// Polygon offsetting using Voronoi diagram prodiced by boost::polygon.
#include "VoronoiOffset.hpp"
#include <cmath>
// #define VORONOI_DEBUG_OUT
#ifdef VORONOI_DEBUG_OUT
#include <libslic3r/VoronoiVisualUtils.hpp>
#endif
namespace Slic3r {
using VD = Geometry::VoronoiDiagram;
namespace detail {
// Intersect a circle with a ray, return the two parameters.
// Currently used for unbounded Voronoi edges only.
double first_circle_segment_intersection_parameter(
const Vec2d ¢er, const double r, const Vec2d &pt, const Vec2d &v)
{
const Vec2d d = pt - center;
#ifndef NDEBUG
double d0 = (pt - center).norm();
double d1 = (pt + v - center).norm();
assert(r < std::max(d0, d1) + EPSILON);
#endif /* NDEBUG */
const double a = v.squaredNorm();
const double b = 2. * d.dot(v);
const double c = d.squaredNorm() - r * r;
std::pair<int, std::array<double, 2>> out;
double u = b * b - 4. * a * c;
assert(u > - EPSILON);
double t;
if (u <= 0) {
// Degenerate to a single closest point.
t = - b / (2. * a);
assert(t >= - EPSILON && t <= 1. + EPSILON);
return Slic3r::clamp(0., 1., t);
} else {
u = sqrt(u);
out.first = 2;
double t0 = (- b - u) / (2. * a);
double t1 = (- b + u) / (2. * a);
// One of the intersections shall be found inside the segment.
assert((t0 >= - EPSILON && t0 <= 1. + EPSILON) || (t1 >= - EPSILON && t1 <= 1. + EPSILON));
if (t1 < 0.)
return 0.;
if (t0 > 1.)
return 1.;
return (t0 > 0.) ? t0 : t1;
}
}
struct Intersections
{
int count;
Vec2d pts[2];
};
// Return maximum two points, that are at distance "d" from both points
Intersections point_point_equal_distance_points(const Point &pt1, const Point &pt2, const double d)
{
// Calculate the two intersection points.
// With the help of Python package sympy:
// res = solve([(x - cx)**2 + (y - cy)**2 - d**2, x**2 + y**2 - d**2], [x, y])
// ccode(cse((res[0][0], res[0][1], res[1][0], res[1][1])))
// where cx, cy is the center of pt1 relative to pt2,
// d is distance from the line and the point (0, 0).
// The result is then shifted to pt2.
auto cx = double(pt1.x() - pt2.x());
auto cy = double(pt1.y() - pt2.y());
double cl = cx * cx + cy * cy;
double discr = 4. * d * d - cl;
if (discr < 0.) {
// No intersection point found, the two circles are too far away.
return Intersections { 0, { Vec2d(), Vec2d() } };
}
// Avoid division by zero if a gets too small.
bool xy_swapped = std::abs(cx) < std::abs(cy);
if (xy_swapped)
std::swap(cx, cy);
double u;
int cnt;
if (discr == 0.) {
cnt = 1;
u = 0;
} else {
cnt = 2;
u = 0.5 * cx * sqrt(cl * discr) / cl;
}
double v = 0.5 * cy - u;
double w = 2. * cy;
double e = 0.5 / cx;
double f = 0.5 * cy + u;
Intersections out { cnt, { Vec2d(-e * (v * w - cl), v),
Vec2d(-e * (w * f - cl), f) } };
if (xy_swapped) {
std::swap(out.pts[0].x(), out.pts[0].y());
std::swap(out.pts[1].x(), out.pts[1].y());
}
out.pts[0] += pt2.cast<double>();
out.pts[1] += pt2.cast<double>();
assert(std::abs((out.pts[0] - pt1.cast<double>()).norm() - d) < SCALED_EPSILON);
assert(std::abs((out.pts[1] - pt1.cast<double>()).norm() - d) < SCALED_EPSILON);
assert(std::abs((out.pts[0] - pt2.cast<double>()).norm() - d) < SCALED_EPSILON);
assert(std::abs((out.pts[1] - pt2.cast<double>()).norm() - d) < SCALED_EPSILON);
return out;
}
// Return maximum two points, that are at distance "d" from both the line and point.
Intersections line_point_equal_distance_points(const Line &line, const Point &ipt, const double d)
{
assert(line.a != ipt && line.b != ipt);
// Calculating two points of distance "d" to a ray and a point.
// Point.
Vec2d pt = ipt.cast<double>();
Vec2d lv = (line.b - line.a).cast<double>();
double l2 = lv.squaredNorm();
Vec2d lpv = (line.a - ipt).cast<double>();
double c = cross2(lpv, lv);
if (c < 0) {
lv = - lv;
c = - c;
}
// Line equation (ax + by + c - d * sqrt(l2)).
auto a = - lv.y();
auto b = lv.x();
// Line point shifted by -ipt is on the line.
assert(std::abs(lpv.x() * a + lpv.y() * b + c) < SCALED_EPSILON);
// Line vector (a, b) points towards ipt.
assert(a * lpv.x() + b * lpv.y() < - SCALED_EPSILON);
#ifndef NDEBUG
{
// Foot point of ipt on line.
Vec2d ft = Geometry::foot_pt(line, ipt);
// Center point between ipt and line, its distance to both line and ipt is equal.
Vec2d centerpt = 0.5 * (ft + pt) - pt;
double dcenter = 0.5 * (ft - pt).norm();
// Verify that the center point
assert(std::abs(centerpt.x() * a + centerpt.y() * b + c - dcenter * sqrt(l2)) < SCALED_EPSILON * sqrt(l2));
}
#endif // NDEBUG
// Calculate the two intersection points.
// With the help of Python package sympy:
// res = solve([a * x + b * y + c - d * sqrt(a**2 + b**2), x**2 + y**2 - d**2], [x, y])
// ccode(cse((res[0][0], res[0][1], res[1][0], res[1][1])))
// where (a, b, c, d) is the line equation, not normalized (vector a,b is not normalized),
// d is distance from the line and the point (0, 0).
// The result is then shifted to ipt.
double dscaled = d * sqrt(l2);
double s = c * (2. * dscaled - c);
if (s < 0.)
// Distance of pt from line is bigger than 2 * d.
return Intersections { 0 };
double u;
int cnt;
// Avoid division by zero if a gets too small.
bool xy_swapped = std::abs(a) < std::abs(b);
if (xy_swapped)
std::swap(a, b);
if (s == 0.) {
// Distance of pt from line is 2 * d.
cnt = 1;
u = 0.;
} else {
// Distance of pt from line is smaller than 2 * d.
cnt = 2;
u = a * sqrt(s) / l2;
}
double e = dscaled - c;
double f = b * e / l2;
double g = f - u;
double h = f + u;
Intersections out { cnt, { Vec2d((- b * g + e) / a, g),
Vec2d((- b * h + e) / a, h) } };
if (xy_swapped) {
std::swap(out.pts[0].x(), out.pts[0].y());
std::swap(out.pts[1].x(), out.pts[1].y());
}
out.pts[0] += pt;
out.pts[1] += pt;
assert(std::abs(Geometry::ray_point_distance<Vec2d>(line.a.cast<double>(), (line.b - line.a).cast<double>(), out.pts[0]) - d) < SCALED_EPSILON);
assert(std::abs(Geometry::ray_point_distance<Vec2d>(line.a.cast<double>(), (line.b - line.a).cast<double>(), out.pts[1]) - d) < SCALED_EPSILON);
assert(std::abs((out.pts[0] - ipt.cast<double>()).norm() - d) < SCALED_EPSILON);
assert(std::abs((out.pts[1] - ipt.cast<double>()).norm() - d) < SCALED_EPSILON);
return out;
}
} // namespace detail
Polygons voronoi_offset(
const Geometry::VoronoiDiagram &vd,
const Lines &lines,
double offset_distance,
double discretization_error)
{
#ifndef NDEBUG
// Verify that twin halfedges are stored next to the other in vd.
for (size_t i = 0; i < vd.num_edges(); i += 2) {
const VD::edge_type &e = vd.edges()[i];
const VD::edge_type &e2 = vd.edges()[i + 1];
assert(e.twin() == &e2);
assert(e2.twin() == &e);
assert(e.is_secondary() == e2.is_secondary());
if (e.is_secondary()) {
assert(e.cell()->contains_point() != e2.cell()->contains_point());
const VD::edge_type &ex = (e.cell()->contains_point() ? e : e2);
// Verify that the Point defining the cell left of ex is an end point of a segment
// defining the cell right of ex.
const Line &line0 = lines[ex.cell()->source_index()];
const Line &line1 = lines[ex.twin()->cell()->source_index()];
const Point &pt = (ex.cell()->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b;
assert(pt == line1.a || pt == line1.b);
}
}
#endif // NDEBUG
enum class EdgeState : unsigned char {
// Initial state, don't know.
Unknown,
// This edge will certainly not be intersected by the offset curve.
Inactive,
// This edge will certainly be intersected by the offset curve.
Active,
// This edge will possibly be intersected by the offset curve.
Possible
};
enum class CellState : unsigned char {
// Initial state, don't know.
Unknown,
// Inactive cell is inside for outside curves and outside for inside curves.
Inactive,
// Active cell is outside for outside curves and inside for inside curves.
Active,
// Boundary cell is intersected by the input segment, part of it is active.
Boundary
};
// Mark edges with outward vertex pointing outside the polygons, thus there is a chance
// that such an edge will have an intersection with our desired offset curve.
bool outside = offset_distance > 0.;
std::vector<EdgeState> edge_state(vd.num_edges(), EdgeState::Unknown);
std::vector<CellState> cell_state(vd.num_cells(), CellState::Unknown);
const VD::edge_type *front_edge = &vd.edges().front();
const VD::cell_type *front_cell = &vd.cells().front();
auto set_edge_state_initial = [&edge_state, front_edge](const VD::edge_type *edge, EdgeState new_edge_type) {
EdgeState &edge_type = edge_state[edge - front_edge];
assert(edge_type == EdgeState::Unknown || edge_type == new_edge_type);
assert(new_edge_type == EdgeState::Possible || new_edge_type == EdgeState::Inactive);
edge_type = new_edge_type;
};
auto set_edge_state_final = [&edge_state, front_edge](const size_t edge_id, EdgeState new_edge_type) {
EdgeState &edge_type = edge_state[edge_id];
assert(edge_type == EdgeState::Possible || edge_type == new_edge_type);
assert(new_edge_type == EdgeState::Active || new_edge_type == EdgeState::Inactive);
edge_type = new_edge_type;
};
auto set_cell_state = [&cell_state, front_cell](const VD::cell_type *cell, CellState new_cell_type) -> bool {
CellState &cell_type = cell_state[cell - front_cell];
assert(cell_type == CellState::Active || cell_type == CellState::Inactive || cell_type == CellState::Boundary || cell_type == CellState::Unknown);
assert(new_cell_type == CellState::Active || new_cell_type == CellState::Inactive || new_cell_type == CellState::Boundary);
switch (cell_type) {
case CellState::Unknown:
break;
case CellState::Active:
if (new_cell_type == CellState::Inactive)
new_cell_type = CellState::Boundary;
break;
case CellState::Inactive:
if (new_cell_type == CellState::Active)
new_cell_type = CellState::Boundary;
break;
case CellState::Boundary:
return false;
}
if (cell_type != new_cell_type) {
cell_type = new_cell_type;
return true;
}
return false;
};
for (const VD::edge_type &edge : vd.edges())
if (edge.vertex1() == nullptr) {
// Infinite Voronoi edge separating two Point sites or a Point site and a Segment site.
// Infinite edge is always outside and it has at least one valid vertex.
assert(edge.vertex0() != nullptr);
set_edge_state_initial(&edge, outside ? EdgeState::Possible : EdgeState::Inactive);
// Opposite edge of an infinite edge is certainly not active.
set_edge_state_initial(edge.twin(), EdgeState::Inactive);
if (edge.is_secondary()) {
// edge.vertex0() must lie on source contour.
const VD::cell_type *cell = edge.cell();
const VD::cell_type *cell2 = edge.twin()->cell();
if (cell->contains_segment())
std::swap(cell, cell2);
// State of a cell containing a boundary point is known.
assert(cell->contains_point());
set_cell_state(cell, outside ? CellState::Active : CellState::Inactive);
// State of a cell containing a boundary edge is Boundary.
assert(cell2->contains_segment());
set_cell_state(cell2, CellState::Boundary);
}
} else if (edge.vertex0() != nullptr) {
// Finite edge.
const VD::cell_type *cell = edge.cell();
const Line *line = cell->contains_segment() ? &lines[cell->source_index()] : nullptr;
if (line == nullptr) {
cell = edge.twin()->cell();
line = cell->contains_segment() ? &lines[cell->source_index()] : nullptr;
}
if (line) {
const VD::vertex_type *v1 = edge.vertex1();
const VD::cell_type *cell2 = (cell == edge.cell()) ? edge.twin()->cell() : edge.cell();
assert(v1);
const Point *pt_on_contour = nullptr;
if (cell == edge.cell() && edge.twin()->cell()->contains_segment()) {
// Constrained bisector of two segments.
// If the two segments share a point, then one end of the current Voronoi edge shares this point as well.
// Find pt_on_contour if it exists.
const Line &line2 = lines[cell2->source_index()];
if (line->a == line2.b)
pt_on_contour = &line->a;
else if (line->b == line2.a)
pt_on_contour = &line->b;
} else if (edge.is_secondary()) {
assert(edge.is_linear());
// One end of the current Voronoi edge shares a point of a contour.
assert(edge.cell()->contains_point() != edge.twin()->cell()->contains_point());
const Line &line2 = lines[cell2->source_index()];
pt_on_contour = &((cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line2.a : line2.b);
}
if (pt_on_contour) {
// One end of the current Voronoi edge shares a point of a contour.
// Find out which one it is.
const VD::vertex_type *v0 = edge.vertex0();
Vec2d vec0(v0->x() - pt_on_contour->x(), v0->y() - pt_on_contour->y());
Vec2d vec1(v1->x() - pt_on_contour->x(), v1->y() - pt_on_contour->y());
double d0 = vec0.squaredNorm();
double d1 = vec1.squaredNorm();
assert(std::min(d0, d1) < SCALED_EPSILON * SCALED_EPSILON);
if (d0 < d1) {
// v0 is equal to pt.
} else {
// Skip secondary edge pointing to a contour point.
set_edge_state_initial(&edge, EdgeState::Inactive);
continue;
}
}
Vec2d l0(line->a.cast<double>());
Vec2d lv((line->b - line->a).cast<double>());
double side = cross2(lv, Vec2d(v1->x(), v1->y()) - l0);
bool edge_active = outside ? (side < 0.) : (side > 0.);
set_edge_state_initial(&edge, edge_active ? EdgeState::Possible : EdgeState::Inactive);
assert(cell->contains_segment());
set_cell_state(cell,
pt_on_contour ? CellState::Boundary :
edge_active ? CellState::Active : CellState::Inactive);
set_cell_state(cell2,
(pt_on_contour && cell2->contains_segment()) ?
CellState::Boundary :
edge_active ? CellState::Active : CellState::Inactive);
}
}
{
// Perform one round of expansion marking Voronoi edges and cells next to boundary cells as active / inactive.
std::vector<const VD::cell_type*> cell_queue;
for (const VD::edge_type &edge : vd.edges())
if (edge_state[&edge - front_edge] == EdgeState::Unknown) {
assert(edge.cell()->contains_point() && edge.twin()->cell()->contains_point());
// Edge separating two point sources, not yet classified as inside / outside.
CellState cs = cell_state[edge.cell() - front_cell];
CellState cs2 = cell_state[edge.twin()->cell() - front_cell];
if (cs != CellState::Unknown || cs2 != CellState::Unknown) {
if (cs == CellState::Unknown) {
cs = cs2;
if (set_cell_state(edge.cell(), cs))
cell_queue.emplace_back(edge.cell());
} else if (set_cell_state(edge.twin()->cell(), cs))
cell_queue.emplace_back(edge.twin()->cell());
EdgeState es = (cs == CellState::Active) ? EdgeState::Possible : EdgeState::Inactive;
set_edge_state_initial(&edge, es);
set_edge_state_initial(edge.twin(), es);
} else {
const VD::edge_type *e = edge.twin()->rot_prev();
do {
EdgeState es = edge_state[e->twin() - front_edge];
if (es != EdgeState::Unknown) {
assert(es == EdgeState::Possible || es == EdgeState::Inactive);
set_edge_state_initial(&edge, es);
CellState cs = (es == EdgeState::Possible) ? CellState::Active : CellState::Inactive;
if (set_cell_state(edge.cell(), cs))
cell_queue.emplace_back(edge.cell());
if (set_cell_state(edge.twin()->cell(), cs))
cell_queue.emplace_back(edge.twin()->cell());
break;
}
e = e->rot_prev();
} while (e != edge.twin());
}
}
// Do a final seed fill over Voronoi cells and unmarked Voronoi edges.
while (! cell_queue.empty()) {
const VD::cell_type *cell = cell_queue.back();
const CellState cs = cell_state[cell - front_cell];
cell_queue.pop_back();
const VD::edge_type *first_edge = cell->incident_edge();
const VD::edge_type *edge = cell->incident_edge();
EdgeState es = (cs == CellState::Active) ? EdgeState::Possible : EdgeState::Inactive;
do {
if (set_cell_state(edge->twin()->cell(), cs)) {
set_edge_state_initial(edge, es);
set_edge_state_initial(edge->twin(), es);
cell_queue.emplace_back(edge->twin()->cell());
}
edge = edge->next();
} while (edge != first_edge);
}
}
if (! outside)
offset_distance = - offset_distance;
#ifdef VORONOI_DEBUG_OUT
BoundingBox bbox;
{
bbox.merge(get_extents(lines));
bbox.min -= (0.01 * bbox.size().cast<double>()).cast<coord_t>();
bbox.max += (0.01 * bbox.size().cast<double>()).cast<coord_t>();
}
static int irun = 0;
++ irun;
{
Lines helper_lines;
for (const VD::edge_type &edge : vd.edges())
if (edge_state[&edge - front_edge] == EdgeState::Possible) {
const VD::vertex_type *v0 = edge.vertex0();
const VD::vertex_type *v1 = edge.vertex1();
assert(v0 != nullptr);
Vec2d pt1(v0->x(), v0->y());
Vec2d pt2;
if (v1 == nullptr) {
// Unconstrained edge. Calculate a trimmed position.
assert(edge.is_linear());
const VD::cell_type *cell = edge.cell();
const VD::cell_type *cell2 = edge.twin()->cell();
const Line &line0 = lines[cell->source_index()];
const Line &line1 = lines[cell2->source_index()];
if (cell->contains_point() && cell2->contains_point()) {
const Point &pt0 = (cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b;
const Point &pt1 = (cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b;
// Direction vector of this unconstrained Voronoi edge.
Vec2d dir(double(pt0.y() - pt1.y()), double(pt1.x() - pt0.x()));
pt2 = Vec2d(v0->x(), v0->y()) + dir.normalized() * scale_(10.);
} else {
// Infinite edges could not be created by two segment sites.
assert(cell->contains_point() != cell2->contains_point());
// Linear edge goes through the endpoint of a segment.
assert(edge.is_secondary());
const Point &ipt = cell->contains_segment() ?
((cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b) :
((cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b);
// Infinite edge starts at an input contour, therefore there is always an intersection with an offset curve.
const Line &line = cell->contains_segment() ? line0 : line1;
assert(line.a == ipt || line.b == ipt);
// dir is perpendicular to line.
Vec2d dir(line.a.y() - line.b.y(), line.b.x() - line.a.x());
assert(dir.norm() > 0.);
if (((line.a == ipt) == cell->contains_point()) == (v0 == nullptr))
dir = - dir;
pt2 = ipt.cast<double>() + dir.normalized() * scale_(10.);
}
} else {
pt2 = Vec2d(v1->x(), v1->y());
// Clip the line by the bounding box, so that the coloring of the line will be visible.
Geometry::liang_barsky_line_clipping(pt1, pt2, BoundingBoxf(bbox.min.cast<double>(), bbox.max.cast<double>()));
}
helper_lines.emplace_back(Line(Point(pt1.cast<coord_t>()), Point(((pt1 + pt2) * 0.5).cast<coord_t>())));
}
dump_voronoi_to_svg(debug_out_path("voronoi-offset-candidates1-%d.svg", irun).c_str(), vd, Points(), lines, Polygons(), helper_lines);
}
#endif // VORONOI_DEBUG_OUT
std::vector<Vec2d> edge_offset_point(vd.num_edges(), Vec2d());
const double offset_distance2 = offset_distance * offset_distance;
for (const VD::edge_type &edge : vd.edges()) {
assert(edge_state[&edge - front_edge] != EdgeState::Unknown);
size_t edge_idx = &edge - front_edge;
if (edge_state[edge_idx] == EdgeState::Possible) {
// Edge candidate, intersection points were not calculated yet.
const VD::vertex_type *v0 = edge.vertex0();
const VD::vertex_type *v1 = edge.vertex1();
assert(v0 != nullptr);
const VD::cell_type *cell = edge.cell();
const VD::cell_type *cell2 = edge.twin()->cell();
const Line &line0 = lines[cell->source_index()];
const Line &line1 = lines[cell2->source_index()];
size_t edge_idx2 = edge.twin() - front_edge;
if (v1 == nullptr) {
assert(edge.is_infinite());
assert(edge.is_linear());
assert(edge_state[edge_idx2] == EdgeState::Inactive);
if (cell->contains_point() && cell2->contains_point()) {
assert(! edge.is_secondary());
const Point &pt0 = (cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b;
const Point &pt1 = (cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b;
double dmin2 = (Vec2d(v0->x(), v0->y()) - pt0.cast<double>()).squaredNorm();
assert(dmin2 >= SCALED_EPSILON * SCALED_EPSILON);
if (dmin2 <= offset_distance2) {
// There shall be an intersection of this unconstrained edge with the offset curve.
// Direction vector of this unconstrained Voronoi edge.
Vec2d dir(double(pt0.y() - pt1.y()), double(pt1.x() - pt0.x()));
Vec2d pt(v0->x(), v0->y());
double t = detail::first_circle_segment_intersection_parameter(Vec2d(pt0.x(), pt0.y()), offset_distance, pt, dir);
edge_offset_point[edge_idx] = pt + t * dir;
set_edge_state_final(edge_idx, EdgeState::Active);
} else
set_edge_state_final(edge_idx, EdgeState::Inactive);
} else {
// Infinite edges could not be created by two segment sites.
assert(cell->contains_point() != cell2->contains_point());
// Linear edge goes through the endpoint of a segment.
assert(edge.is_secondary());
const Point &ipt = cell->contains_segment() ?
((cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b) :
((cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b);
#ifndef NDEBUG
if (cell->contains_segment()) {
const Point &pt1 = (cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b;
assert((pt1.x() == line0.a.x() && pt1.y() == line0.a.y()) ||
(pt1.x() == line0.b.x() && pt1.y() == line0.b.y()));
} else {
const Point &pt0 = (cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b;
assert((pt0.x() == line1.a.x() && pt0.y() == line1.a.y()) ||
(pt0.x() == line1.b.x() && pt0.y() == line1.b.y()));
}
assert((Vec2d(v0->x(), v0->y()) - ipt.cast<double>()).norm() < SCALED_EPSILON);
#endif /* NDEBUG */
// Infinite edge starts at an input contour, therefore there is always an intersection with an offset curve.
const Line &line = cell->contains_segment() ? line0 : line1;
assert(line.a == ipt || line.b == ipt);
edge_offset_point[edge_idx] = ipt.cast<double>() + offset_distance * Vec2d(line.b.y() - line.a.y(), line.a.x() - line.b.x()).normalized();
set_edge_state_final(edge_idx, EdgeState::Active);
}
// The other edge of an unconstrained edge starting with null vertex shall never be intersected.
set_edge_state_final(edge_idx2, EdgeState::Inactive);
} else if (edge.is_secondary()) {
assert(edge.is_linear());
assert(cell->contains_point() != cell2->contains_point());
const Line &line0 = lines[edge.cell()->source_index()];
const Line &line1 = lines[edge.twin()->cell()->source_index()];
const Point &pt = cell->contains_point() ?
((cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b) :
((cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b);
#ifndef NDEBUG
const Line &line = cell->contains_segment() ? line0 : line1;
assert(pt == line.a || pt == line.b);
assert((pt.cast<double>() - Vec2d(v0->x(), v0->y())).norm() < SCALED_EPSILON);
#endif // NDEBUG
Vec2d dir(v1->x() - v0->x(), v1->y() - v0->y());
double l2 = dir.squaredNorm();
if (offset_distance2 <= l2) {
edge_offset_point[edge_idx] = pt.cast<double>() + (offset_distance / sqrt(l2)) * dir;
set_edge_state_final(edge_idx, EdgeState::Active);
} else {
set_edge_state_final(edge_idx, EdgeState::Inactive);
}
set_edge_state_final(edge_idx2, EdgeState::Inactive);
} else {
// Finite edge has valid points at both sides.
bool done = false;
if (cell->contains_segment() && cell2->contains_segment()) {
// This edge is a bisector of two line segments. Project v0, v1 onto one of the line segments.
Vec2d pt(line0.a.cast<double>());
Vec2d dir(line0.b.cast<double>() - pt);
Vec2d vec0 = Vec2d(v0->x(), v0->y()) - pt;
Vec2d vec1 = Vec2d(v1->x(), v1->y()) - pt;
double l2 = dir.squaredNorm();
assert(l2 > 0.);
double dmin = (dir * (vec0.dot(dir) / l2) - vec0).squaredNorm();
double dmax = (dir * (vec1.dot(dir) / l2) - vec1).squaredNorm();
bool flip = dmin > dmax;
if (flip)
std::swap(dmin, dmax);
if (offset_distance2 >= dmin && offset_distance2 <= dmax) {
// Intersect. Maximum one intersection will be found.
// This edge is a bisector of two line segments. Distance to the input polygon increases/decreases monotonically.
dmin = sqrt(dmin);
dmax = sqrt(dmax);
assert(offset_distance > dmin - EPSILON && offset_distance < dmax + EPSILON);
double ddif = dmax - dmin;
if (ddif == 0.) {
// line, line2 are exactly parallel. This is a singular case, the offset curve should miss it.
} else {
if (flip) {
std::swap(edge_idx, edge_idx2);
std::swap(v0, v1);
}
double t = clamp(0., 1., (offset_distance - dmin) / ddif);
edge_offset_point[edge_idx] = Vec2d(lerp(v0->x(), v1->x(), t), lerp(v0->y(), v1->y(), t));
set_edge_state_final(edge_idx, EdgeState::Active);
set_edge_state_final(edge_idx2, EdgeState::Inactive);
done = true;
}
}
} else {
assert(cell->contains_point() || cell2->contains_point());
bool point_vs_segment = cell->contains_point() != cell2->contains_point();
const Point &pt0 = cell->contains_point() ?
((cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b) :
((cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b);
// Project p0 to line segment <v0, v1>.
Vec2d p0(v0->x(), v0->y());
Vec2d p1(v1->x(), v1->y());
Vec2d px(pt0.x(), pt0.y());
double d0 = (p0 - px).squaredNorm();
double d1 = (p1 - px).squaredNorm();
double dmin = std::min(d0, d1);
double dmax = std::max(d0, d1);
bool has_intersection = false;
bool possibly_two_points = false;
if (offset_distance2 <= dmax) {
if (offset_distance2 >= dmin) {
has_intersection = true;
} else {
double dmin_new = dmin;
if (point_vs_segment) {
// Project on the source segment.
const Line &line = cell->contains_segment() ? line0 : line1;
const Vec2d pt_line = line.a.cast<double>();
const Vec2d v_line = (line.b - line.a).cast<double>();
double t0 = (p0 - pt_line).dot(v_line);
double t1 = (p1 - pt_line).dot(v_line);
double tx = (px - pt_line).dot(v_line);
if ((tx >= t0 && tx <= t1) || (tx >= t1 && tx <= t0)) {
// Projection of the Point site falls between the projections of the Voronoi edge end points
// onto the Line site.
Vec2d ft = pt_line + (tx / v_line.squaredNorm()) * v_line;
dmin_new = (ft - px).squaredNorm() * 0.25;
}
} else {
// Point-Point Voronoi sites. Project point site onto the current Voronoi edge.
Vec2d v = p1 - p0;
auto l2 = v.squaredNorm();
assert(l2 > 0);
auto t = v.dot(px - p0);
if (t >= 0. && t <= l2) {
// Projection falls onto the Voronoi edge. Calculate foot point and distance.
Vec2d ft = p0 + (t / l2) * v;
dmin_new = (ft - px).squaredNorm();
}
}
assert(dmin_new < dmax + SCALED_EPSILON);
assert(dmin_new < dmin + SCALED_EPSILON);
if (dmin_new < dmin) {
dmin = dmin_new;
has_intersection = possibly_two_points = offset_distance2 >= dmin;
}
}
}
if (has_intersection) {
detail::Intersections intersections;
if (point_vs_segment) {
assert(cell->contains_point() || cell2->contains_point());
intersections = detail::line_point_equal_distance_points(cell->contains_segment() ? line0 : line1, pt0, offset_distance);
} else {
const Point &pt1 = (cell2->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line1.a : line1.b;
intersections = detail::point_point_equal_distance_points(pt0, pt1, offset_distance);
}
// If the span of distances of start / end point / foot point to the point site indicate an intersection,
// we should find one.
assert(intersections.count > 0);
if (intersections.count == 2) {
// Now decide which points fall on this Voronoi edge.
// Tangential points (single intersection) are ignored.
if (possibly_two_points) {
Vec2d v = p1 - p0;
double l2 = v.squaredNorm();
double t0 = v.dot(intersections.pts[0] - p0);
double t1 = v.dot(intersections.pts[1] - p0);
if (t0 > t1) {
std::swap(t0, t1);
std::swap(intersections.pts[0], intersections.pts[1]);
}
// Remove points outside of the line range.
if (t0 < 0. || t0 > l2) {
if (t1 < 0. || t1 > l2)
intersections.count = 0;
else {
-- intersections.count;
t0 = t1;
intersections.pts[0] = intersections.pts[1];
}
} else if (t1 < 0. || t1 > l2)
-- intersections.count;
} else {
// Take the point furthest from the end points of the Voronoi edge or a Voronoi parabolic arc.
double d0 = std::max((intersections.pts[0] - p0).squaredNorm(), (intersections.pts[0] - p1).squaredNorm());
double d1 = std::max((intersections.pts[1] - p0).squaredNorm(), (intersections.pts[1] - p1).squaredNorm());
if (d0 > d1)
intersections.pts[0] = intersections.pts[1];
-- intersections.count;
}
assert(intersections.count > 0);
if (intersections.count == 2) {
set_edge_state_final(edge_idx, EdgeState::Active);
set_edge_state_final(edge_idx2, EdgeState::Active);
edge_offset_point[edge_idx] = intersections.pts[1];
edge_offset_point[edge_idx2] = intersections.pts[0];
done = true;
} else if (intersections.count == 1) {
if (d1 < d0)
std::swap(edge_idx, edge_idx2);
set_edge_state_final(edge_idx, EdgeState::Active);
set_edge_state_final(edge_idx2, EdgeState::Inactive);
edge_offset_point[edge_idx] = intersections.pts[0];
done = true;
}
}
}
}
if (! done) {
set_edge_state_final(edge_idx, EdgeState::Inactive);
set_edge_state_final(edge_idx2, EdgeState::Inactive);
}
}
}
}
#ifndef NDEBUG
for (const VD::edge_type &edge : vd.edges()) {
assert(edge_state[&edge - front_edge] == EdgeState::Inactive || edge_state[&edge - front_edge] == EdgeState::Active);
// None of a new edge candidate may start with null vertex.
assert(edge_state[&edge - front_edge] == EdgeState::Inactive || edge.vertex0() != nullptr);
assert(edge_state[edge.twin() - front_edge] == EdgeState::Inactive || edge.twin()->vertex0() != nullptr);
}
#endif // NDEBUG
#ifdef VORONOI_DEBUG_OUT
{
Lines helper_lines;
for (const VD::edge_type &edge : vd.edges())
if (edge_state[&edge - front_edge] == EdgeState::Active)
helper_lines.emplace_back(Line(Point(edge.vertex0()->x(), edge.vertex0()->y()), Point(edge_offset_point[&edge - front_edge].cast<coord_t>())));
dump_voronoi_to_svg(debug_out_path("voronoi-offset-candidates2-%d.svg", irun).c_str(), vd, Points(), lines, Polygons(), helper_lines);
}
#endif // VORONOI_DEBUG_OUT
auto next_offset_edge = [&edge_state, front_edge](const VD::edge_type *start_edge) -> const VD::edge_type* {
for (const VD::edge_type *edge = start_edge->next(); edge != start_edge; edge = edge->next())
if (edge_state[edge->twin() - front_edge] == EdgeState::Active)
return edge->twin();
// assert(false);
return nullptr;
};
#ifndef NDEBUG
auto dist_to_site = [&lines](const VD::cell_type &cell, const Vec2d &point) {
const Line &line = lines[cell.source_index()];
return cell.contains_point() ?
(((cell.source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line.a : line.b).cast<double>() - point).norm() :
(Geometry::foot_pt<Vec2d>(line.a.cast<double>(), (line.b - line.a).cast<double>(), point) - point).norm();
};
#endif /* NDEBUG */
// Track the offset curves.
Polygons out;
double angle_step = 2. * acos((offset_distance - discretization_error) / offset_distance);
double cos_threshold = cos(angle_step);
for (size_t seed_edge_idx = 0; seed_edge_idx < vd.num_edges(); ++ seed_edge_idx)
if (edge_state[seed_edge_idx] == EdgeState::Active) {
const VD::edge_type *start_edge = &vd.edges()[seed_edge_idx];
const VD::edge_type *edge = start_edge;
Polygon poly;
do {
// find the next edge
const VD::edge_type *next_edge = next_offset_edge(edge);
#ifdef VORONOI_DEBUG_OUT
if (next_edge == nullptr) {
Lines helper_lines;
dump_voronoi_to_svg(debug_out_path("voronoi-offset-open-loop-%d.svg", irun).c_str(), vd, Points(), lines, Polygons(), to_lines(poly));
}
#endif // VORONOI_DEBUG_OUT
assert(next_edge);
//std::cout << "offset-output: "; print_edge(edge); std::cout << " to "; print_edge(next_edge); std::cout << "\n";
// Interpolate a circular segment or insert a linear segment between edge and next_edge.
const VD::cell_type *cell = edge->cell();
edge_state[next_edge - front_edge] = EdgeState::Inactive;
Vec2d p1 = edge_offset_point[edge - front_edge];
Vec2d p2 = edge_offset_point[next_edge - front_edge];
#ifndef NDEBUG
{
double err = dist_to_site(*cell, p1) - offset_distance;
double err2 = dist_to_site(*cell, p2) - offset_distance;
#ifdef VORONOI_DEBUG_OUT
if (std::max(err, err2) >= SCALED_EPSILON) {
Lines helper_lines;
dump_voronoi_to_svg(debug_out_path("voronoi-offset-incorrect_pt-%d.svg", irun).c_str(), vd, Points(), lines, Polygons(), to_lines(poly));
}
#endif // VORONOI_DEBUG_OUT
assert(std::abs(err) < SCALED_EPSILON);
assert(std::abs(err2) < SCALED_EPSILON);
}
#endif /* NDEBUG */
if (cell->contains_point()) {
// Discretize an arc from p1 to p2 with radius = offset_distance and discretization_error.
// The extracted contour is CCW oriented, extracted holes are CW oriented.
// The extracted arc will have the same orientation. As the Voronoi regions are convex, the angle covered by the arc will be convex as well.
const Line &line0 = lines[cell->source_index()];
const Vec2d ¢er = ((cell->source_category() == boost::polygon::SOURCE_CATEGORY_SEGMENT_START_POINT) ? line0.a : line0.b).cast<double>();
const Vec2d v1 = p1 - center;
const Vec2d v2 = p2 - center;
bool ccw = cross2(v1, v2) > 0;
double cos_a = v1.dot(v2);
double norm = v1.norm() * v2.norm();
assert(norm > 0.);
if (cos_a < cos_threshold * norm) {
// Angle is bigger than the threshold, therefore the arc will be discretized.
cos_a /= norm;
assert(cos_a > -1. - EPSILON && cos_a < 1. + EPSILON);
double angle = acos(std::max(-1., std::min(1., cos_a)));
size_t n_steps = size_t(ceil(angle / angle_step));
double astep = angle / n_steps;
if (! ccw)
astep *= -1.;
double a = astep;
for (size_t i = 1; i < n_steps; ++ i, a += astep) {
double c = cos(a);
double s = sin(a);
Vec2d p = center + Vec2d(c * v1.x() - s * v1.y(), s * v1.x() + c * v1.y());
poly.points.emplace_back(Point(coord_t(p.x()), coord_t(p.y())));
}
}
}
{
Point pt_last(coord_t(p2.x()), coord_t(p2.y()));
if (poly.empty() || poly.points.back() != pt_last)
poly.points.emplace_back(pt_last);
}
edge = next_edge;
} while (edge != start_edge);
out.emplace_back(std::move(poly));
}
return out;
}
} // namespace Slic3r