wrap), "Cannot combine caps with row_wrap")
    assert(in_list(style,["default","alt","quincunx", "convex","concave", "min_edge"]))
    assert(is_matrix(points[0], n=3),"Point array has the wrong shape or points are not 3d")
    assert(is_consistent(points), "Non-rectangular or invalid point array")
    let(
        pts = flatten(points),
        pcnt = len(pts),
        rows = len(points),
        cols = len(points[0])
    )
    rows<=1 || cols<=1 ? EMPTY_VNF :
    let(
        cap1 = first_defined([cap1,caps,false]),
        cap2 = first_defined([cap2,caps,false]),
        colcnt = cols - (col_wrap?0:1),
        rowcnt = rows - (row_wrap?0:1),
        verts = [
            each pts,
            if (style=="quincunx") 
                for (r = [0:1:rowcnt-1], c = [0:1:colcnt-1]) 
                   let(
                       i1 = ((r+0)%rows)*cols + ((c+0)%cols),
                       i2 = ((r+1)%rows)*cols + ((c+0)%cols),
                       i3 = ((r+1)%rows)*cols + ((c+1)%cols),
                       i4 = ((r+0)%rows)*cols + ((c+1)%cols)
                   )
                   mean([pts[i1], pts[i2], pts[i3], pts[i4]])
        ],
        allfaces = [
            if (cap1) count(cols,reverse=!reverse),
            if (cap2) count(cols,(rows-1)*cols, reverse=reverse),
            for (r = [0:1:rowcnt-1], c=[0:1:colcnt-1])
               each
               let(
                   i1 = ((r+0)%rows)*cols + ((c+0)%cols),
                   i2 = ((r+1)%rows)*cols + ((c+0)%cols),
                   i3 = ((r+1)%rows)*cols + ((c+1)%cols),
                   i4 = ((r+0)%rows)*cols + ((c+1)%cols),
                   faces =
                        style=="quincunx"? 
                          let(i5 = pcnt + r*colcnt + c)
                          [[i1,i5,i2],[i2,i5,i3],[i3,i5,i4],[i4,i5,i1]]
                      : style=="alt"? 
                          [[i1,i4,i2],[i2,i4,i3]]
                      : style=="min_edge"?
                          let(
                               d42=norm(pts[i4]-pts[i2]),
                               d13=norm(pts[i1]-pts[i3]),
                               shortedge = d42<=d13 ? [[i1,i4,i2],[i2,i4,i3]]
                                                    : [[i1,i3,i2],[i1,i4,i3]]
                          )
                          shortedge
                      : style=="convex"?  
                          let(   // Find normal for 3 of the points.  Is the other point above or below?
                              n = (reverse?-1:1)*cross(pts[i2]-pts[i1],pts[i3]-pts[i1]),
                              convexfaces = n==0 ? [[i1,i4,i3]]
                                          : n*pts[i4] > n*pts[i1] ? [[i1,i4,i2],[i2,i4,i3]]
                                                                  : [[i1,i3,i2],[i1,i4,i3]]
                          )
                          convexfaces
                      : style=="concave"?  
                          let(   // Find normal for 3 of the points.  Is the other point above or below?
                              n = (reverse?-1:1)*cross(pts[i2]-pts[i1],pts[i3]-pts[i1]),
                              concavefaces = n==0 ? [[i1,i4,i3]]
                                          : n*pts[i4] <= n*pts[i1] ? [[i1,i4,i2],[i2,i4,i3]]
                                                                  : [[i1,i3,i2],[i1,i4,i3]]
                          )
                          concavefaces
                      : [[i1,i3,i2],[i1,i4,i3]],
                   // remove degenerate faces 
                   culled_faces= [for(face=faces)
                       if (norm(verts[face[0]]-verts[face[1]])>EPSILON &&
                           norm(verts[face[1]]-verts[face[2]])>EPSILON && 
                           norm(verts[face[2]]-verts[face[0]])>EPSILON) 
                           face
                   ],
                   rfaces = reverse? [for (face=culled_faces) reverse(face)] : culled_faces
               )
               rfaces,
        ]
    )
    [verts,allfaces];


// Function: vnf_tri_array()
// Usage:
//   vnf = vnf_tri_array(points, [row_wrap], [reverse])
// Description:
//   Produces a vnf from an array of points where each row length can differ from the adjacent rows by up to 2 in length.  This enables
//   the construction of triangular VNF patches.  The resulting VNF can be wrapped along the rows by setting `row_wrap` to true.
//   You cannot wrap columns: if you need to do that you'll need to merge two VNF arrays that share edges.  Degenerate faces
//   are not included in the output, but if this results in unused vertices they will still appear in the output.  
// Arguments:
//   points = List of point lists for each row
//   row_wrap = If true then add faces connecting the first row and last row.  These rows must differ by at most 2 in length.
//   reverse = Set this to reverse the direction of the faces
// Example(3D,NoAxes): Each row has one more point than the preceeding one.
//   pts = [for(y=[1:1:10]) [for(x=[0:y-1]) [x,y,y]]];
//   vnf = vnf_tri_array(pts);
//   vnf_wireframe(vnf,width=0.1);
//   color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
// Example(3D,NoAxes): Each row has two more points than the preceeding one.
//   pts = [for(y=[0:2:10]) [for(x=[-y/2:y/2]) [x,y,y]]];
//   vnf = vnf_tri_array(pts);
//   vnf_wireframe(vnf,width=0.1);
//   color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
// Example(3D): Merging two VNFs to construct a cone with one point length change between rows.
//   pts1 = [for(z=[0:10]) path3d(arc(3+z,r=z/2+1, angle=[0,180]),10-z)];
//   pts2 = [for(z=[0:10]) path3d(arc(3+z,r=z/2+1, angle=[180,360]),10-z)];
//   vnf = vnf_join([vnf_tri_array(pts1),
//                     vnf_tri_array(pts2)]);
//   color("green")vnf_wireframe(vnf,width=0.1);
//   vnf_polyhedron(vnf);
// Example(3D): Cone with length change two between rows
//   pts1 = [for(z=[0:1:10]) path3d(arc(3+2*z,r=z/2+1, angle=[0,180]),10-z)];
//   pts2 = [for(z=[0:1:10]) path3d(arc(3+2*z,r=z/2+1, angle=[180,360]),10-z)];
//   vnf = vnf_join([vnf_tri_array(pts1),
//                    vnf_tri_array(pts2)]);
//   color("green")vnf_wireframe(vnf,width=0.1);
//   vnf_polyhedron(vnf);
// Example(3D,NoAxes): Point count can change irregularly
//   lens = [10,9,7,5,6,8,8,10];
//   pts = [for(y=idx(lens)) lerpn([-lens[y],y,y],[lens[y],y,y],lens[y])];
//   vnf = vnf_tri_array(pts);
//   vnf_wireframe(vnf,width=0.1);
//   color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
function vnf_tri_array(points, row_wrap=false, reverse=false) = 
    let(
       lens = [for(row=points) len(row)],
       rowstarts = [0,each cumsum(lens)],
       faces =
          [for(i=[0:1:len(points) - 1 - (row_wrap ? 0 : 1)]) each
            let(
                rowstart = rowstarts[i],
                nextrow = select(rowstarts,i+1),
                delta = select(lens,i+1)-lens[i]
            )
            delta == 0 ?
              [for(j=[0:1:lens[i]-2]) reverse ? [j+rowstart+1, j+rowstart, j+nextrow] : [j+rowstart, j+rowstart+1, j+nextrow],
               for(j=[0:1:lens[i]-2]) reverse ? [j+rowstart+1, j+nextrow, j+nextrow+1] : [j+rowstart+1, j+nextrow+1, j+nextrow]] :
            delta == 1 ?
              [for(j=[0:1:lens[i]-2]) reverse ? [j+rowstart+1, j+rowstart, j+nextrow+1] : [j+rowstart, j+rowstart+1, j+nextrow+1],
               for(j=[0:1:lens[i]-1]) reverse ? [j+rowstart, j+nextrow, j+nextrow+1] : [j+rowstart, j+nextrow+1, j+nextrow]] :
            delta == -1 ?
              [for(j=[0:1:lens[i]-3]) reverse ? [j+rowstart+1, j+nextrow, j+nextrow+1]: [j+rowstart+1, j+nextrow+1, j+nextrow],
               for(j=[0:1:lens[i]-2]) reverse ? [j+rowstart+1, j+rowstart, j+nextrow] : [j+rowstart, j+rowstart+1, j+nextrow]] :
            let(count = floor((lens[i]-1)/2))
            delta == 2 ?
              [
               for(j=[0:1:count-1]) reverse ? [j+rowstart+1, j+rowstart, j+nextrow+1] : [j+rowstart, j+rowstart+1, j+nextrow+1],       // top triangles left
               for(j=[count:1:lens[i]-2]) reverse ? [j+rowstart+1, j+rowstart, j+nextrow+2] : [j+rowstart, j+rowstart+1, j+nextrow+2], // top triangles right
               for(j=[0:1:count]) reverse ? [j+rowstart, j+nextrow, j+nextrow+1] : [j+rowstart, j+nextrow+1, j+nextrow],                        // bot triangles left
               for(j=[count+1:1:select(lens,i+1)-2]) reverse ? [j+rowstart-1, j+nextrow, j+nextrow+1] : [j+rowstart-1, j+nextrow+1, j+nextrow], // bot triangles right
              ] :
            delta == -2 ?
              [
               for(j=[0:1:count-2]) reverse ? [j+nextrow, j+nextrow+1, j+rowstart+1] : [j+nextrow, j+rowstart+1, j+nextrow+1],
               for(j=[count-1:1:lens[i]-4]) reverse ? [j+nextrow,j+nextrow+1,j+rowstart+2] : [j+nextrow,j+rowstart+2, j+nextrow+1],
               for(j=[0:1:count-1]) reverse ? [j+nextrow, j+rowstart+1, j+rowstart] : [j+nextrow, j+rowstart, j+rowstart+1],
               for(j=[count:1:select(lens,i+1)]) reverse ? [ j+nextrow-1, j+rowstart+1, j+rowstart]: [ j+nextrow-1, j+rowstart, j+rowstart+1],
              ] :
            assert(false,str("Unsupported row length difference of ",delta, " between row ",i," and ",(i+1)%len(points)))
          ],
       verts = flatten(points),
       culled_faces=
           [for(face=faces)
               if (norm(verts[face[0]]-verts[face[1]])>EPSILON &&
                   norm(verts[face[1]]-verts[face[2]])>EPSILON && 
                   norm(verts[face[2]]-verts[face[0]])>EPSILON) 
                   face
           ]
    )
    [flatten(points), culled_faces];



// Function: vnf_join()
// Usage:
//   vnf = vnf_join([VNF, VNF, VNF, ...]);
// Description:
//   Given a list of VNF structures, merges them all into a single VNF structure.
//   Combines all the points of the input VNFs and labels the faces appropriately.
//   All the points in the input VNFs will appear in the output, even if they are
//   duplicates of each other.  It is valid to repeat points in a VNF, but if you
//   with to remove the duplicates that will occur along joined edges, use {{vnf_merge_points()}}.
// Arguments:
//   vnfs = a list of the VNFs to joint into one VNF.
function vnf_join(vnfs) = 
    assert(is_vnf_list(vnfs) , "Input must be a list of VNFs")
    len(vnfs)==1 ? vnfs[0]
    :
    let (
        offs  = cumsum([ 0, for (vnf = vnfs) len(vnf[0]) ]),
        verts = [for (vnf=vnfs) each vnf[0]],
        faces =
            [ for (i = idx(vnfs)) 
                let( faces = vnfs[i][1] )
                for (face = faces) 
                    if ( len(face) >= 3 )
                        [ for (j = face) 
                            assert( j>=0 && j<len(vnfs[i][0]), 
                                    str("VNF number ", i, " has a face indexing an nonexistent vertex") )
                            offs[i] + j ]
            ]
    )
    [verts,faces];



// Function: vnf_from_polygons()
// Usage:
//   vnf = vnf_from_polygons(polygons);
// Description:
//   Given a list of 3d polygons, produces a VNF containing those polygons.  
//   It is up to the caller to make sure that the points are in the correct order to make the face
//   normals point outwards.  No checking for duplicate vertices is done.  If you want to
//   remove duplicate vertices use {{vnf_merge_points()}}.
// Arguments:
//   polygons = The list of 3d polygons to turn into a VNF
function vnf_from_polygons(polygons) =
   assert(is_list(polygons) && is_path(polygons[0]),"Input should be a list of polygons")
   let(
       offs = cumsum([0, for(p=polygons) len(p)]),
       faces = [for(i=idx(polygons))
                  [for (j=idx(polygons[i])) offs[i]+j]
               ]
   )
   [flatten(polygons), faces];




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(column(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(list_rotate(path1,v1)),
        path2 = ccw_polygon(list_rotate(path2,v2))
    )
    [
        each path1,
        if (repeat_start) path1[0],
        each path2,
        if (repeat_start) path2[0],
    ];                      

        
/// Internal Function: _cleave_connected_region(region, eps)
/// Description:
///   Given a region that is connected and has its outer border in region[0],
///   produces a overlapping connected path to join internal holes to  
///   the outer border without adding points. Output is a single non-simple polygon. 
/// Requirements:
///   It expects that all region paths be simple closed paths, with region[0] CW and 
///   the other paths CCW and encircled by region[0]. The input region paths are also 
///   supposed to be disjoint except for common vertices and common edges but with 
///   no crossings. It may return `undef` if these conditions are not met.
///   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(region[0]),             // 
        holes   = [for(i=[1:1:len(region)-1])         // deduplication possibly unneeded
                      deduplicate( region[i] ) ],     //
        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
    )
    brdg == undef ? undef :
    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 = 
            let( edges = pair(outer,wrap=true) )
            [for( i = idx(edges) )
                let( edge = edges[i] )
                // consider just descending outer edges at right of pt crossing ordinate pt.y
                if(    (edge[0].y >  pt.y+eps) 
                    && (edge[1].y <= pt.y) 
                    && _is_at_left(pt, [edge[1], edge[0]], eps) ) 
                    [ 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]
                    ]
             ]
    )
    crxs == [] ? undef :
    let( 
        // the intersection point of the nearest edge to pt with minimum slope
        minX    = min([for(p=crxs) p[1].x]),
        crxcand = [for(crx=crxs) if(crx[1].x < minX+eps) crx ], // nearest edges
        nearest = min_index([for(crx=crxcand) 
                                (outer[crx[0]].x - pt.x) / (outer[crx[0]].y - pt.y) ]), // minimum slope
        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
    )
    norm(pt-vert1) < eps ? (proj[0]+1)%l : // if pt touches an outer vertex, return its index
    // as vert0.y > pt.y then pt!=vert0 
    norm(pt-isect) < eps ? undef :         // if pt touches the middle of an outer edge -> error
    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()
// Usage:
//   vnf = vnf_from_region(region, [transform], [reverse]);
// Description:
//   Given a (two-dimensional) region, applies the given transformation matrix to it and makes a (three-dimensional) triangulated VNF of
//   faces for that region, reversed if desired. 
// Arguments:
//   region = The region to conver to a vnf.
//   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
// Example(3D):
//   region = [square([20,10],center=true),
//             right(5,square(4,center=true)),
//             left(5,square(6,center=true))];
//   vnf = vnf_from_region(region);
//   color("gray")down(.125)
//        linear_extrude(height=.125)region(region);
//   vnf_wireframe(vnf,width=.25);
function vnf_from_region(region, transform, reverse=false) =
    let (
        regions = region_parts(force_region(region)),
        vnfs =
            [ for (rgn = regions) 
                let( cleaved = path3d(_cleave_connected_region(rgn)) ) 
                assert( cleaved, "The region is invalid")
                let(
                    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_join(vnfs)
    )
    vnf_triangulate(outvnf);



// Section: VNF Testing and Access


// Function: is_vnf()
// Usage:
//   bool = is_vnf(x);
// Description:
//   Returns true if the given value looks like a VNF structure.
function is_vnf(x) =
    is_list(x) &&
    len(x)==2 &&
    is_list(x[0]) &&
    is_list(x[1]) &&
    (x[0]==[] || (len(x[0])>=3 && is_vector(x[0][0],3))) &&
    (x[1]==[] || is_vector(x[1][0]));


// Function: is_vnf_list()
// Description: Returns true if the given value looks passingly like a list of VNF structures.
function is_vnf_list(x) = is_list(x) && all([for (v=x) is_vnf(v)]);


// Function: vnf_vertices()
// Description: Given a VNF structure, returns the list of vertex points.
function vnf_vertices(vnf) = vnf[0];


// Function: vnf_faces()
// Description: Given a VNF structure, returns the list of faces, where each face is a list of indices into the VNF vertex list.
function vnf_faces(vnf) = vnf[1];



// Section: Altering the VNF Internals


// Function: vnf_reverse_faces()
// Usage:
//   rvnf = vnf_reverse_faces(vnf);
// Description:
//   Reverses the orientation of all the faces in the given VNF.
function vnf_reverse_faces(vnf) =
    [vnf[0], [for (face=vnf[1]) reverse(face)]];


// Function: vnf_quantize()
// Usage:
//   vnf2 = vnf_quantize(vnf,[q]);
// Description:
//   Quantizes the vertex coordinates of the VNF to the given quanta `q`.
// Arguments:
//   vnf = The VNF to quantize.
//   q = The quanta to quantize the VNF coordinates to.
function vnf_quantize(vnf,q=pow(2,-12)) =
    [[for (pt = vnf[0]) quant(pt,q)], vnf[1]];



// Function: vnf_merge_points()
// Usage:
//   new_vnf = vnf_merge_points(vnf, [eps]);
// Description:
//   Given a VNF, consolidates all duplicate vertices with a tolerance `eps`, relabeling the faces as necessary,
//   and eliminating any face with fewer than 3 vertices.  Unreferenced vertices of the input VNF are not dropped.
//   To remove such vertices uses {{vnf_drop_unused_points()}}.  
// Arguments:
//   vnf = a VNF to consolidate
//   eps = the tolerance in finding duplicates. Default: EPSILON
function vnf_merge_points(vnf,eps=EPSILON) = 
    let(
        verts = vnf[0], 
        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] ],    // this doesn't eliminate unreferenced vertices
        nfaces = 
            [ for(face=vnf[1]) 
                let(
                    nface = [ for(vi=face) map2[map[vi]] ],
                    dface = [for (i=idx(nface)) 
                                if( nface[i]!=nface[(i+1)%len(nface)]) 
                                    nface[i] ] 
                )
                if(len(dface) >= 3) dface 
            ]
    ) 
    [nverts, nfaces];


// Function: vnf_drop_unused_points()
// Usage:
//   clean_vnf=vnf_drop_unused_points(vnf);
// Description:
//   Remove all unreferenced vertices from a VNF.  Note that in most
//   cases unreferenced vertices cause no harm, and this function may
//   be slow on large VNFs.
function vnf_drop_unused_points(vnf) =
    let(
        flat = flatten(vnf[1]),
        ind  = _link_indicator(flat,0,len(vnf[0])-1),
        verts = [for(i=idx(vnf[0])) if(ind[i]==1) vnf[0][i] ], 
        map   = cumsum(ind) 
    )
    [ verts, [for(face=vnf[1]) [for(v=face) map[v]-1 ] ] ];

function _link_indicator(l,imin,imax) =
    len(l) == 0  ? repeat(imax-imin+1,0) :
    imax-imin<100 || len(l)<400 ? [for(si=search(list([imin:1:imax]),l,1)) si!=[] ? 1: 0 ] : 
    let( 
        pivot   = floor((imax+imin)/2),
        lesser  = [ for(li=l) if( li< pivot) li ],
        greater = [ for(li=l) if( li> pivot) li ] 
    )
    concat( _link_indicator(lesser ,imin,pivot-1), 
            search(pivot,l,1) ? 1 : 0 ,
            _link_indicator(greater,pivot+1,imax) ) ;

// Function: vnf_triangulate()
// Usage:
//   vnf2 = vnf_triangulate(vnf);
// Description:
//   Triangulates faces in the VNF that have more than 3 vertices.
// Arguments:
//   vnf = vnf to triangulate
// Example(3D):
//   include <BOSL2/polyhedra.scad>
//   vnf = zrot(33,regular_polyhedron_info("vnf", "dodecahedron", side=12));
//   vnf_polyhedron(vnf);
//   triangulated = vnf_triangulate(vnf);
//   color("red")vnf_wireframe(triangulated,width=.3);
function vnf_triangulate(vnf) =
    let(
        verts = vnf[0],
        faces = [for (face=vnf[1]) 
                    each (len(face)==3 ? [face] : 
                    let( tris = polygon_triangulate(verts, face) )
                    assert( tris!=undef, "Some `vnf` face cannot be triangulated.")
                    tris ) ]
    ) 
    [verts, faces]; 



// Function: vnf_slice()
// Usage:
//   sliced = vnf_slice(vnf, dir, cuts);
// Description:
//   Slice the faces of a VNF along a specified axis direction at a given list
//   of cut points.  The cut points can appear in any order.  You can use this to refine the faces of a VNF before applying
//   a nonlinear transformation to its vertex set.
// Arguments:
//   vnf = vnf to slice
//   dir = normal direction to the slices, either "X", "Y" or "Z"
//   cuts = X, Y or Z values where cuts occur
// Example(3D):
//   include <BOSL2/polyhedra.scad>
//   vnf = regular_polyhedron_info("vnf", "dodecahedron", side=12);
//   vnf_polyhedron(vnf);
//   sliced = vnf_slice(vnf, "X", [-6,-1,10]);
//   color("red")vnf_wireframe(sliced,width=.3);
function vnf_slice(vnf,dir,cuts) =
  let(
       vert = vnf[0],
       faces = [for(face=vnf[1]) select(vert,face)],
       poly_list = _slice_3dpolygons(faces, dir, cuts)
  )
  vnf_merge_points(vnf_from_polygons(poly_list));


function _split_polygon_at_x(poly, x) =
    let(
        xs = column(poly,0)
    ) (min(xs) >= x || max(xs) <= x)? [poly] :
    let(
        poly2 = [
            for (p = pair(poly,true)) each [
                p[0],
                if(
                    (p[0].x < x && p[1].x > x) ||
                    (p[1].x < x && p[0].x > x)
                ) let(
                    u = (x - p[0].x) / (p[1].x - p[0].x)
                ) [
                    x,  // Important for later exact match tests
                    u*(p[1].y-p[0].y)+p[0].y
                ]
            ]
        ],
        out1 = [for (p = poly2) if(p.x <= x) p],
        out2 = [for (p = poly2) if(p.x >= x) p],
        out3 = [
            if (len(out1)>=3) each split_path_at_self_crossings(out1),
            if (len(out2)>=3) each split_path_at_self_crossings(out2),
        ],
        out = [for (p=out3) if (len(p) > 2) cleanup_path(p)]
    ) out;


function _split_2dpolygons_at_each_x(polys, xs, _i=0) =
    _i>=len(xs)? polys :
    _split_2dpolygons_at_each_x(
        [
            for (poly = polys)
            each _split_polygon_at_x(poly, xs[_i])
        ], xs, _i=_i+1
    );

/// Internal Function: _slice_3dpolygons()
/// Usage:
///   splitpolys = _slice_3dpolygons(polys, dir, cuts);
/// Topics: Geometry, Polygons, Intersections
/// Description:
///   Given a list of 3D polygons, a choice of X, Y, or Z, and a cut list, `cuts`, splits all of the polygons where they cross
///   X/Y/Z at any value given in cuts.  
/// Arguments:
///   polys = A list of 3D polygons to split.
///   dir_ind = slice direction, 0=X, 1=Y, or 2=Z
///   cuts = A list of scalar values for locating the cuts
function _slice_3dpolygons(polys, dir, cuts) =
    assert( [for (poly=polys) if (!is_path(poly,3)) 1] == [], "Expects list of 3D paths.")
    assert( is_vector(cuts), "The split list must be a vector.")
    assert( in_list(dir, ["X", "Y", "Z"]))
    let(
        I = ident(3),
        dir_ind = ord(dir)-ord("X")
    )
    flatten([for (poly = polys)
        let(
            plane = plane_from_polygon(poly)
        )
        assert(plane,"Found non-coplanar face.")
        let(
            normal = point3d(plane),
            pnormal = normal - (normal*I[dir_ind])*I[dir_ind]
        )
        approx(pnormal,[0,0,0]) ? [poly] :
        let (
            pind = max_index(v_abs(pnormal)),  // project along this direction
            otherind = 3-pind-dir_ind,         // keep dir_ind and this direction
            keep = [I[dir_ind], I[otherind]],  // dir ind becomes the x dir
            poly2d = poly*transpose(keep),     // project to 2d, putting selected direction in the X position
            poly_list = [for(p=_split_2dpolygons_at_each_x([poly2d], cuts))
                            let(
                                a = p*keep,    // unproject, but pind dimension data is missing
                                ofs = outer_product((repeat(plane[3], len(a))-a*normal)/plane[pind],I[pind])
                             )
                             a+ofs]    // ofs computes the missing pind dimension data and adds it back in
        )
        poly_list
    ]);





// Section: Turning a VNF into geometry


// Module: vnf_polyhedron()
// Usage:
//   vnf_polyhedron(vnf);
//   vnf_polyhedron([VNF, VNF, VNF, ...]);
// Description:
//   Given a VNF structure, or a list of VNF structures, creates a polyhedron from them.
// Arguments:
//   vnf = A VNF structure, or list of VNF structures.
//   convexity = Max number of times a line could intersect a wall of the shape.
//   extent = If true, calculate anchors by extents, rather than intersection.  Default: true.
//   cp = Centerpoint for determining intersection anchors or centering the shape.  Determintes the base of the anchor vector.  Can be "centroid", "mean", "box" or a 3D point.  Default: "centroid"
//   anchor = Translate so anchor point is at origin (0,0,0).  See [anchor](attachments.scad#subsection-anchor).  Default: `"origin"`
//   spin = Rotate this many degrees around the Z axis after anchor.  See [spin](attachments.scad#subsection-spin).  Default: `0`
//   orient = Vector to rotate top towards, after spin.  See [orient](attachments.scad#subsection-orient).  Default: `UP`
//   atype = Select "hull" or "intersect" anchor type.  Default: "hull"
module vnf_polyhedron(vnf, convexity=2, extent=true, cp="centroid", anchor="origin", spin=0, orient=UP, atype="hull") {
    vnf = is_vnf_list(vnf)? vnf_join(vnf) : vnf;
    assert(in_list(atype, _ANCHOR_TYPES), "Anchor type must be \"hull\" or \"intersect\"");
    attachable(anchor,spin,orient, vnf=vnf, extent=atype=="hull", cp=cp) {
        polyhedron(vnf[0], vnf[1], convexity=convexity);
        children();
    }
}


// Module: vnf_wireframe()
// Usage:
//   vnf_wireframe(vnf, [width]);
// Description:
//   Given a VNF, creates a wire frame ball-and-stick model of the polyhedron with a cylinder for
//   each edge and a sphere at each vertex.  The width parameter specifies the width of the sticks
//   that form the wire frame and the diameter of the balls.  
// Arguments:
//   vnf = A vnf structure
//   width = width of the cylinders forming the wire frame.  Default: 1
// Example:
//   $fn=32;
//   ball = sphere(r=20, $fn=6);
//   vnf_wireframe(ball,width=1);
// Example:
//   include <BOSL2/polyhedra.scad>
//   $fn=32;
//   cube_oct = regular_polyhedron_info("vnf",
//                      name="cuboctahedron", or=20);
//   vnf_wireframe(cube_oct);
// Example: The spheres at the vertex are imperfect at aligning with the cylinders, so especially at low $fn things look prety ugly.  This is normal.
//   include <BOSL2/polyhedra.scad>
//   $fn=8;
//   octahedron = regular_polyhedron_info("vnf",
//                         name="octahedron", or=20);
//   vnf_wireframe(octahedron,width=5);
module vnf_wireframe(vnf, width=1)
{
  vertex = vnf[0];
  edges = unique([for (face=vnf[1], i=idx(face))
                    sort([face[i], select(face,i+1)])
                 ]);
  for (e=edges) extrude_from_to(vertex[e[0]],vertex[e[1]]) circle(d=width);
  // Identify vertices actually used and draw them
  vertused = search(count(len(vertex)), flatten(edges), 1);
  for(i=idx(vertex)) if(vertused[i]!=[]) move(vertex[i]) sphere(d=width); 
}


// Section: Operations on VNFs

// Function: vnf_volume()
// Usage:
//   vol = vnf_volume(vnf);
// Description:
//   Returns the volume enclosed by the given manifold VNF.   The VNF must describe a valid polyhedron with consistent face direction and
//   no holes; otherwise the results are undefined.  Returns a positive volume if face direction is clockwise and a negative volume
//   if face direction is counter-clockwise.

// Divide the polyhedron into tetrahedra with the origin as one vertex and sum up the signed volume.
function vnf_volume(vnf) =
    let(verts = vnf[0])
    sum([
         for(face=vnf[1], j=[1:1:len(face)-2])
             cross(verts[face[j+1]], verts[face[j]]) * verts[face[0]]
    ])/6;


// Function: vnf_area()
// Usage:
//   area = vnf_area(vnf);
// Description:
//   Returns the surface area in any VNF by adding up the area of all its faces.  The VNF need not be a manifold.  
function vnf_area(vnf) =
    let(verts=vnf[0])
    sum([for(face=vnf[1]) polygon_area(select(verts,face))]);


/// Internal Function: _vnf_centroid()
/// Usage:
///   vol = _vnf_centroid(vnf);
/// Description:
///   Returns the centroid of the given manifold VNF.  The VNF must describe a valid polyhedron with consistent face direction and
///   no holes; otherwise the results are undefined.

/// Divide the solid up into tetrahedra with the origin as one vertex.  
/// The centroid of a tetrahedron is the average of its vertices.
/// The centroid of the total is the volume weighted average.
function _vnf_centroid(vnf,eps=EPSILON) =
    assert(is_vnf(vnf) && len(vnf[0])!=0 && len(vnf[1])!=0,"Invalid or empty VNF given to centroid") 
    let(
        verts = vnf[0],
        pos = sum([
            for(face=vnf[1], j=[1:1:len(face)-2]) let(
                v0  = verts[face[0]],
                v1  = verts[face[j]],
                v2  = verts[face[j+1]],
                vol = cross(v2,v1)*v0
            )
            [ vol, (v0+v1+v2)*vol ]
        ])
    )
    assert(!approx(pos[0],0, eps), "The vnf has self-intersections.")
    pos[1]/pos[0]/4;


// Function: vnf_halfspace()
// Usage:
//   newvnf = vnf_halfspace(plane, vnf, [closed]);
// Description:
//   Returns the intersection of the vnf with a half space.  The half space is defined by
//   plane = [A,B,C,D], taking the side where the normal [A,B,C] points: Ax+By+Cz≥D.
//   If closed is set to false then the cut face is not included in the vnf.  This could
//   allow further extension of the vnf by merging with other vnfs.  
// Arguments:
//   plane = plane defining the boundary of the half space
//   vnf = vnf to cut
//   closed = if false do not return include cut face(s).  Default: true
// Example(3D):
//   vnf = cube(10,center=true);
//   cutvnf = vnf_halfspace([-1,1,-1,0], vnf);
//   vnf_polyhedron(cutvnf);
// Example(3D):  Cut face has 2 components
//   vnf = path_sweep(circle(r=4, $fn=16),
//                    circle(r=20, $fn=64),closed=true);
//   cutvnf = vnf_halfspace([-1,1,-4,0], vnf);
//   vnf_polyhedron(cutvnf);
// Example(3D): Cut face is not simply connected
//   vnf = path_sweep(circle(r=4, $fn=16),
//                    circle(r=20, $fn=64),closed=true);
//   cutvnf = vnf_halfspace([0,0.7,-4,0], vnf);
//   vnf_polyhedron(cutvnf);
// Example(3D): Cut object has multiple components
//   function knot(a,b,t) =   // rolling knot 
//        [ a * cos (3 * t) / (1 - b* sin (2 *t)), 
//          a * sin( 3 * t) / (1 - b* sin (2 *t)), 
//        1.8 * b * cos (2 * t) /(1 - b* sin (2 *t))]; 
//   a = 0.8; b = sqrt (1 - a * a); 
//   ksteps = 400;
//   knot_path = [for (i=[0:ksteps-1]) 50 * knot(a,b,(i/ksteps)*360)];
//   ushape = [[-10, 0],[-10, 10],[ -7, 10],[ -7, 2],[  7, 2],[  7, 7],[ 10, 7],[ 10, 0]];
//   knot=path_sweep(ushape, knot_path, closed=true, method="incremental");
//   cut_knot = vnf_halfspace([1,0,0,0], knot);
//   vnf_polyhedron(cut_knot);
function vnf_halfspace(plane, vnf, closed=true) =
    assert(_valid_plane(plane), "Invalid plane")
    assert(is_vnf(vnf), "Invalid vnf")
    let(
         inside = [for(x=vnf[0]) plane*[each x,-1] >= 0 ? 1 : 0],
         vertexmap = [0,each cumsum(inside)],
         faces_edges_vertices = _vnfcut(plane, vnf[0],vertexmap,inside, vnf[1], last(vertexmap)),
         newvert = concat(bselect(vnf[0],inside), faces_edges_vertices[2])
    )
    closed==false ? [newvert, faces_edges_vertices[0]] :
    let(
        allpaths = _assemble_paths(newvert, faces_edges_vertices[1]),
        newpaths = [for(p=allpaths) if (len(p)>=3) p
                                    else assert(approx(p[0],p[1]),"Orphan edge found when assembling cut edges.")
           ]
    )
    len(newpaths)<=1 ? [newvert, concat(faces_edges_vertices[0], newpaths)] 
    :
      let(
           M = project_plane(plane),
           faceregion = [for(path=newpaths) path2d(apply(M,select(newvert,path)))],
           facevnf = vnf_from_region(faceregion,transform=rot_inverse(M),reverse=true)
      )
      vnf_join([[newvert, faces_edges_vertices[0]], facevnf]);


function _assemble_paths(vertices, edges, paths=[],i=0) =
     i==len(edges) ? paths :
     norm(vertices[edges[i][0]]-vertices[edges[i][1]])<EPSILON ? _assemble_paths(vertices,edges,paths,i+1) :
     let(    // Find paths that connects on left side and right side of the edges (if one exists)
         left = [for(j=idx(paths)) if (approx(vertices[last(paths[j])],vertices[edges[i][0]])) j],
         right = [for(j=idx(paths)) if (approx(vertices[edges[i][1]],vertices[paths[j][0]])) j]
     )
     assert(len(left)<=1 && len(right)<=1)
     let(              
          keep_path = list_remove(paths,concat(left,right)),
          update_path = left==[] && right==[] ? edges[i] 
                      : left==[] ? concat([edges[i][0]],paths[right[0]])
                      : right==[] ?  concat(paths[left[0]],[edges[i][1]])
                      : left != right ? concat(paths[left[0]], paths[right[0]])
                      : paths[left[0]]
     )
     _assemble_paths(vertices, edges, concat(keep_path, [update_path]), i+1);


function _vnfcut(plane, vertices, vertexmap, inside, faces, vertcount, newfaces=[], newedges=[], newvertices=[], i=0) =
   i==len(faces) ? [newfaces, newedges, newvertices] :
   let(
        pts_inside = select(inside,faces[i])
   )
   all(pts_inside) ? _vnfcut(plane, vertices, vertexmap, inside, faces, vertcount,
                             concat(newfaces, [select(vertexmap,faces[i])]), newedges, newvertices, i+1):
   !any(pts_inside) ? _vnfcut(plane, vertices, vertexmap,inside, faces, vertcount, newfaces, newedges, newvertices, i+1):
   let(
        first = search([[1,0]],pair(pts_inside,wrap=true),0)[0],
        second = search([[0,1]],pair(pts_inside,wrap=true),0)[0]
   )
   assert(len(first)==1 && len(second)==1, "Found concave face in VNF.  Run vnf_triangulate first to ensure convex faces.")
   let(
        newface = [each select(vertexmap,select(faces[i],second[0]+1,first[0])),vertcount, vertcount+1],
        newvert = [plane_line_intersection(plane, select(vertices,select(faces[i],first[0],first[0]+1)),eps=0),
                   plane_line_intersection(plane, select(vertices,select(faces[i],second[0],second[0]+1)),eps=0)]
   )
   true //!approx(newvert[0],newvert[1])
       ? _vnfcut(plane, vertices, vertexmap, inside, faces, vertcount+2,
                 concat(newfaces, [newface]), concat(newedges,[[vertcount+1,vertcount]]),concat(newvertices,newvert),i+1)
   :len(newface)>3
       ? _vnfcut(plane, vertices, vertexmap, inside, faces, vertcount+1,
                 concat(newfaces, [list_head(newface)]), newedges,concat(newvertices,[newvert[0]]),i+1)
   :
   _vnfcut(plane, vertices, vertexmap, inside, faces, vertcount,newfaces, newedges, newvert, i+1);
 



function _triangulate_planar_convex_polygons(polys) =
    polys==[]? [] :
    let(
        tris = [for (poly=polys) if (len(poly)==3) poly],
        bigs = [for (poly=polys) if (len(poly)>3) poly],
        newtris = [for (poly=bigs) select(poly,-2,0)],
        newbigs = [for (poly=bigs) select(poly,0,-2)],
        newtris2 = _triangulate_planar_convex_polygons(newbigs),
        outtris = concat(tris, newtris, newtris2)
    ) outtris;

//**
// this function may produce degenerate triangles:
//    _triangulate_planar_convex_polygons([ [for(i=[0:1]) [i,i],
//                                           [1,-1], [-1,-1],
//                                           for(i=[-1:0]) [i,i] ] ] )
//    == [[[-1, -1], [ 0,  0], [0,  0]]
//        [[-1, -1], [-1, -1], [0,  0]]
//        [[ 1, -1], [-1, -1], [0,  0]]
//        [[ 0,  0], [ 1,  1], [1, -1]] ]
//

// Function: vnf_bend()
// Usage:
//   bentvnf = vnf_bend(vnf,r,d,[axis]);
// Description:
//   Bend a VNF around the X, Y or Z axis, splitting up faces as necessary.  Returns the bent
//   VNF.  For bending around the Z axis the input VNF must not cross the Y=0 plane.  For bending
//   around the X or Y axes the VNF must not cross the Z=0 plane.  Note that if you wrap a VNF all the way around
//   it may intersect itself, which produces an invalid polyhedron.  It is your responsibility to
//   avoid this situation.  The 1:1
//   radius is where the curved length of the bent VNF matches the length of the original VNF.  If the
//   `r` or `d` arguments are given, then they will specify the 1:1 radius or diameter.  If they are
//   not given, then the 1:1 radius will be defined by the distance of the furthest vertex in the
//   original VNF from the Z=0 plane.  You can adjust the granularity of the bend using the standard
//   `$fa`, `$fs`, and `$fn` variables.
// Arguments:
//   vnf = The original VNF to bend.
//   r = If given, the radius where the size of the original shape is the same as in the original.
//   ---
//   d = If given, the diameter where the size of the original shape is the same as in the original.
//   axis = The axis to wrap around.  "X", "Y", or "Z".  Default: "Z"
// Example(3D):
//   vnf0 = cube([100,40,10], center=true);
//   vnf1 = up(50, p=vnf0);
//   vnf2 = down(50, p=vnf0);
//   bent1 = vnf_bend(vnf1, axis="Y");
//   bent2 = vnf_bend(vnf2, axis="Y");
//   vnf_polyhedron([bent1,bent2]);
// Example(3D):
//   vnf0 = linear_sweep(star(n=5,step=2,d=100), height=10);
//   vnf1 = up(50, p=vnf0);
//   vnf2 = down(50, p=vnf0);
//   bent1 = vnf_bend(vnf1, axis="Y");
//   bent2 = vnf_bend(vnf2, axis="Y");
//   vnf_polyhedron([bent1,bent2]);
// Example(3D):
//   rgn = union(rect([100,20]),
//               rect([20,100]));
//   vnf0 = linear_sweep(zrot(45,p=rgn), height=10);
//   vnf1 = up(50, p=vnf0);
//   vnf2 = down(50, p=vnf0);
//   bent1 = vnf_bend(vnf1, axis="Y");
//   bent2 = vnf_bend(vnf2, axis="Y");
//   vnf_polyhedron([bent1,bent2]);
// Example(3D): Bending Around X Axis.
//   rgnr = union(
//       rect([20,100]),
//       back(50, p=trapezoid(w1=40, w2=0, h=20, anchor=FRONT))
//   );
//   vnf0 = xrot(00,p=linear_sweep(rgnr, height=10));
//   vnf1 = up(50, p=vnf0);
//   #vnf_polyhedron(vnf1);
//   bent1 = vnf_bend(vnf1, axis="X");
//   vnf_polyhedron([bent1]);
// Example(3D): Bending Around Y Axis.
//   rgn = union(
//       rect([20,100]),
//       back(50, p=trapezoid(w1=40, w2=0, h=20, anchor=FRONT))
//   );
//   rgnr = zrot(-90, p=rgn);
//   vnf0 = xrot(00,p=linear_sweep(rgnr, height=10));
//   vnf1 = up(50, p=vnf0);
//   #vnf_polyhedron(vnf1);
//   bent1 = vnf_bend(vnf1, axis="Y");
//   vnf_polyhedron([bent1]);
// Example(3D): Bending Around Z Axis.
//   rgn = union(
//       rect([20,100]),
//       back(50, p=trapezoid(w1=40, w2=0, h=20, anchor=FRONT))
//   );
//   rgnr = zrot(90, p=rgn);
//   vnf0 = xrot(90,p=linear_sweep(rgnr, height=10));
//   vnf1 = fwd(50, p=vnf0);
//   #vnf_polyhedron(vnf1);
//   bent1 = vnf_bend(vnf1, axis="Z");
//   vnf_polyhedron([bent1]);
// Example(3D): Bending more than once around the cylinder
//   $fn=32;
//   vnf = apply(fwd(5)*yrot(30),cube([100,2,5],center=true));
//   bent = vnf_bend(vnf, axis="Z");
//   vnf_polyhedron(bent);
function vnf_bend(vnf,r,d,axis="Z") =
    let(
        chk_axis = assert(in_list(axis,["X","Y","Z"])),
        verts = vnf[0],
        bounds = pointlist_bounds(verts),
        bmin = bounds[0],
        bmax = bounds[1],
        dflt = axis=="Z"?
            max(abs(bmax.y), abs(bmin.y)) :
            max(abs(bmax.z), abs(bmin.z)),
        r = get_radius(r=r,d=d,dflt=dflt),
        extent = axis=="X" ? [bmin.y, bmax.y] : [bmin.x, bmax.x]
    )
    let(
        span_chk = axis=="Z"?
            assert(bmin.y > 0 || bmax.y < 0, "Entire shape MUST be completely in front of or behind y=0.") :
            assert(bmin.z > 0 || bmax.z < 0, "Entire shape MUST be completely above or below z=0."),
        steps = ceil(segs(r) * (extent[1]-extent[0])/(2*PI*r)),
        step = (extent[1]-extent[0]) / steps,
        bend_at = [for(i = [1:1:steps-1]) i*step+extent[0]],
        slicedir = axis=="X"? "Y" : "X",   // slice in y dir for X axis case, and x dir otherwise
        sliced = vnf_slice(vnf, slicedir, bend_at),
        coord = axis=="X" ? [0,sign(bmax.z),0] : axis=="Y" ? [sign(bmax.z),0,0] : [sign(bmax.y),0,0],
        new_vert = [for(p=sliced[0])
                       let(a=coord*p*180/(PI*r))
                       axis=="X"? [p.x, p.z*sin(a), p.z*cos(a)] :
                       axis=="Y"? [p.z*sin(a), p.y, p.z*cos(a)] :
                       [p.y*sin(a), p.y*cos(a), p.z]]
                         
   ) [new_vert,sliced[1]];


// Section: Debugging Polyhedrons

/// Internal Module: _show_vertices()
/// Usage:
///   _show_vertices(vertices, [size])
/// Description:
///   Draws all the vertices in an array, at their 3D position, numbered by their
///   position in the vertex array.  Also draws any children of this module with
///   transparency.
/// Arguments:
///   vertices = Array of point vertices.
///   size = The size of the text used to label the vertices.  Default: 1
/// Example:
///   verts = [for (z=[-10,10], y=[-10,10], x=[-10,10]) [x,y,z]];
///   faces = [[0,1,2], [1,3,2], [0,4,5], [0,5,1], [1,5,7], [1,7,3], [3,7,6], [3,6,2], [2,6,4], [2,4,0], [4,6,7], [4,7,5]];
///   _show_vertices(vertices=verts, size=2) {
///       polyhedron(points=verts, faces=faces);
///   }
module _show_vertices(vertices, size=1) {
    color("blue") {
        dups = vector_search(vertices, EPSILON, vertices);
        for (ind = dups){
            numstr = str_join([for(i=ind) str(i)],",");
            v = vertices[ind[0]];
            translate(v) {
                rot($vpr) back(size/8){
                   linear_extrude(height=size/10, center=true, convexity=10) {
                      text(text=numstr, size=size, halign="center");
                   }
                }
                sphere(size/10);
            }
        }
    }
}


/// Internal Module: _show_faces()
/// Usage:
///   _show_faces(vertices, faces, [size=]);
/// Description:
///   Draws all the vertices at their 3D position, numbered in blue by their
///   position in the vertex array.  Each face will have their face number drawn
///   in red, aligned with the center of face.  All children of this module are drawn
///   with transparency.
/// Arguments:
///   vertices = Array of point vertices.
///   faces = Array of faces by vertex numbers.
///   size = The size of the text used to label the faces and vertices.  Default: 1
/// Example(EdgesMed):
///   verts = [for (z=[-10,10], y=[-10,10], x=[-10,10]) [x,y,z]];
///   faces = [[0,1,2], [1,3,2], [0,4,5], [0,5,1], [1,5,7], [1,7,3], [3,7,6], [3,6,2], [2,6,4], [2,4,0], [4,6,7], [4,7,5]];
///   _show_faces(vertices=verts, faces=faces, size=2) {
///       polyhedron(points=verts, faces=faces);
///   }
module _show_faces(vertices, faces, size=1) {
    vlen = len(vertices);
    color("red") {
        for (i = [0:1:len(faces)-1]) {
            face = faces[i];
            if (face[0] < 0 || face[1] < 0 || face[2] < 0 || face[0] >= vlen || face[1] >= vlen || face[2] >= vlen) {
                echo("BAD FACE: ", vlen=vlen, face=face);
            } else {
                verts = select(vertices,face);
                c = mean(verts);
                v0 = verts[0];
                v1 = verts[1];
                v2 = verts[2];
                dv0 = unit(v1 - v0);
                dv1 = unit(v2 - v0);
                nrm0 = cross(dv0, dv1);
                nrm1 = UP;
                axis = vector_axis(nrm0, nrm1);
                ang = vector_angle(nrm0, nrm1);
                theta = atan2(nrm0[1], nrm0[0]);
                translate(c) {
                    rotate(a=180-ang, v=axis) {
                        zrot(theta-90)
                        linear_extrude(height=size/10, center=true, convexity=10) {
                            union() {
                                text(text=str(i), size=size, halign="center");
                                text(text=str("_"), size=size, halign="center");
                            }
                        }
                    }
                }
            }
        }
    }
}



// Module: debug_vnf()
// Usage:
//   debug_vnf(vnfs, [faces], [vertices], [opacity], [size], [convexity]);
// Description:
//   A drop-in module to replace `vnf_polyhedron()` to help debug vertices and faces.
//   Draws all the vertices at their 3D position, numbered in blue by their
//   position in the vertex array.  Each face will have its face number drawn
//   in red, aligned with the center of face.  All given faces are drawn with
//   transparency. All children of this module are drawn with transparency.
//   Works best with Thrown-Together preview mode, to see reversed faces.
//   You can set opacity to 0 if you want to supress the display of the polyhedron faces.  
//   .
//   The vertex numbers are shown rotated to face you.  As you rotate your polyhedron you
//   can rerun the preview to display them oriented for viewing from a different viewpoint.
// Topics: Polyhedra, Debugging
// Arguments:
//   vnf = vnf to display
//   ---
//   faces = if true display face numbers.  Default: true
//   vertices = if true display vertex numbers.  Default: true
//   opacity = Opacity of the polyhedron faces.  Default: 0.5
//   convexity = The max number of walls a ray can pass through the given polygon paths.
//   size = The size of the text used to label the faces and vertices.  Default: 1
// Example(EdgesMed):
//   verts = [for (z=[-10,10], a=[0:120:359.9]) [10*cos(a),10*sin(a),z]];
//   faces = [[0,1,2], [5,4,3], [0,3,4], [0,4,1], [1,4,5], [1,5,2], [2,5,3], [2,3,0]];
//   debug_vnf([verts,faces], size=2);
module debug_vnf(vnf, faces=true, vertices=true, opacity=0.5, size=1, convexity=6 ) {
    no_children($children);
    if (faces)
      _show_faces(vertices=vnf[0], faces=vnf[1], size=size);
    if (vertices)
      _show_vertices(vertices=vnf[0], size=size);
    color([0.2, 1.0, 0, opacity])
       vnf_polyhedron(vnf,convexity=convexity);
}


// Function&Module: vnf_validate()
// Usage: As Function
//   fails = vnf_validate(vnf);
// Usage: As Module
//   vnf_validate(vnf, [size]);
// Description:
//   When called as a function, returns a list of non-manifold errors with the given VNF.
//   Each error has the format `[ERR_OR_WARN,CODE,MESG,POINTS,COLOR]`.
//   When called as a module, echoes the non-manifold errors to the console, and color hilites the
//   bad edges and vertices, overlaid on a transparent gray polyhedron of the VNF.
//   .
//   Currently checks for these problems:
//   .
//   Type    | Color    | Code         | Message
//   ------- | -------- | ------------ | ---------------------------------
//   WARNING | Yellow   | BIG_FACE     | Face has more than 3 vertices, and may confuse CGAL.
//   WARNING | Brown    | NULL_FACE    | Face has zero area.
//   ERROR   | Cyan     | NONPLANAR    | Face vertices are not coplanar.
//   ERROR   | Brown    | DUP_FACE     | Multiple instances of the same face.
//   ERROR   | Orange   | MULTCONN     | Multiply Connected Geometry. Too many faces attached at Edge.
//   ERROR   | Violet   | REVERSAL     | Faces reverse across edge.
//   ERROR   | Red      | T_JUNCTION   | Vertex is mid-edge on another Face.
//   ERROR   | Blue     | FACE_ISECT   | Faces intersect.
//   ERROR   | Magenta  | HOLE_EDGE    | Edge bounds Hole.
//   .
//   Still to implement:
//   - Overlapping coplanar faces.
// Arguments:
//   vnf = The VNF to validate.
//   size = The width of the lines and diameter of points used to highlight edges and vertices.  Module only.  Default: 1
//   check_isects = If true, performs slow checks for intersecting faces.  Default: false
// Example: BIG_FACE Warnings; Faces with More Than 3 Vertices.  CGAL often will fail to accept that a face is planar after a rotation, if it has more than 3 vertices.
//   vnf = skin([
//       path3d(regular_ngon(n=3, d=100),0),
//       path3d(regular_ngon(n=5, d=100),100)
//   ], slices=0, caps=true, method="tangent");
//   vnf_validate(vnf);
// Example: NONPLANAR Errors; Face Vertices are Not Coplanar
//   a = [  0,  0,-50];
//   b = [-50,-50, 50];
//   c = [-50, 50, 50];
//   d = [ 50, 50, 60];
//   e = [ 50,-50, 50];
//   vnf = vnf_from_polygons([
//       [a, b, e], [a, c, b], [a, d, c], [a, e, d], [b, c, d, e]
//   ]);
//   vnf_validate(vnf);
// Example: MULTCONN Errors; More Than Two Faces Attached to the Same Edge.  This confuses CGAL, and can lead to failed renders.
//   vnf = vnf_triangulate(linear_sweep(union(square(50), square(50,anchor=BACK+RIGHT)), height=50));
//   vnf_validate(vnf);
// Example: REVERSAL Errors; Faces Reversed Across Edge
//   vnf1 = skin([
//       path3d(square(100,center=true),0),
//       path3d(square(100,center=true),100),
//   ], slices=0, caps=false);
//   vnf = vnf_join([vnf1, vnf_from_polygons([
//       [[-50,-50,  0], [ 50, 50,  0], [-50, 50,  0]],
//       [[-50,-50,  0], [ 50,-50,  0], [ 50, 50,  0]],
//       [[-50,-50,100], [-50, 50,100], [ 50, 50,100]],
//       [[-50,-50,100], [ 50,-50,100], [ 50, 50,100]],
//   ])]);
//   vnf_validate(vnf);
// Example: T_JUNCTION Errors; Vertex is Mid-Edge on Another Face.
//   vnf1 = skin([
//       path3d(square(100,center=true),0),
//       path3d(square(100,center=true),100),
//   ], slices=0, caps=false);
//   vnf = vnf_join([vnf1, vnf_from_polygons([
//       [[-50,-50,0], [50,50,0], [-50,50,0]],
//       [[-50,-50,0], [50,-50,0], [50,50,0]],
//       [[-50,-50,100], [-50,50,100], [0,50,100]],
//       [[-50,-50,100], [0,50,100], [0,-50,100]],
//       [[0,-50,100], [0,50,100], [50,50,100]],
//       [[0,-50,100], [50,50,100], [50,-50,100]],
//   ])]);
//   vnf_validate(vnf);
// Example: FACE_ISECT Errors; Faces Intersect
//   vnf = vnf_join([
//       vnf_triangulate(linear_sweep(square(100,center=true), height=100)),
//       move([75,35,30],p=vnf_triangulate(linear_sweep(square(100,center=true), height=100)))
//   ]);
//   vnf_validate(vnf,size=2,check_isects=true);
// Example: HOLE_EDGE Errors; Edges Adjacent to Holes.
//   vnf = skin([
//       path3d(regular_ngon(n=4, d=100),0),
//       path3d(regular_ngon(n=5, d=100),100)
//   ], slices=0, caps=false);
//   vnf_validate(vnf,size=2);
function vnf_validate(vnf, show_warns=true, check_isects=false) =
    assert(is_vnf(vnf), "Invalid VNF")
    let(
        vnf = vnf_merge_points(vnf),
        varr = vnf[0],
        faces = vnf[1],
        lvarr = len(varr),
        edges = sort([
            for (face=faces, edge=pair(face,true))
            edge[0]<edge[1]? edge : [edge[1],edge[0]]
        ]),
        dfaces = [
            for (face=faces) let(
                face=deduplicate_indexed(varr,face,closed=true)
            ) if(len(face)>=3)
            face
        ],
        face_areas = [
            for (face = faces)
            len(face) < 3? 0 :
            polygon_area([for (k=face) varr[k]])
        ],
        edgecnts = unique_count(edges),
        uniq_edges = edgecnts[0],
        issues = []
    )
    let(
        big_faces = !show_warns? [] : [
            for (face = faces)
            if (len(face) > 3)
            _vnf_validate_err("BIG_FACE", [for (i=face) varr[i]])
        ],
        null_faces = !show_warns? [] : [
            for (i = idx(faces)) let(
                face = faces[i],
                area = face_areas[i],
                faceverts = [for (k=face) varr[k]]
            )
            if (is_num(area) && abs(area) < EPSILON)
            _vnf_validate_err("NULL_FACE", faceverts)
        ],
        issues = concat(big_faces, null_faces)
    )
    let(
        bad_indices = [
            for (face = faces, idx = face)
            if (idx < 0 || idx >= lvarr)
            _vnf_validate_err("BAD_INDEX", [idx])
        ],
        issues = concat(issues, bad_indices)
    ) bad_indices? issues :
    let(
        repeated_faces = [
            for (i=idx(dfaces), j=idx(dfaces))
            if (i!=j) let(
                face1 = dfaces[i],
                face2 = dfaces[j]
            ) if (min(face1) == min(face2)) let(
                min1 = min_index(face1),
                min2 = min_index(face2)
            ) if (min1 == min2) let(
                sface1 = list_rotate(face1,min1),
                sface2 = list_rotate(face2,min2)
            ) if (sface1 == sface2)
            _vnf_validate_err("DUP_FACE", [for (i=sface1) varr[i]])
        ],
        issues = concat(issues, repeated_faces)
    ) repeated_faces? issues :
    let(
        multconn_edges = unique([
            for (i = idx(uniq_edges))
            if (edgecnts[1][i]>2)
            _vnf_validate_err("MULTCONN", [for (i=uniq_edges[i]) varr[i]])
        ]),
        issues = concat(issues, multconn_edges)
    ) multconn_edges? issues :
    let(
        reversals = unique([
            for(i = idx(dfaces), j = idx(dfaces)) if(i != j)
            for(edge1 = pair(faces[i],true))
            for(edge2 = pair(faces[j],true))
            if(edge1 == edge2)  // Valid adjacent faces will never have the same vertex ordering.
            if(_edge_not_reported(edge1, varr, multconn_edges))
            _vnf_validate_err("REVERSAL", [for (i=edge1) varr[i]])
        ]),
        issues = concat(issues, reversals)
    ) reversals? issues :
    let(
        t_juncts = unique([
            for (v=idx(varr), edge=uniq_edges) let(
                ia = edge[0],
                ib = v,
                ic = edge[1]
            )
            if (ia!=ib && ib!=ic && ia!=ic) let(
                a = varr[ia],
                b = varr[ib],
                c = varr[ic]
            )
            if (!approx(a,b) && !approx(b,c) && !approx(a,c)) let(
                pt = line_closest_point([a,c],b,SEGMENT)
            )
            if (approx(pt,b))
            _vnf_validate_err("T_JUNCTION", [b])
        ]),
        issues = concat(issues, t_juncts)
    ) t_juncts? issues :
    let(
        isect_faces = !check_isects? [] : unique([
            for (i = [0:1:len(faces)-2]) let(
                f1 = faces[i],
                poly1   = select(varr, faces[i]),
                plane1  = plane3pt(poly1[0], poly1[1], poly1[2]),
                normal1 = [plane1[0], plane1[1], plane1[2]]
            )
            for (j = [i+1:1:len(faces)-1]) let(
                f2 = faces[j],
                poly2 = select(varr, f2),
                val = poly2 * normal1
            )
            if( min(val)<=plane1[3] && max(val)>=plane1[3] ) let(
                plane2  = plane_from_polygon(poly2),
                normal2 = [plane2[0], plane2[1], plane2[2]],
                val = poly1 * normal2
            )
            if( min(val)<=plane2[3] && max(val)>=plane2[3] ) let(
                shared_edges = [
                    for (edge1 = pair(f1, true), edge2 = pair(f2, true))
                    if (edge1 == [edge2[1], edge2[0]]) 1
                ]
            )
            if (!shared_edges) let(
                line = plane_intersection(plane1, plane2)
            )
            if (!is_undef(line)) let(
                isects = polygon_line_intersection(poly1, line)
            )
            if (!is_undef(isects))
            for (isect = isects)
            if (len(isect) > 1) let(
                isects2 = polygon_line_intersection(poly2, isect, bounded=true)
            )
            if (!is_undef(isects2))
            for (seg = isects2)
            if (seg[0] != seg[1])
            _vnf_validate_err("FACE_ISECT", seg)
        ]),
        issues = concat(issues, isect_faces)
    ) isect_faces? issues :
    let(
        hole_edges = unique([
            for (i=idx(uniq_edges))
            if (edgecnts[1][i]<2)
            if (_pts_not_reported(uniq_edges[i], varr, t_juncts))
            if (_pts_not_reported(uniq_edges[i], varr, isect_faces))
            _vnf_validate_err("HOLE_EDGE", [for (i=uniq_edges[i]) varr[i]])
        ]),
        issues = concat(issues, hole_edges)
    ) hole_edges? issues :
    let(
        nonplanars = unique([
            for (i = idx(faces)) let(
                face = faces[i],
                area = face_areas[i],
                faceverts = [for (k=face) varr[k]]
            )
            if (is_num(area) && abs(area) > EPSILON)
            if (!is_coplanar(faceverts))
            _vnf_validate_err("NONPLANAR", faceverts)
        ]),
        issues = concat(issues, nonplanars)
    ) issues;


_vnf_validate_errs = [
    ["BIG_FACE",    "WARNING", "cyan",    "Face has more than 3 vertices, and may confuse CGAL"],
    ["NULL_FACE",   "WARNING", "blue",    "Face has zero area."],
    ["BAD_INDEX",   "ERROR",   "cyan",    "Invalid face vertex index."],
    ["NONPLANAR",   "ERROR",   "yellow",  "Face vertices are not coplanar"],
    ["DUP_FACE",    "ERROR",   "brown",   "Multiple instances of the same face."],
    ["MULTCONN",    "ERROR",   "orange",  "Multiply Connected Geometry. Too many faces attached at Edge"],
    ["REVERSAL",    "ERROR",   "violet",  "Faces Reverse Across Edge"],
    ["T_JUNCTION",  "ERROR",   "magenta", "Vertex is mid-edge on another Face"],
    ["FACE_ISECT",  "ERROR",   "brown",   "Faces intersect"],
    ["HOLE_EDGE",   "ERROR",   "red",     "Edge bounds Hole"]
];


function _vnf_validate_err(name, extra) =
    let(
        info = [for (x = _vnf_validate_errs) if (x[0] == name) x][0]
    ) concat(info, [extra]);


function _pts_not_reported(pts, varr, reports) =
    [
        for (i = pts, report = reports, pt = report[3])
        if (varr[i] == pt) 1
    ] == [];


function _edge_not_reported(edge, varr, reports) =
    let(
        edge = sort([for (i=edge) varr[i]])
    ) [
        for (report = reports) let(
            pts = sort(report[3])
        ) if (len(pts)==2 && edge == pts) 1
    ] == [];


module vnf_validate(vnf, size=1, show_warns=true, check_isects=false) {
    faults = vnf_validate(
        vnf, show_warns=show_warns,
        check_isects=check_isects
    );
    for (fault = faults) {
        err = fault[0];
        typ = fault[1];
        clr = fault[2];
        msg = fault[3];
        pts = fault[4];
        echo(str(typ, " ", err, " (", clr ,"): ", msg, " at ", pts));
        color(clr) {
            if (is_vector(pts[0])) {
                if (len(pts)==2) {
                    stroke(pts, width=size, closed=true, endcaps="butt", hull=false, $fn=8);
                } else if (len(pts)>2) {
                    stroke(pts, width=size, closed=true, hull=false, $fn=8);
                    polyhedron(pts,[[for (i=idx(pts)) i]]);
                } else {
                    move_copies(pts) sphere(d=size*3, $fn=18);
                }
            }
        }
    }
    color([0.5,0.5,0.5,0.67]) vnf_polyhedron(vnf);
}



// vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap