grey/BOSL2
1
0
Fork 0
mirror of https://github.com/BelfrySCAD/BOSL2.git synced 2024-12-29 00:09:41 +00:00
Code Issues Projects Releases Packages Wiki Activity Actions

Correct bugs in polygon_triangulation and _cleave_connected_region(

Browse source
This commit is contained in:
RonaldoCMP 2021-11-01 04:42:02 +00:00
parent 94033a0bfd
commit c5da41ed8c
5 changed files with 307 additions and 80 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

213
geometry.scad
View file

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

32
tests/test_all.scad Normal file
View file

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

8
tests/test_geometry.scad
View file

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

9
vectors.scad
View file

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

125
vnf.scad
View file

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