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vnf.scad
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//////////////////////////////////////////////////////////////////////
// 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. It can find faults in your polyhedrons. Note that this file is for low level manipulation
// of lists of vertices and faces: it can perform some simple transformations on VNF structures
// but cannot perform boolean operations on the polyhedrons represented by VNFs.
// 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()
// Synopsis: Returns a VNF structure from a rectangular vertex list.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_tri_array(), vnf_join(), vnf_from_polygons(), vnf_from_region()
// 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]);
// Example(3D): Building a Multi-Stage Cylindrical Ramp
// include <BOSL2/rounding.scad>
// major_r = 50;
// groove_profile = [
// [-10,0], each arc(points=[[-7,0],[0,-3],[7,0]]), [10,0]
// ];
// ramp_profile = [ [-10,25], [90,25], [180,5], [190,5] ];
// rgroove = apply(right(major_r) * xrot(90), path3d(groove_profile));
// rprofile = round_corners(ramp_profile, radius=20, closed=false, $fn=72);
// vnf = vnf_vertex_array([
// for (a = [ramp_profile[0].x : 1 : last(ramp_profile).x]) let(
// z = lookup(a,rprofile),
// m = zrot(a) * up(z)
// )
// apply(m, [ [rgroove[0].x,0,-z], each rgroove, [last(rgroove).x,0,-z] ])
// ], caps=true, col_wrap=true, reverse=true);
// vnf_polyhedron(vnf, convexity=8);
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","min_area"]))
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_area"?
let(
area42 = norm(cross(pts[i2]-pts[i1], pts[14]-pts[i1]))+norm(cross(pts[i4]-pts[i3], pts[i2]-pts[i3])),
area13 = norm(cross(pts[i1]-pts[i4], pts[i3]-pts[i4]))+norm(cross(pts[i3]-pts[i2], pts[i1]-pts[i2])),
minarea_edge = area42 < area13 + EPSILON
? [[i1,i4,i2],[i2,i4,i3]]
: [[i1,i3,i2],[i1,i4,i3]]
)
minarea_edge
: style=="min_edge"?
let(
d42=norm(pts[i4]-pts[i2]),
d13=norm(pts[i1]-pts[i3]),
shortedge = d42<d13+EPSILON
? [[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(cross(verts[face[1]]-verts[face[0]],
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()
// Synopsis: Returns a VNF from an array of points.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_vertex_array(), vnf_join(), vnf_from_polygons(), vnf_from_region()
// 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()
// Synopsis: Returns a single VNF structure from a list of VNF structures.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_tri_array(), vnf_vertex_array(), vnf_from_polygons(), vnf_from_region()
// 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()}}.
// .
// Note that this is a tool for manipulating polyhedron data. It is for
// building up a full polyhedron from partial polyhedra.
// It is *not* a union operator for VNFs. The VNFs to be joined must not intersect each other,
// except at edges, or the result will be an invalid polyhedron. Similarly the
// result must not have any other illegal polyhedron characteristics, such as creating
// more than two faces sharing the same edge.
// If you want a valid result it is your responsibility to ensure that the polyhedron
// has no holes, no intersecting faces or edges, and obeys all the requirements
// that CGAL expects.
// .
// For example, if you combine two pyramids to try to make an octahedron, the result will
// be invalid because of the two internal faces created by the pyramid bases. A valid
// use would be to build a cube missing one face and a pyramid missing its base and
// then join them into a cube with a point.
// Arguments:
// vnfs = a list of the VNFs to joint into one VNF.
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): Here is a VNF where the top face is missing. It is not a valid polyhedron like this, but we can use it as a building block to make a polyhedron.
// bottom = vnf_vertex_array([path3d(rect(8)), path3d(rect(5),4)],col_wrap=true,cap1=true);
// vnf_polyhedron(bottom);
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): Here is a VNF that also has a missing face.
// triangle = yrot(-90,path3d(regular_ngon(n=3,side=5,anchor=LEFT)));
// top = up(4,vnf_vertex_array([list_set(right(2.5,triangle),0,[0,0,7]),
// right(6,triangle)
// ], col_wrap=true, cap2=true));
// vnf_polyhedron(zrot(90,top));
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): Using vnf_join combines the two VNFs into a single VNF. Note that they share an edge. But the result still isn't closed, so it is not yet a valid polyhedron.
// bottom = vnf_vertex_array([path3d(rect(8)), path3d(rect(5),4)],col_wrap=true,cap1=true);
// triangle = yrot(-90,path3d(regular_ngon(n=3,side=5,anchor=LEFT)));
// top = up(4,vnf_vertex_array([list_set(right(2.5,triangle),0,[0,0,7]),
// right(6,triangle)
// ], col_wrap=true, cap2=true));
// full = vnf_join([bottom,zrot(90,top)]);
// vnf_polyhedron(full);
// Example(3D,VPR=[60,0,26],VPD=55,VPT=[5.6,-5.3,9.8]): If we add enough pieces, and the pieces are all consistent with each other, then we can arrive at a valid polyhedron like this one. To be valid you need to meet all the CGAL requirements: every edge has exactly two faces, all faces are in clockwise order, no intersections of edges.
// bottom = vnf_vertex_array([path3d(rect(8)), path3d(rect(5),4)],col_wrap=true,cap1=true);
// triangle = yrot(-90,path3d(regular_ngon(n=3,side=5,anchor=LEFT)));
// top = up(4,vnf_vertex_array([list_set(right(2.5,triangle),0,[0,0,7]),
// right(6,triangle)
// ], col_wrap=true, cap2=true));
// full = vnf_join([bottom,
// for(theta=[0:90:359]) zrot(theta,top)
// ]);
// vnf_polyhedron(full);
// Example(3D): The vnf_join function is not a union operator for polyhedra. If any faces intersect, like they do in this example where we combine the faces of two cubes, the result is invalid and will give rise to CGAL errors when you add more objects into the model.
// cube1 = cube(5);
// cube2 = move([2,2,2],cube1);
// badvnf = vnf_join([cube1,cube2]);
// vnf_polyhedron(badvnf);
// right(2.5)up(3)color("red")
// text3d("Invalid",size=1,anchor=CENTER,
// orient=FRONT,h=.1);
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()
// Synopsis: Returns a VNF from a list of 3D polygons.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_tri_array(), vnf_join(), vnf_vertex_array(), vnf_from_region()
// 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()
// Synopsis: Returns a 3D VNF given a 2D region.
// SynTags: VNF
// Topics: VNF Generators, Lists
// See Also: vnf_vertex_array(), vnf_tri_array(), vnf_join(), vnf_from_polygons()
// 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 convert 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 (
region = [for (path = region) deduplicate(path, closed=true)],
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()
// Synopsis: Returns true given a VNF-like structure.
// Topics: VNF Manipulation
// See Also: is_vnf_list(), vnf_vertices(), vnf_faces()
// 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()
// Synopsis: Returns true given a list of VNF-like structures.
// Topics: VNF Manipulation
// See Also: is_vnf(), vnf_vertices(), vnf_faces()
//
// 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()
// Synopsis: Returns the list of vertex points from a VNF.
// Topics: VNF Manipulation
// See Also: is_vnf(), is_vnf_list(), vnf_faces()
// Description: Given a VNF structure, returns the list of vertex points.
function vnf_vertices(vnf) = vnf[0];
// Function: vnf_faces()
// Synopsis: Returns the list of faces from a VNF.
// Topics: VNF Manipulation
// See Also: is_vnf(), is_vnf_list(), vnf_vertices()
// 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()
// Synopsis: Reverses the faces of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_quantize(), vnf_merge_points(), vnf_drop_unused_points(), vnf_triangulate(), vnf_slice()
// 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()
// Synopsis: Quantizes the vertex coordinates of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_merge_points(), vnf_drop_unused_points(), vnf_triangulate(), vnf_slice()
// 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()
// Synopsis: Consolidates duplicate vertices of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_drop_unused_points(), vnf_triangulate(), vnf_slice()
// 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()
// Synopsis: Removes unreferenced vertices from a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_merge_points(), vnf_triangulate(), vnf_slice()
// 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()
// Synopsis: Triangulates the faces of a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_merge_points(), vnf_drop_unused_points(), vnf_slice()
// 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_sort_vertices(vnf, idx=[2,1,0]) =
let(
verts = vnf[0],
faces = vnf[1],
vidx = sortidx(verts, idx=idx),
rvidx = sortidx(vidx),
sorted_vnf = [
[ for (i = vidx) verts[i] ],
[ for (face = faces) [ for (i = face) rvidx[i] ] ],
]
) sorted_vnf;
// Function: vnf_slice()
// Synopsis: Slice the faces of a VNF along an axis.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_reverse_faces(), vnf_quantize(), vnf_merge_points(), vnf_drop_unused_points(), vnf_triangulate()
// 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(
cuts = [for (cut=cuts) _shift_cut_plane(vnf,dir,cut)],
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 _shift_cut_plane(vnf,dir,cut,off=0.001) =
let(
I = ident(3),
dir_ind = ord(dir)-ord("X"),
verts = vnf[0],
on_cut = [for (x = verts * I[dir_ind]) if(approx(x,cut,eps=1e-4)) 1] != []
) !on_cut? cut :
_shift_cut_plane(vnf,dir,cut+off);
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) list_unwrap(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()
// Synopsis: Returns a polyhedron from a VNF or list of VNFs.
// SynTags: Geom
// Topics: VNF Manipulation
// See Also: vnf_wireframe()
// Usage:
// vnf_polyhedron(vnf) [ATTACHMENTS];
// vnf_polyhedron([VNF, VNF, VNF, ...]) [ATTACHMENTS];
// 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"
// Anchor Types:
// "hull" = Anchors to the virtual convex hull of the shape.
// "intersect" = Anchors to the surface of the shape.
// Extra Anchors:
// "origin" = Anchor at the origin, oriented UP.
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()
// Synopsis: Creates a wireframe model from a VNF.
// SynTags: VNF
// Topics: VNF Manipulation
// See Also: vnf_polyhedron()
// 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);