tippecanoe/geometry.cpp

#include <iostream>
#include <fstream>
#include <string>
#include <vector>
#include <map>
#include <algorithm>
#include <cstdio>
#include <unistd.h>
#include <cmath>
#include <limits.h>
#include <sqlite3.h>
#include <mapbox/geometry/point.hpp>
#include <mapbox/geometry/multi_polygon.hpp>
#include <mapbox/geometry/snap_rounding.hpp>
#include "geometry.hpp"
#include "projection.hpp"
#include "serial.hpp"
#include "main.hpp"
#include "options.hpp"
#include "errors.hpp"

drawvec decode_geometry(const char **meta, int z, unsigned tx, unsigned ty, long long *bbox, unsigned initial_x, unsigned initial_y) {
	drawvec out;

	bbox[0] = LLONG_MAX;
	bbox[1] = LLONG_MAX;
	bbox[2] = LLONG_MIN;
	bbox[3] = LLONG_MIN;

	long long wx = initial_x, wy = initial_y;

	while (1) {
		draw d;

		deserialize_byte(meta, &d.op);
		if (d.op == VT_END) {
			break;
		}

		if (d.op == VT_MOVETO || d.op == VT_LINETO) {
			long long dx, dy;

			deserialize_long_long(meta, &dx);
			deserialize_long_long(meta, &dy);

			wx += dx * (1 << geometry_scale);
			wy += dy * (1 << geometry_scale);

			long long wwx = wx;
			long long wwy = wy;

			if (z != 0) {
				wwx -= tx << (32 - z);
				wwy -= ty << (32 - z);
			}

			bbox[0] = std::min(wwx, bbox[0]);
			bbox[1] = std::min(wwy, bbox[1]);
			bbox[2] = std::max(wwx, bbox[2]);
			bbox[3] = std::max(wwy, bbox[3]);

			d.x = wwx;
			d.y = wwy;
		}

		out.push_back(d);
	}

	return out;
}

void check_polygon(drawvec &geom) {
	geom = remove_noop(geom, VT_POLYGON, 0);

	mapbox::geometry::multi_polygon<long long> mp;
	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].op == VT_MOVETO) {
			size_t j;
			for (j = i + 1; j < geom.size(); j++) {
				if (geom[j].op != VT_LINETO) {
					break;
				}
			}

			if (j >= i + 4) {
				mapbox::geometry::linear_ring<long long> lr;

				for (size_t k = i; k < j; k++) {
					lr.push_back({geom[k].x, geom[k].y});
				}

				if (lr.size() >= 3) {
					mapbox::geometry::polygon<long long> p;
					p.push_back(std::move(lr));
					mp.push_back(std::move(p));
				}
			}

			i = j - 1;
		}
	}

	mapbox::geometry::multi_polygon<long long> mp2 = mapbox::geometry::snap_round(mp, true, true);
	if (mp != mp2) {
		fprintf(stderr, "Internal error: self-intersecting polygon\n");
	}

	size_t outer_start = -1;
	size_t outer_len = 0;

	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].op == VT_MOVETO) {
			size_t j;
			for (j = i + 1; j < geom.size(); j++) {
				if (geom[j].op != VT_LINETO) {
					break;
				}
			}

			double area = get_area(geom, i, j);

			if (area > 0) {
				outer_start = i;
				outer_len = j - i;
			} else {
				for (size_t k = i; k < j; k++) {
					if (!pnpoly(geom, outer_start, outer_len, geom[k].x, geom[k].y)) {
						bool on_edge = false;

						for (size_t l = outer_start; l < outer_start + outer_len; l++) {
							if (geom[k].x == geom[l].x || geom[k].y == geom[l].y) {
								on_edge = true;
								break;
							}
						}

						if (!on_edge) {
							fprintf(stderr, "%lld,%lld at %lld not in outer ring (%lld to %lld)\n", (long long) geom[k].x, (long long) geom[k].y, (long long) k, (long long) outer_start, (long long) (outer_start + outer_len));
						}
					}
				}
			}
		}
	}
}

int quick_check(const long long *bbox, int z, long long buffer) {
	long long min = 0;
	long long area = 1LL << (32 - z);

	// bbox entirely within the tile proper
	if (bbox[0] > min && bbox[1] > min && bbox[2] < area && bbox[3] < area) {
		return 1;
	}

	min -= buffer * area / 256;
	area += buffer * area / 256;

	// bbox entirely within the tile, including its buffer
	if (bbox[0] > min && bbox[1] > min && bbox[2] < area && bbox[3] < area) {
		return 3;
	}

	// bbox entirely outside the tile
	if (bbox[0] > area || bbox[1] > area) {
		return 0;
	}
	if (bbox[2] < min || bbox[3] < min) {
		return 0;
	}

	// some overlap of edge
	return 2;
}

bool point_within_tile(long long x, long long y, int z) {
	// No adjustment for buffer, because the point must be
	// strictly within the tile to appear exactly once

	long long area = 1LL << (32 - z);

	return x >= 0 && y >= 0 && x < area && y < area;
}

// If any line segment crosses a tile boundary, add a node there
// that cannot be simplified away, to prevent the edge of any
// feature from jumping abruptly at the tile boundary.
drawvec impose_tile_boundaries(const drawvec &geom, long long extent) {
	drawvec out;

	for (size_t i = 0; i < geom.size(); i++) {
		if (i > 0 && geom[i].op == VT_LINETO && (geom[i - 1].op == VT_MOVETO || geom[i - 1].op == VT_LINETO)) {
			long long x1 = geom[i - 1].x;
			long long y1 = geom[i - 1].y;

			long long x2 = geom[i - 0].x;
			long long y2 = geom[i - 0].y;

			int c = clip(&x1, &y1, &x2, &y2, 0, 0, extent, extent);

			if (c > 1) {  // clipped
				if (x1 != geom[i - 1].x || y1 != geom[i - 1].y) {
					out.emplace_back(VT_LINETO, x1, y1);
					out[out.size() - 1].necessary = 1;
				}
				if (x2 != geom[i - 0].x || y2 != geom[i - 0].y) {
					out.emplace_back(VT_LINETO, x2, y2);
					out[out.size() - 1].necessary = 1;
				}
			}
		}

		out.push_back(geom[i]);
	}

	return out;
}

drawvec simplify_lines(drawvec &geom, int z, int tx, int ty, int detail, bool mark_tile_bounds, double simplification, size_t retain, drawvec const &shared_nodes, struct node *shared_nodes_map, size_t nodepos, std::string const &shared_nodes_bloom) {
	int res = 1 << (32 - detail - z);
	long long area = 1LL << (32 - z);

	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].op == VT_MOVETO) {
			geom[i].necessary = 1;
		} else if (geom[i].op == VT_LINETO) {
			geom[i].necessary = 0;
			// if this is actually the endpoint, not an intermediate point,
			// it will be marked as necessary below
		} else {
			geom[i].necessary = 1;
		}

		if (prevent[P_SIMPLIFY_SHARED_NODES]) {
			// This is kind of weird, because we have two lists of shared nodes to look through:
			// * the drawvec, which is nodes that were introduced during clipping to the tile edge,
			//   and which are in local tile coordinates
			// * the shared_nodes_map, which was made globally before tiling began, and which
			//   is in global quadkey coordinates.
			// To look through the latter, we need to offset and encode the coordinates
			// of the feature we are simplifying.

			auto pt = std::lower_bound(shared_nodes.begin(), shared_nodes.end(), geom[i]);
			if (pt != shared_nodes.end() && *pt == geom[i]) {
				geom[i].necessary = true;
			}

			if (nodepos > 0) {
				// offset to global
				draw d = geom[i];
				if (z != 0) {
					d.x += tx * (1LL << (32 - z));
					d.y += ty * (1LL << (32 - z));
				}

				struct node n;
				n.index = encode_vertex((unsigned) d.x, (unsigned) d.y);
				size_t bloom_ix = n.index % (shared_nodes_bloom.size() * 8);
				unsigned char bloom_mask = 1 << (bloom_ix & 7);
				bloom_ix >>= 3;

				if (shared_nodes_bloom[bloom_ix] & bloom_mask) {
					if (bsearch(&n, shared_nodes_map, nodepos / sizeof(node), sizeof(node), nodecmp) != NULL) {
						geom[i].necessary = true;
					}
				}
			}
		}
	}

	if (mark_tile_bounds) {
		geom = impose_tile_boundaries(geom, area);
	}

	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].op == VT_MOVETO) {
			size_t j;
			for (j = i + 1; j < geom.size(); j++) {
				if (geom[j].op != VT_LINETO) {
					break;
				}
			}

			geom[i].necessary = 1;
			geom[j - 1].necessary = 1;

			// empirical mapping from douglas-peucker simplifications
			// to visvalingam simplifications that yield similar
			// output sizes
			double sim = simplification * (0.1596 * z + 0.878);
			double scale = (res * sim) * (res * sim);
			scale = exp(1.002 * log(scale) + 0.3043);

			if (j - i > 1) {
				if (additional[A_VISVALINGAM]) {
					visvalingam(geom, i, j, scale, retain);
				} else {
					douglas_peucker(geom, i, j - i, res * simplification, 2, retain, prevent[P_SIMPLIFY_SHARED_NODES]);
				}
			}
			i = j - 1;
		}
	}

	size_t out = 0;
	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].necessary) {
			geom[out++] = geom[i];
		}
	}
	geom.resize(out);
	return geom;
}

drawvec reorder_lines(const drawvec &geom) {
	// Only reorder simple linestrings with a single moveto

	if (geom.size() == 0) {
		return geom;
	}

	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].op == VT_MOVETO) {
			if (i != 0) {
				// moveto is not at the start, so it is not simple
				return geom;
			}
		} else if (geom[i].op == VT_LINETO) {
			if (i == 0) {
				// lineto is at the start: can't happen
				return geom;
			}
		} else {
			// something other than moveto or lineto: can't happen
			return geom;
		}
	}

	// Reorder anything that goes up and to the left
	// instead of down and to the right
	// so that it will coalesce better

	unsigned long long l1 = encode_index(geom[0].x, geom[0].y);
	unsigned long long l2 = encode_index(geom[geom.size() - 1].x, geom[geom.size() - 1].y);

	if (l1 > l2) {
		drawvec out;
		for (size_t i = 0; i < geom.size(); i++) {
			out.push_back(geom[geom.size() - 1 - i]);
		}
		out[0].op = VT_MOVETO;
		if (out.size() > 1) {
			out[out.size() - 1].op = VT_LINETO;
		}
		return out;
	}

	return geom;
}

#if 0
std::vector<drawvec> chop_polygon(std::vector<drawvec> &geoms) {
	while (1) {
		bool again = false;
		std::vector<drawvec> out;

		for (size_t i = 0; i < geoms.size(); i++) {
			if (geoms[i].size() > 700) {
				static bool warned = false;
				if (!warned) {
					fprintf(stderr, "Warning: splitting up polygon with more than 700 sides\n");
					warned = true;
				}

				long long midx = 0, midy = 0, count = 0;
				long long maxx = LLONG_MIN, maxy = LLONG_MIN, minx = LLONG_MAX, miny = LLONG_MAX;

				for (size_t j = 0; j < geoms[i].size(); j++) {
					if (geoms[i][j].op == VT_MOVETO || geoms[i][j].op == VT_LINETO) {
						midx += geoms[i][j].x;
						midy += geoms[i][j].y;
						count++;

						if (geoms[i][j].x > maxx) {
							maxx = geoms[i][j].x;
						}
						if (geoms[i][j].y > maxy) {
							maxy = geoms[i][j].y;
						}
						if (geoms[i][j].x < minx) {
							minx = geoms[i][j].x;
						}
						if (geoms[i][j].y < miny) {
							miny = geoms[i][j].y;
						}
					}
				}

				midx /= count;
				midy /= count;

				drawvec c1, c2;

				if (maxy - miny > maxx - minx) {
					c1 = simple_clip_poly(geoms[i], minx, miny, maxx, midy, prevent[P_SIMPLIFY_EDGE_NODES]);
					c2 = simple_clip_poly(geoms[i], minx, midy, maxx, maxy, prevent[P_SIMPLIFY_EDGE_NODES]);
				} else {
					c1 = simple_clip_poly(geoms[i], minx, miny, midx, maxy, prevent[P_SIMPLIFY_EDGE_NODES]);
					c2 = simple_clip_poly(geoms[i], midx, miny, maxx, maxy, prevent[P_SIMPLIFY_EDGE_NODES]);
				}

				if (c1.size() >= geoms[i].size()) {
					fprintf(stderr, "Subdividing complex polygon failed\n");
				} else {
					out.push_back(c1);
				}
				if (c2.size() >= geoms[i].size()) {
					fprintf(stderr, "Subdividing complex polygon failed\n");
				} else {
					out.push_back(c2);
				}

				again = true;
			} else {
				out.push_back(geoms[i]);
			}
		}

		if (!again) {
			return out;
		}

		geoms = out;
	}
}
#endif

drawvec stairstep(drawvec &geom, int z, int detail) {
	drawvec out;
	double scale = 1 << (32 - detail - z);

	for (size_t i = 0; i < geom.size(); i++) {
		geom[i].x = std::round(geom[i].x / scale);
		geom[i].y = std::round(geom[i].y / scale);
	}

	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].op == VT_MOVETO) {
			out.push_back(geom[i]);
		} else if (out.size() > 0) {
			long long x0 = out[out.size() - 1].x;
			long long y0 = out[out.size() - 1].y;
			long long x1 = geom[i].x;
			long long y1 = geom[i].y;
			bool swap = false;

			if (y0 < y1) {
				swap = true;
				std::swap(x0, x1);
				std::swap(y0, y1);
			}

			long long xx = x0, yy = y0;
			long long dx = std::abs(x1 - x0);
			long long sx = (x0 < x1) ? 1 : -1;
			long long dy = std::abs(y1 - y0);
			long long sy = (y0 < y1) ? 1 : -1;
			long long err = ((dx > dy) ? dx : -dy) / 2;
			int last = -1;

			drawvec tmp;
			tmp.push_back(draw(VT_LINETO, xx, yy));

			while (xx != x1 || yy != y1) {
				long long e2 = err;

				if (e2 > -dx) {
					err -= dy;
					xx += sx;
					if (last == 1) {
						tmp[tmp.size() - 1] = draw(VT_LINETO, xx, yy);
					} else {
						tmp.push_back(draw(VT_LINETO, xx, yy));
					}
					last = 1;
				}
				if (e2 < dy) {
					err += dx;
					yy += sy;
					if (last == 2) {
						tmp[tmp.size() - 1] = draw(VT_LINETO, xx, yy);
					} else {
						tmp.push_back(draw(VT_LINETO, xx, yy));
					}
					last = 2;
				}
			}

			if (swap) {
				for (size_t j = tmp.size(); j > 0; j--) {
					out.push_back(tmp[j - 1]);
				}
			} else {
				for (size_t j = 0; j < tmp.size(); j++) {
					out.push_back(tmp[j]);
				}
			}

			// out.push_back(draw(VT_LINETO, xx, yy));
		} else {
			fprintf(stderr, "Can't happen: stairstepping lineto with no moveto\n");
			exit(EXIT_IMPOSSIBLE);
		}
	}

	for (size_t i = 0; i < out.size(); i++) {
		out[i].x *= 1 << (32 - detail - z);
		out[i].y *= 1 << (32 - detail - z);
	}

	return out;
}

// https://github.com/Turfjs/turf/blob/master/packages/turf-center-of-mass/index.ts
//
// The MIT License (MIT)
//
// Copyright (c) 2019 Morgan Herlocker
//
// Permission is hereby granted, free of charge, to any person obtaining a copy of
// this software and associated documentation files (the "Software"), to deal in
// the Software without restriction, including without limitation the rights to
// use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of
// the Software, and to permit persons to whom the Software is furnished to do so,
// subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in all
// copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS
// FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR
// COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER
// IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN
// CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
draw centerOfMass(const drawvec &dv, size_t start, size_t end, draw centre) {
	std::vector<draw> coords;
	for (size_t i = start; i < end; i++) {
		coords.push_back(dv[i]);
	}

	// First, we neutralize the feature (set it around coordinates [0,0]) to prevent rounding errors
	// We take any point to translate all the points around 0
	draw translation = centre;
	double sx = 0;
	double sy = 0;
	double sArea = 0;
	draw pi, pj;
	double xi, xj, yi, yj, a;

	std::vector<draw> neutralizedPoints;
	for (size_t i = 0; i < coords.size(); i++) {
		neutralizedPoints.push_back(draw(coords[i].op, coords[i].x - translation.x, coords[i].y - translation.y));
	}

	for (size_t i = 0; i < coords.size() - 1; i++) {
		// pi is the current point
		pi = neutralizedPoints[i];
		xi = pi.x;
		yi = pi.y;

		// pj is the next point (pi+1)
		pj = neutralizedPoints[i + 1];
		xj = pj.x;
		yj = pj.y;

		// a is the common factor to compute the signed area and the final coordinates
		a = xi * yj - xj * yi;

		// sArea is the sum used to compute the signed area
		sArea += a;

		// sx and sy are the sums used to compute the final coordinates
		sx += (xi + xj) * a;
		sy += (yi + yj) * a;
	}

	// Shape has no area: fallback on turf.centroid
	if (sArea == 0) {
		return centre;
	} else {
		// Compute the signed area, and factorize 1/6A
		double area = sArea * 0.5;
		double areaFactor = 1 / (6 * area);

		// Compute the final coordinates, adding back the values that have been neutralized
		return draw(VT_MOVETO, translation.x + areaFactor * sx, translation.y + areaFactor * sy);
	}
}

draw center_of_mass_mp(const drawvec &dv) {
	double ringx = 0, ringy = 0;
	size_t ringcount = 0;

	for (size_t i = 0; i < dv.size(); i++) {
		if (dv[i].op == VT_MOVETO) {
			double xsum = dv[i].x, ysum = dv[i].y;
			ssize_t count = 1;
			size_t j;
			for (j = i + 1; j < dv.size(); j++) {
				if (dv[j].op != VT_LINETO) {
					break;
				} else {
					xsum += dv[j].x;
					ysum += dv[j].y;
					count++;
				}
			}

			double area = get_area(dv, i, j);
			draw centroid(VT_MOVETO, std::llround(xsum / count), std::llround(ysum / count));

			draw center = centerOfMass(dv, i, j, centroid);
			ringx += center.x * area;
			ringy += center.y * area;
			ringcount += area;

			i = j - 1;
		}
	}

	draw center(VT_MOVETO, ringx / ringcount, ringy / ringcount);
	return center;
}

double label_goodness(const drawvec &dv, long long x, long long y) {
	int nesting = 0;

	for (size_t i = 0; i < dv.size(); i++) {
		if (dv[i].op == VT_MOVETO) {
			size_t j;
			for (j = i + 1; j < dv.size(); j++) {
				if (dv[j].op != VT_LINETO) {
					break;
				}
			}

			// if it's inside the ring, and it's an outer ring,
			// we are nested more; if it's an inner ring, we are
			// nested less.
			if (pnpoly(dv, i, j - i, x, y)) {
				if (get_area(dv, i, j) >= 0) {
					nesting++;
				} else {
					nesting--;
				}
			}

			i = j - 1;
		}
	}

	if (nesting < 1) {
		return 0;  // outside the polygon is as bad as it gets
	}

	double closest = INFINITY;  // closest distance to the border

	for (size_t i = 0; i < dv.size(); i++) {
		double dx = dv[i].x - x;
		double dy = dv[i].y - y;
		double dist = sqrt(dx * dx + dy * dy);
		if (dist < closest) {
			closest = dist;
		}

		if (i > 0 && dv[i].op == VT_LINETO) {
			dist = distance_from_line(x, y, dv[i - 1].x, dv[i - 1].y, dv[i].x, dv[i].y);
			if (dist < closest) {
				closest = dist;
			}
		}
	}

	return closest;
}

struct sorty {
	long long x;
	long long y;
};

struct sorty_sorter {
	int kind;
	sorty_sorter(int k)
	    : kind(k){};

	bool operator()(const sorty &a, const sorty &b) const {
		long long xa, ya, xb, yb;

		if (kind == 0) {  // Y first
			xa = a.x;
			ya = a.y;

			xb = b.x;
			yb = b.y;
		} else if (kind == 1) {	 // X first
			xa = a.y;
			ya = a.x;

			xb = b.y;
			yb = b.x;
		} else if (kind == 2) {	 // diagonal
			xa = a.x + a.y;
			ya = a.x - a.y;

			xb = b.x + b.y;
			yb = b.x - b.y;
		} else {  // other diagonal
			xa = a.x - a.y;
			ya = a.x + a.y;

			xb = b.x - b.y;
			yb = b.x + b.y;
		}

		if (ya < yb) {
			return true;
		} else if (ya == yb && xa < xb) {
			return true;
		} else {
			return false;
		}
	};
};

struct candidate {
	long long x;
	long long y;
	double dist;

	bool operator<(const candidate &c) const {
		// largest distance sorts first
		return dist > c.dist;
	};
};

// Generate a label point for a polygon feature.
//
// A good label point will be near the center of the feature and far from any border.
//
// Polylabel is supposed to be able to do this optimally, but can be quite slow
// and sometimes still produces some odd results.
//
// The centroid is often off-center because edges with many curves will be
// weighted higher than edges with straight lines.
//
// Turf's center-of-mass algorithm generally does a good job, but can sometimes
// find a point that is outside the bounds of the polygon or quite close to the edge.
//
// So prefer the center of mass, but if it produces something too close to the border
// or outside the polygon, try a series of gridded points within the feature's bounding box
// until something works well, or if nothing does after several iterations, use the
// least-bad option.

drawvec polygon_to_anchor(const drawvec &geom) {
	size_t start = 0, end = 0;
	size_t best_area = 0;
	std::vector<sorty> points;

	// find the largest outer ring, which will be the best thing
	// to label if we can do it.

	for (size_t i = 0; i < geom.size(); i++) {
		if (geom[i].op == VT_MOVETO) {
			size_t j;
			for (j = i + 1; j < geom.size(); j++) {
				if (geom[j].op != VT_LINETO) {
					break;
				}

				sorty sy;
				sy.x = geom[j].x;
				sy.y = geom[j].y;
				points.push_back(sy);
			}

			double area = get_area(geom, i, j);
			if (area > best_area) {
				start = i;
				end = j;
				best_area = area;
			}

			i = j - 1;
		}
	}

	// If there are no outer rings, don't generate a label point

	if (best_area > 0) {
		long long xsum = 0;
		long long ysum = 0;
		size_t count = 0;
		long long xmin = LLONG_MAX, ymin = LLONG_MAX, xmax = LLONG_MIN, ymax = LLONG_MIN;

		// Calculate centroid and bounding box of biggest ring.
		// start + 1 to exclude the first point, which is duplicated as the last
		for (size_t k = start + 1; k < end; k++) {
			xsum += geom[k].x;
			ysum += geom[k].y;
			count++;

			xmin = std::min(xmin, geom[k].x);
			ymin = std::min(ymin, geom[k].y);
			xmax = std::max(xmax, geom[k].x);
			ymax = std::max(ymax, geom[k].y);
		}

		if (count > 0) {
			// We want label points that are at least a moderate distance from
			// the edge of the feature. The threshold for what is too close
			// is derived from the area of the feature.

			double radius = sqrt(best_area / M_PI);
			double goodness_threshold = radius / 5;

			// First choice: Turf's center of mass.

			draw centroid(VT_MOVETO, xsum / count, ysum / count);
			draw d = centerOfMass(geom, start, end, centroid);
			double goodness = label_goodness(geom, d.x, d.y);
			const char *kind = "mass";

			if (goodness < goodness_threshold) {
				// Label is too close to the border or outside it,
				// so try some other possible points. Sort the vertices
				// both by Y and X coordinate and then by diagonals,
				// and walk through each set
				// in sorted order. Adjacent pairs of coordinates should
				// tend to bounce back and forth between rings, so the
				// midpoint of each pair will hopefully be somewhere in the
				// interior of the polygon.

				std::vector<candidate> candidates;

				for (size_t pass = 0; pass < 4; pass++) {
					std::stable_sort(points.begin(), points.end(), sorty_sorter(pass));

					for (size_t i = 1; i < points.size(); i++) {
						double dx = points[i].x - points[i - 1].x;
						double dy = points[i].y - points[i - 1].y;

						double dist = sqrt(dx * dx + dy * dy);
						if (dist > 2 * goodness_threshold) {
							candidate c;

							c.x = (points[i].x + points[i - 1].x) / 2;
							c.y = (points[i].y + points[i - 1].y) / 2;
							c.dist = dist;

							candidates.push_back(c);
						}
					}
				}

				// Now sort the accumulate list of segment midpoints by the lengths
				// of the segments. Starting from the longest
				// segment, if we find one whose midpoint is inside the polygon and
				// far enough from any edge to be good enough, stop looking.

				std::stable_sort(candidates.begin(), candidates.end());
				// only check the top 50 stride midpoints, since this list can be quite large
				for (size_t i = 0; i < candidates.size() && i < 50; i++) {
					double maybe_goodness = label_goodness(geom, candidates[i].x, candidates[i].y);

					if (maybe_goodness > goodness) {
						d.x = candidates[i].x;
						d.y = candidates[i].y;

						goodness = maybe_goodness;
						kind = "diagonal";
						if (goodness > goodness_threshold) {
							break;
						}
					}
				}
			}

			// We may still not have anything decent, so the next thing to look at
			// is points from gridding the bounding box of the largest ring.

			if (goodness < goodness_threshold) {
				for (long long sub = 2;
				     sub < 32 && (xmax - xmin) > 2 * sub && (ymax - ymin) > 2 * sub;
				     sub *= 2) {
					for (long long x = 1; x < sub; x++) {
						for (long long y = 1; y < sub; y++) {
							draw maybe(VT_MOVETO,
								   xmin + x * (xmax - xmin) / sub,
								   ymin + y * (ymax - ymin) / sub);

							double maybe_goodness = label_goodness(geom, maybe.x, maybe.y);
							if (maybe_goodness > goodness) {
								// better than the previous
								d = maybe;
								goodness = maybe_goodness;
								kind = "grid";
							}
						}
					}

					if (goodness > goodness_threshold) {
						break;
					}
				}

				// There is nothing really good. Is the centroid maybe better?
				// If not, we're stuck with whatever the best we found was.
				double maybe_goodness = label_goodness(geom, centroid.x, centroid.y);
				if (maybe_goodness > goodness) {
					d = centroid;
					goodness = maybe_goodness;
					kind = "centroid";
				}

				if (goodness <= 0) {
					double lon, lat;
					tile2lonlat(d.x, d.y, 32, &lon, &lat);

					static std::atomic<long long> warned(0);
					if (warned++ < 10) {
						fprintf(stderr, "could not find good label point: %s %f,%f\n", kind, lat, lon);
					}
				}
			}

			drawvec dv;
			dv.push_back(d);
			return dv;
		}
	}

	return drawvec();
}

drawvec checkerboard_anchors(drawvec const &geom, int tx, int ty, int z, long long wx, long long wy, int buffer) {
	drawvec out;

	// upper left of tile in world coordinates
	long long tx1 = 0, ty1 = 0;
	// lower right of tile in world coordinates;
	long long tx2 = 1LL << 32, ty2 = 1LL << 32;
	if (z != 0) {
		tx1 = (long long) tx << (32 - z);
		ty1 = (long long) ty << (32 - z);

		tx2 = (long long) (tx + 1) << (32 - z);
		ty2 = (long long) (ty + 1) << (32 - z);
	}

	long long tx1_buffer = tx1 - (1LL << (32 - z)) * buffer / 256;
	long long ty1_buffer = ty1 - (1LL << (32 - z)) * buffer / 256;
	long long tx2_buffer = tx2 + (1LL << (32 - z)) * buffer / 256;
	long long ty2_buffer = ty2 + (1LL << (32 - z)) * buffer / 256;

	// if the central label point is within this tile, always prefer that.
	// none of the label repetition code below matters much any more,
	// now that we truncate the tile pyramid once we have all the features
	if (wx >= tx1_buffer && wx <= tx2_buffer && wy >= ty1_buffer && wy <= ty2_buffer) {
		// offset to tile origin
		out.push_back(draw(VT_MOVETO, wx - tx1, wy - ty1));
		return out;
	}

	// upper left of feature in world coordinates
	long long bx1 = LLONG_MAX, by1 = LLONG_MAX;
	// lower right of feature in world coordinates;
	long long bx2 = LLONG_MIN, by2 = LLONG_MIN;

	for (auto const &g : geom) {
		bx1 = std::min(bx1, g.x + tx1);
		by1 = std::min(by1, g.y + ty1);

		bx2 = std::max(bx2, g.x + tx1);
		by2 = std::max(by2, g.y + ty1);
	}

	if (bx1 > bx2 || by1 > by2) {
		return out;
	}

	// labels repeat every 0.3 tiles at z0
	double spiral_dist = 0.3;
	if (z > 0) {
		// only every ~6 tiles by the time we get to z15
		spiral_dist = spiral_dist * exp(log(z) * 1.2);
	}

	const long long label_spacing = spiral_dist * (tx2 - tx1);

	long long x1 = floor(std::min(bx1 - wx, bx2 - wx) / label_spacing);
	long long x2 = ceil(std::max(bx1 - wx, bx2 - wx) / label_spacing);

	long long y1 = floor(std::min(by1 - wy, by2 - wy) / label_spacing - 0.5);
	long long y2 = ceil(std::max(by1 - wy, by2 - wy) / label_spacing);

	for (long long lx = x1; lx <= x2; lx++) {
		for (long long ly = y1; ly <= y2; ly++) {
			long long x = lx * label_spacing + wx;
			long long y = ly * label_spacing + wy;

			if (((unsigned long long) lx & 1) == 1) {
				y += label_spacing / 2;
			}

			if (x < bx1 || x > bx2 || y < by1 || y > by2) {
				continue;
			}

			// If it's the central label, it's the best we've got,
			// so accept it in any case. If it's from the outer spiral,
			// don't use it if it's too close to a border.
			if (lx == 0 && ly == 0) {
				out.push_back(draw(VT_MOVETO, x - tx1, y - ty1));
				break;
			} else {
				double tilesize = 1LL << (32 - z);
				double goodness_threshold = tilesize / 100;
				if (label_goodness(geom, x - tx1, y - ty1) > goodness_threshold) {
					out.push_back(draw(VT_MOVETO, x - tx1, y - ty1));
					break;
				}
			}
		}
	}

	return out;
}