vnf.scad

//////////////////////////////////////////////////////////////////////
// LibFile: vnf.scad
//   VNF structures, holding Vertices 'N' Faces for use with `polyhedron().`
// Includes:
//   include <BOSL2/std.scad>
//////////////////////////////////////////////////////////////////////


include <triangulation.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.


EMPTY_VNF = [[],[]];  // The standard empty VNF with no vertices or faces.


// 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]))) &&
    (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];


// 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_get_vertex()
// Usage:
//   vvnf = vnf_get_vertex(vnf, p);
// Description:
//   Finds the index number of the given vertex point `p` in the given VNF structure `vnf`.
//   If said point does not already exist in the VNF vertex list, it is added to the returned VNF.
//   Returns: `[INDEX, VNF]` where INDEX is the index of the point in the returned VNF's vertex list,
//   and VNF is the possibly modified new VNF structure.  If `p` is given as a list of points, then
//   the returned INDEX will be a list of indices.
// Arguments:
//   vnf = The VNF structue to get the point index from.
//   p = The point, or list of points to get the index of.
// Example:
//   vnf1 = vnf_get_vertex(p=[3,5,8]);  // Returns: [0, [[[3,5,8]],[]]]
//   vnf2 = vnf_get_vertex(vnf1, p=[3,2,1]);  // Returns: [1, [[[3,5,8],[3,2,1]],[]]]
//   vnf3 = vnf_get_vertex(vnf2, p=[3,5,8]);  // Returns: [0, [[[3,5,8],[3,2,1]],[]]]
//   vnf4 = vnf_get_vertex(vnf3, p=[[1,3,2],[3,2,1]]);  // Returns: [[1,2], [[[3,5,8],[3,2,1],[1,3,2]],[]]]
function vnf_get_vertex(vnf=EMPTY_VNF, p) =
    let(
        isvec = is_vector(p),
        pts = isvec? [p] : p,
        res = set_union(vnf[0], pts, get_indices=true)
    ) [
        (isvec? res[0][0] : res[0]),
        [ res[1], vnf[1] ]
    ];


// Function: vnf_add_face()
// Usage:
//   vnf_add_face(vnf, pts);
// Description:
//   Given a VNF structure and a list of face vertex points, adds the face to the VNF structure.
//   Returns the modified VNF structure `[VERTICES, FACES]`.  It is up to the caller to make
//   sure that the points are in the correct order to make the face normal point outwards.
// Arguments:
//   vnf = The VNF structure to add a face to.
//   pts = The vertex points for the face.
function vnf_add_face(vnf=EMPTY_VNF, pts) =
    assert(is_vnf(vnf))
    assert(is_path(pts))
    let(
        res = set_union(vnf[0], pts, get_indices=true),
        face = deduplicate(res[0], closed=true)
    ) [
        res[1],
        concat(vnf[1], len(face)>2? [face] : [])
    ];


// Function: vnf_add_faces()
// Usage:
//   vnf_add_faces(vnf, faces);
// Description:
//   Given a VNF structure and a list of faces, where each face is given as a list of vertex points,
//   adds the faces to the VNF structure.  Returns the modified VNF structure `[VERTICES, FACES]`.
//   It is up to the caller to make sure that the points are in the correct order to make the face
//   normals point outwards.
// Arguments:
//   vnf = The VNF structure to add a face to.
//   faces = The list of faces, where each face is given as a list of vertex points.
function vnf_add_faces(vnf=EMPTY_VNF, faces) =
    assert(is_vnf(vnf))
    assert(is_list(faces))
    let(
        res = set_union(vnf[0], flatten(faces), get_indices=true),
        idxs = res[0],
        nverts = res[1],
        offs = cumsum([0, for (face=faces) len(face)]),
        ifaces = [
            for (i=idx(faces)) [
                for (j=idx(faces[i]))
                idxs[offs[i]+j]
            ]
        ]
    ) [
        nverts,
        concat(vnf[1],ifaces)
    ];


// Function: vnf_merge()
// Usage:
//   vnf = vnf_merge([VNF, VNF, VNF, ...], [cleanup],[eps]);
// Description:
//   Given a list of VNF structures, merges them all into a single VNF structure.
//   When cleanup=true, it consolidates all duplicate vertices with a tolerance `eps`,
//   drops unreferenced vertices and any final face with less than 3 vertices. 
//   Unreferenced vertices of the input VNFs that doesn't duplicate any other vertex 
//   are not dropped.
// Arguments:
//   vnfs - a list of the VNFs to merge in one VNF.
//   cleanup - when true, consolidates the duplicate vertices of the merge. Default: false
//   eps - the tolerance in finding duplicates when cleanup=true. Default: EPSILON
function vnf_merge(vnfs, cleanup=false, eps=EPSILON) =
    is_vnf(vnfs) ? vnf_merge([vnfs], cleanup, eps) :
    assert( is_vnf_list(vnfs) , "Improper vnf or vnf list")  
    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 ]
            ]
    )
    ! cleanup ? [verts, faces] :
    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
        nfaces = 
            [ for(face=faces) 
                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_reverse_faces()
// Usage:
//   rvnf = vnf_reverse_faces(vnf);
// Description:
//   Reverses the facing of all the faces in the given VNF.
function vnf_reverse_faces(vnf) =
    [vnf[0], [for (face=vnf[1]) reverse(face)]];


// Function: vnf_triangulate()
// Usage:
//   vnf2 = vnf_triangulate(vnf);
// Description:
//   Forces triangulation of faces in the VNF that have more than 3 vertices.
function vnf_triangulate(vnf) =
    let(
        vnf = is_vnf_list(vnf)? vnf_merge(vnf) : vnf,
        verts = vnf[0]
    ) [verts, triangulate_faces(verts, vnf[1])];


// Function: vnf_vertex_array()
// Usage:
//   vnf = vnf_vertex_array(points, [caps], [cap1], [cap2], [reverse], [col_wrap], [row_wrap], [vnf]);
// Description:
//   Creates a VNF structure from a 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.  
// 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".
//   vnf = If given, add all the vertices and faces to this existing VNF structure.
// 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",
    vnf=EMPTY_VNF
) = 
    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_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 ? 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]])
        ]
    )
    vnf_merge(cleanup=false, [
        vnf,
        [
              verts,
              [
               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,
                if (cap1) count(cols,reverse=!reverse),
                if (cap2) count(cols,(rows-1)*cols, reverse=reverse)
              ] 
        ]
    ]);


// 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.
// 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
// Examples:  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,d=.1);
//   color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
// Examples:  Each row has one more point than the preceeding one.
//   pts = [for(y=[0:2:10]) [for(x=[-y/2:y/2]) [x,y,y]]];
//   vnf = vnf_tri_array(pts);
//   vnf_wireframe(vnf,d=.1);
//   color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
// Example: Chaining 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_tri_array(pts1,
//                       vnf=vnf_tri_array(pts2));
//   color("green")vnf_wireframe(vnf,d=.1);
//   vnf_polyhedron(vnf);
// Example: 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_tri_array(pts1,
//                       vnf=vnf_tri_array(pts2));
//   color("green")vnf_wireframe(vnf,d=.1);
//   vnf_polyhedron(vnf);
// Example: 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,d=.1);
//   color("red")move_copies(flatten(pts)) sphere(r=.15,$fn=9);
function vnf_tri_array(points, row_wrap=false, reverse=false, vnf=EMPTY_VNF) = 
   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)))
        ])
    vnf_merge(cleanup=true, [vnf, [flatten(points), faces]]);



// 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 of VNF to use for anchoring when `extent` is false.  Default: `[0, 0, 0]`
//   anchor = Translate so anchor point is at origin (0,0,0).  See [anchor](attachments.scad#anchor).  Default: `"origin"`
//   spin = Rotate this many degrees around the Z axis after anchor.  See [spin](attachments.scad#spin).  Default: `0`
//   orient = Vector to rotate top towards, after spin.  See [orient](attachments.scad#orient).  Default: `UP`
module vnf_polyhedron(vnf, convexity=2, extent=true, cp=[0,0,0], anchor="origin", spin=0, orient=UP) {
    vnf = is_vnf_list(vnf)? vnf_merge(vnf) : vnf;
    cp = is_def(cp) ? cp : vnf_centroid(vnf);
    attachable(anchor,spin,orient, vnf=vnf, extent=extent, cp=cp) {
        polyhedron(vnf[0], vnf[1], convexity=convexity);
        children();
    }
}



// Module: vnf_wireframe()
// Usage:
//   vnf_wireframe(vnf, <r|d>);
// 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.
// Arguments:
//   vnf = A vnf structure
//   r|d = radius or diameter of the cylinders forming the wire frame.  Default: r=1
// Example:
//   $fn=32;
//   ball = sphere(r=20, $fn=6);
//   vnf_wireframe(ball,d=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,r=5);
module vnf_wireframe(vnf, r, d)
{
  r = get_radius(r=r,d=d,dflt=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(r=r);
  move_copies(vertex) sphere(r=r);
}


// 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_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) =
    assert(is_vnf(vnf) && len(vnf[0])!=0 ) 
    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, EPSILON), "The vnf has self-intersections.")
    pos[1]/pos[0]/4;


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:
//   Given a VNF that is entirely above, or entirely below the Z=0 plane, bends the VNF around the
//   Y axis, splitting up faces as necessary.  Returns the bent VNF.  Will error out if the VNF
//   straddles the Z=0 plane, or if the bent VNF would wrap more than completely around.  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],center=true), rect([20,100],center=true));
//   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],center=true),
//       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],center=true),
//       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],center=true),
//       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]);
function vnf_bend(vnf,r,d,axis="Z") =
    let(
        chk_axis = assert(in_list(axis,["X","Y","Z"])),
        vnf = vnf_triangulate(vnf),
        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),
        width = axis=="X"? (bmax.y-bmin.y) : (bmax.x - bmin.x)
    )
    assert(width <= 2*PI*r, "Shape would wrap more than completely around the cylinder.")
    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."),
        min_ang = 180 * bmin.x / (PI * r),
        max_ang = 180 * bmax.x / (PI * r),
        ang_span = max_ang-min_ang,
        steps = ceil(segs(r) * ang_span/360),
        step = width / steps,
        bend_at = axis=="X"? [for(i = [1:1:steps-1]) i*step+bmin.y] :
            [for(i = [1:1:steps-1]) i*step+bmin.x],
        facepolys = [for (face=vnf[1]) select(verts,face)],
        splits = axis=="X"?
            split_polygons_at_each_y(facepolys, bend_at) :
            split_polygons_at_each_x(facepolys, bend_at),
        newtris = _triangulate_planar_convex_polygons(splits),
        bent_faces = [
            for (tri = newtris) [
                for (p = tri) let(
                    a = axis=="X"? 180*p.y/(r*PI) * sign(bmax.z) :
                        axis=="Y"? 180*p.x/(r*PI) * sign(bmax.z) :
                        180*p.x/(r*PI) * sign(bmax.y)
                )
                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]
            ]
        ]
    ) vnf_add_faces(faces=bent_faces);


// 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_add_faces(faces=[
//       [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_add_faces(vnf=vnf1, faces=[
//       [[-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_add_faces(vnf=vnf1, faces=[
//       [[-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_merge([
//       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_path(vnf[0]))
    let(
        vnf = vnf_merge(vnf, cleanup=true),
        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 = segment_closest_point([a,c],b)
            )
            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 (!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);
}


// Section: VNF Transformations

// 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:
//   vnf = cube(10,center=true);
//   cutvnf = vnf_halfspace([-1,1,-1,0], vnf);
//   vnf_polyhedron(cutvnf);
// Example:  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: 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: 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) =
    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(
           faceregion = project_plane(plane, newpaths),
           facevnf = region_faces(faceregion,reverse=true)
      )
      vnf_merge([[newvert, faces_edges_vertices[0]], lift_plane(plane, facevnf)]);


function _assemble_paths(vertices, edges, paths=[],i=0) =
     i==len(edges) ? paths :
     norm(vertices[edges[i][0]]-vertices[edges[i][1]])<EPSILON ? echo(degen=i)_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);
 

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