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Merge pull request #693 from RonaldoCMP/master

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Minor changes in triangulate code and docs,
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Revar Desmera 2021-10-13 22:28:11 -07:00 committed by GitHub
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3 changed files with 101 additions and 104 deletions
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176
geometry.scad
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@ -1603,17 +1603,25 @@ function point_in_polygon(point, poly, nonzero=false, eps=EPSILON) =
// Description: // Description:
// Given a simple polygon in 2D or 3D, triangulates it and returns a list // Given a simple polygon in 2D or 3D, triangulates it and returns a list
// of triples indexing into the polygon vertices. When the optional argument `ind` is // of triples indexing into the polygon vertices. When the optional argument `ind` is
// given, the it is used as an index list into `poly` to define the polygon. In that case, // given, it is used as an index list into `poly` to define the polygon. In that case,
// `poly` may have a length greater than `ind`. Otherwise, all points in `poly` // `poly` may have a length greater than `ind`. When `ind` is undefined, all points in `poly`
// are considered as vertices of the polygon. // are considered as vertices of the polygon.
// . // .
// The function may issue an error if it finds that the polygon is not simple
// (self-intersecting) or its vertices are collinear. It can work for 3d non-planar polygons
// if they are close enough to planar but may otherwise issue an error for this case.
// .
// For 2d polygons, the output triangles will have the same winding (CW or CCW) of // For 2d polygons, the output triangles will have the same winding (CW or CCW) of
// the input polygon. For 3d polygons, the triangle windings will induce a normal // the input polygon. For 3d polygons, the triangle windings will induce a normal
// vector with the same direction of the polygon normal. // vector with the same direction of the polygon normal.
// .
// 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
// (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, returns an empty list.
// .
// 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
// can work for 3d non-planar polygons if they are close enough to planar but may otherwise
// issue an error for this case.
// Arguments: // Arguments:
// poly = Array of vertices for the polygon. // poly = Array of vertices for the polygon.
// ind = A list indexing the vertices of the polygon in `poly`. // ind = A list indexing the vertices of the polygon in `poly`.
@ -1621,7 +1629,28 @@ function point_in_polygon(point, poly, nonzero=false, eps=EPSILON) =
// Example(2D,NoAxes): // Example(2D,NoAxes):
// poly = star(id=10, od=15,n=11); // poly = star(id=10, od=15,n=11);
// tris = polygon_triangulate(poly); // tris = polygon_triangulate(poly);
// for(tri=tris) stroke(select(poly,tri), width=.2, closed=true); // color("lightblue") for(tri=tris) polygon(select(poly,tri));
// color("magenta") up(2) stroke(poly,.25,closed=true);
// color("black") up(3) vnf_debug([poly,[]],faces=false,size=1);
// Example(2D,NoAxes): a polygon with a hole and one "contact" edge
// poly = [ [-10,0], [10,0], [0,10], [-10,0], [-4,4], [4,4], [0,2], [-4,4] ];
// tris = polygon_triangulate(poly);
// color("lightblue") for(tri=tris) polygon(select(poly,tri));
// color("magenta") up(2) stroke(poly,.25,closed=true);
// color("black") up(3) vnf_debug([poly,[]],faces=false,size=1);
// Example(2D,NoAxes): a polygon with "touching" vertices and no holes
// poly = [ [0,0], [5,5], [-5,5], [0,0], [-5,-5], [5,-5] ];
// tris = polygon_triangulate(poly);
// color("lightblue") for(tri=tris) polygon(select(poly,tri));
// color("magenta") up(2) stroke(poly,.25,closed=true);
// color("black") up(3) vnf_debug([poly,[]],faces=false,size=1);
// 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
// color("lightblue") for(tri=tris) polygon(select(poly,tri));
// color("magenta") up(2) stroke(poly,.25,closed=true);
// color("black") up(3) vnf_debug([poly,[]],faces=false,size=1);
// Example(3D): // Example(3D):
// include <BOSL2/polyhedra.scad> // include <BOSL2/polyhedra.scad>
// vnf = regular_polyhedron_info(name="dodecahedron",side=5,info="vnf"); // vnf = regular_polyhedron_info(name="dodecahedron",side=5,info="vnf");
@ -1630,82 +1659,43 @@ function point_in_polygon(point, poly, nonzero=false, eps=EPSILON) =
// color("blue") // color("blue")
// vnf_wireframe(vnf_tri, width=.15); // vnf_wireframe(vnf_tri, width=.15);
function polygon_triangulate(poly, ind, eps=EPSILON) = function polygon_triangulate(poly, ind, eps=EPSILON) =
assert(is_path(poly), "Polygon `poly` should be a list of 2d or 3d points") assert(is_path(poly) && len(poly)>=3, "Polygon `poly` should be a list of at least three 2d or 3d points")
assert(is_undef(ind) assert(is_undef(ind)
|| (is_vector(ind) && min(ind)>=0 && max(ind)<len(poly) ), || (is_vector(ind) && min(ind)>=0 && max(ind)<len(poly) ),
"Improper or out of bounds list of indices") "Improper or out of bounds list of indices")
let( ind = deduplicate_indexed(poly,is_undef(ind) ? count(len(poly)) : ind) ) let( ind = is_undef(ind) ? count(len(poly)) : ind )
len(ind) == 3 ? [ind] : len(ind) == 3
len(ind) < 3 ? [] : ? _is_degenerate([poly[ind[0]], poly[ind[1]], poly[ind[2]]], eps) ? [] :
len(poly[ind[0]]) == 3 // no zero area
? // represents the polygon projection on its plane as a 2d polygon assert( norm(scalar_vec3(cross(poly[ind[1]]-poly[ind[0]], poly[ind[2]]-poly[ind[0]]))) > 2*eps,
let( "The polygon vertices are collinear.")
pts = select(poly,ind), [ind]
nrm = polygon_normal(pts) : len(poly[ind[0]]) == 3
) ? // represents the polygon projection on its plane as a 2d polygon
// here, instead of an error, it might return [] or undef let(
assert( nrm!=undef, ind = deduplicate_indexed(poly, ind, eps)
"The polygon has self-intersections or its vertices are collinear or non coplanar.") )
let( len(ind)<3 ? [] :
imax = max_index([for(p=pts) norm(p-pts[0]) ]), let(
v1 = unit( pts[imax] - pts[0] ), pts = select(poly,ind),
v2 = cross(v1,nrm), nrm = polygon_normal(pts)
prpts = pts*transpose([v1,v2]) )
) // here, instead of an error, it might return [] or undef
[for(tri=_triangulate(prpts, count(len(ind)), eps)) select(ind,tri) ] assert( nrm!=undef,
: let( cw = is_polygon_clockwise(select(poly, ind)) ) "The polygon has self-intersections or its vertices are collinear or non coplanar.")
cw let(
? [for(tri=_triangulate( poly, reverse(ind), eps )) reverse(tri) ] imax = max_index([for(p=pts) norm(p-pts[0]) ]),
: _triangulate( poly, ind, eps ); v1 = unit( pts[imax] - pts[0] ),
v2 = cross(v1,nrm),
prpts = pts*transpose([v1,v2])
)
[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 );
// requires ccw 2d polygons
// returns ccw triangles
function _old_triangulate(poly, ind, eps=EPSILON, tris=[]) =
len(ind)==3 ? concat(tris,[ind]) :
let( ear = _get_ear(poly,ind,eps) )
assert( ear!=undef,
"The polygon has self-intersections or its vertices are collinear or non coplanar.")
let(
ear_tri = select(ind,ear,ear+2),
indr = select(ind,ear+2, ear) // indices of the remaining points
)
_triangulate(poly, indr, eps, concat(tris,[ear_tri]));
// search a valid ear from the remaining polygon
function _old_get_ear(poly, ind, eps, _i=0) =
_i>=len(ind) ? undef : // poly has no ears
let( // the _i-th ear candidate
p0 = poly[ind[_i]],
p1 = poly[ind[(_i+1)%len(ind)]],
p2 = poly[ind[(_i+2)%len(ind)]]
)
// if it is not a convex vertex, try 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
to_tst = select(ind,_i+3, _i-1),
pt2tst = select(poly,to_tst), // points other than p0, p1 and p2
q = [(p0-p2).y, (p2-p0).x], // orthogonal to ray [p0,p2] pointing right
q0 = q*p0,
atleft = [for(p=pt2tst) if(p*q<=q0) p ]
)
atleft==[] ? _i : // no point inside -> an ear
let(
q = [(p2-p1).y, (p1-p2).x], // orthogonal to ray [p1,p2] pointing right
q0 = q*p2,
atleft = [for(p=atleft) if(p*q<=q0) p ]
)
atleft==[] ? _i : // no point inside -> an ear
let(
q = [(p1-p0).y, (p0-p1).x], // orthogonal to ray [p1,p0] pointing right
q0 = q*p1,
atleft = [for(p=atleft) if(p*q<=q0) p ]
)
atleft==[] ? _i : // no point inside -> an ear
// check the next ear candidate
_get_ear(poly, ind, eps, _i=_i+1);
function _triangulate(poly, ind, eps=EPSILON, tris=[]) = function _triangulate(poly, ind, eps=EPSILON, tris=[]) =
len(ind)==3 len(ind)==3
? _is_degenerate(select(poly,ind),eps) ? _is_degenerate(select(poly,ind),eps)
@ -1714,18 +1704,18 @@ function _triangulate(poly, ind, eps=EPSILON, tris=[]) =
: let( ear = _get_ear(poly,ind,eps) ) : let( ear = _get_ear(poly,ind,eps) )
assert( ear!=undef, assert( ear!=undef,
"The polygon has self-intersections or its vertices are collinear or non coplanar.") "The polygon has self-intersections or its vertices are collinear or non coplanar.")
ear<0 // degenerate ear is_list(ear) // degenerate ear
? let( indr = select(ind,-ear+1, -ear-1) ) // discard it ? _triangulate(poly, select(ind,ear[0]+2, ear[0]), eps, tris) // discard it
_triangulate(poly, indr, eps, tris)
: let( : let(
ear_tri = select(ind,ear,ear+2), ear_tri = select(ind,ear,ear+2),
indr = select(ind,ear+2, ear) // indices of the remaining points indr = select(ind,ear+2, ear) // remaining point indices
) )
_triangulate(poly, indr, eps, concat(tris,[ear_tri])); _triangulate(poly, indr, eps, concat(tris,[ear_tri]));
// a returned ear will be: // a returned ear will be:
// 1. a CCW triangle without points inside except possibly at its vertices // 1. a CCW (non-degenerate) triangle, made of subsequent vertices, without other
// points inside except possibly at its vertices
// 2. or a degenerate triangle where two vertices are coincident // 2. or a degenerate triangle where two vertices are coincident
// the returned ear is specified by the index of `ind` of its first vertex // the returned ear is specified by the index of `ind` of its first vertex
function _get_ear(poly, ind, eps, _i=0) = function _get_ear(poly, ind, eps, _i=0) =
@ -1735,29 +1725,25 @@ function _get_ear(poly, ind, eps, _i=0) =
p1 = poly[ind[(_i+1)%len(ind)]], p1 = poly[ind[(_i+1)%len(ind)]],
p2 = poly[ind[(_i+2)%len(ind)]] p2 = poly[ind[(_i+2)%len(ind)]]
) )
// if it is a degenerate triangle, return it (codified) // degenerate triangles are returned codified
_is_degenerate([p0,p1,p2],eps) ? -(_i+1) : _is_degenerate([p0,p1,p2],eps) ? [_i] :
// if it is not a convex vertex, try the next one // 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) : _is_cw2(p0,p1,p2,eps) ? _get_ear(poly,ind,eps, _i=_i+1) :
let( // vertex p1 is convex let( // vertex p1 is convex
// check if the triangle contains any other point // check if the triangle contains any other point
// except possibly its own vertices // except possibly its own vertices
to_tst = select(ind,_i+3, _i-1), to_tst = select(ind,_i+3, _i-1),
pt2tst = select(poly,to_tst), // points other than p0, p1 and p2
q = [(p0-p2).y, (p2-p0).x], // orthogonal to ray [p0,p2] pointing right q = [(p0-p2).y, (p2-p0).x], // orthogonal to ray [p0,p2] pointing right
q0 = q*p0,
r = [(p2-p1).y, (p1-p2).x], // orthogonal to ray [p2,p1] pointing right r = [(p2-p1).y, (p1-p2).x], // orthogonal to ray [p2,p1] pointing right
r0 = r*p2,
s = [(p1-p0).y, (p0-p1).x], // orthogonal to ray [p1,p0] pointing right s = [(p1-p0).y, (p0-p1).x], // orthogonal to ray [p1,p0] pointing right
s0 = s*p1, inside = [for(p=select(poly,to_tst)) // for vertices other than p0, p1 and p2
inside = [for(p=pt2tst) if( (p-p0)*q<=0 && (p-p2)*r<=0 && (p-p1)*s<=0 // p is on the triangle
if( p*q<=q0 && p*r<=r0 && p*s<=s0 ) // p is in the triangle && norm(p-p0)>eps // but not on any vertex of it
if( norm(p-p0)>eps // and doesn't coincide with && norm(p-p1)>eps
&& norm(p-p1)>eps // any of its vertices
&& norm(p-p2)>eps ) && norm(p-p2)>eps )
p ] p ]
) )
inside==[] ? _i : // no point inside -> an ear inside==[] ? _i : // found an ear
// check the next ear candidate // check the next ear candidate
_get_ear(poly, ind, eps, _i=_i+1); _get_ear(poly, ind, eps, _i=_i+1);

13
math.scad
View file

@ -689,17 +689,12 @@ function _sum(v,_total,_i=0) = _i>=len(v) ? _total : _sum(v,_total+v[_i], _i+1);
// cumsum([[1,2,3], [3,4,5], [5,6,7]]); // returns [[1,2,3], [4,6,8], [9,12,15]] // cumsum([[1,2,3], [3,4,5], [5,6,7]]); // returns [[1,2,3], [4,6,8], [9,12,15]]
function cumsum(v) = function cumsum(v) =
assert(is_consistent(v), "The input is not consistent." ) assert(is_consistent(v), "The input is not consistent." )
_cumsum(v,_i=0,_acc=[]); len(v)<=1 ? v :
_cumsum(v,_i=1,_acc=[v[0]]);
function _cumsum(v,_i=0,_acc=[]) = function _cumsum(v,_i=0,_acc=[]) =
_i==len(v) ? _acc : _i>=len(v) ? _acc :
_cumsum( _cumsum( v, _i+1, [ each _acc, _acc[len(_acc)-1] + v[_i] ] );
v, _i+1,
concat(
_acc,
[_i==0 ? v[_i] : last(_acc) + v[_i]]
)
);
// Function: sum_of_sines() // Function: sum_of_sines()

16
tests/test_geometry.scad
View file

@ -48,6 +48,7 @@ test_reindex_polygon();
test_align_polygon(); test_align_polygon();
test_polygon_centroid(); test_polygon_centroid();
test_point_in_polygon(); test_point_in_polygon();
test_polygon_triangulate();
test_is_polygon_clockwise(); test_is_polygon_clockwise();
test_clockwise_polygon(); test_clockwise_polygon();
test_ccw_polygon(); test_ccw_polygon();
@ -78,6 +79,21 @@ function info_str(list,i=0,string=chr(10)) =
: info_str(list,i+1,str(string,str(list[i][0],_valstr(list[i][1]),chr(10)))); : info_str(list,i+1,str(string,str(list[i][0],_valstr(list[i][1]),chr(10))));
module test_polygon_triangulate() {
poly0 = [ [0,0,1], [10,0,2], [10,10,0] ];
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));
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]])));
tris2 = (polygon_triangulate(poly2));
assert(approx(tris2,([[0, 1, 2], [3, 4, 5]])));
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]])));
}
module test__normalize_plane(){ module test__normalize_plane(){
plane = rands(-5,5,4,seed=333)+[10,0,0,0]; plane = rands(-5,5,4,seed=333)+[10,0,0,0];
plane2 = _normalize_plane(plane); plane2 = _normalize_plane(plane);