polygon.hpp

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#pragma once
 
#include "shape/polyline.hpp"
 
#include <algorithm>
#include <vector>
#include <cassert>
#include <compare>
#include <concepts>
#include <cstddef>
#include <functional>
#include <iterator>
#include <ranges>
#include <limits>
#include <optional>
#include <ostream>
#include <span>
#include <type_traits>
#include <utility>
 
 
namespace pgl {
 
template <class PointType = Point<>, class Label>
struct Polygon;
 
Polygon() -> Polygon<Point<>, NoLabel>;
 
template <std::ranges::input_range Range>
requires detail::is_point_v<std::ranges::range_value_t<Range>>
Polygon(Range&&) -> Polygon<std::remove_cvref_t<std::ranges::range_value_t<Range>>, NoLabel>;
 
template <class Number>
requires (!detail::is_point_v<Number>)
Polygon(std::initializer_list<Number>) -> Polygon<Point<Number>, NoLabel>;
 
template <class Number>
requires (!detail::is_point_v<Number>)
Polygon(std::initializer_list<Number>, bool) -> Polygon<Point<Number>, NoLabel>;
 

template <class PointType_, class TLabel>
struct Polygon {
    using PointType = PointType_;
    using NumberType = PointType::NumberType;
    using LabelType = TLabel;
    static_assert(detail::is_point_v<PointType>, "Polygon requires pgl::Point vertices");
 
    template <bool Oriented>
    using BoundaryType = std::conditional_t<Oriented, OrientedSegment<PointType>, Segment<PointType>>;
 
    template <bool Oriented>
    class BoundaryIterator;
 
    using EdgeIterator = BoundaryIterator<false>;
    using OrientedEdgeIterator = BoundaryIterator<true>;

    constexpr Polygon() = default;

    template<std::ranges::input_range Range = std::initializer_list<PointType>>
    requires std::ranges::common_range<Range> &&
             std::convertible_to<std::ranges::range_value_t<Range>, PointType>
    constexpr explicit Polygon(Range&& points, bool trusted = false) {
        for (const auto& p : points) {
            points_.push_back(p);
        }
        if (!trusted) {
            normalize();
        }
    }

    constexpr explicit Polygon(std::initializer_list<NumberType> coords, bool trusted = false) {
        assert(coords.size() % 2 == 0);
        points_.reserve(coords.size() / 2);
        for (auto it = coords.begin(); it != coords.end(); ) {
            NumberType x = *it++;
            NumberType y = *it++;
            points_.emplace_back(x, y);
        }
        if (!trusted) {
            normalize();
        }
    }

    template<PointConcept OtherPointType, class OtherLabelType>
        requires(std::constructible_from<PointType, const OtherPointType&>)
    constexpr Polygon(const Polygon<OtherPointType, OtherLabelType>& other)
        : points_(other.begin(), other.end()), label_(detail::copyLabel<LabelType>(other)) {}

    template <class A = LabelType>
        requires(detail::has_label_v<A>)
    constexpr A& label() const {
        return label_;
    }

    constexpr const PointType operator[](std::size_t index) const {
        assert(index < size());
        return points_[index] + translation_;
    }

    constexpr PointType get(std::ptrdiff_t index) const {
        const std::ptrdiff_t n = static_cast<std::ptrdiff_t>(size());
        return (*this)[static_cast<std::size_t>(((index % n) + n) % n)];
    }

    constexpr std::ptrdiff_t index(const PointType& point) const {
        for (std::ptrdiff_t i = 0; i < static_cast<std::ptrdiff_t>(size()); ++i) {
            if ((*this)[static_cast<std::size_t>(i)] == point) {
                return i;
            }
        }
        return -1;
    }

    constexpr auto begin() const {
        return Iterator(points_.begin(), translation_);
    }

    constexpr auto cbegin() const {
        return Iterator(points_.cbegin(), translation_);
    }

    constexpr auto end() const {
        return Iterator(points_.end(), translation_);
    }

    constexpr auto cend() const {
        return Iterator(points_.cend(), translation_);
    }

    constexpr auto operator<=>(const Polygon& other) const {
        if (auto cmp = points_.size() <=> other.points_.size(); cmp != 0) {
            return cmp;
        }
        for (std::size_t i = 0; i < points_.size(); ++i) {
            if (auto cmp = points_[i] + translation_ <=> other.points_[i] + other.translation_; cmp != 0) {
                return cmp;
            }
        }
        return std::strong_ordering::equal;
    }

    constexpr bool operator==(const Polygon& other) const {
        if (points_.size() != other.points_.size()) {
            return false;
        }
        for (std::size_t i = 0; i < points_.size(); ++i) {
            if (points_[i] + translation_ != other.points_[i] + other.translation_) {
                return false;
            }
        }
        return true;
    }

    template<AnyShapeConcept OtherShape>
    [[nodiscard]] constexpr bool samePointSet(const OtherShape& other) const;

    constexpr std::size_t size() const {
        return points_.size();
    }

    template <class ResultNumber = NumberType>
    constexpr ResultNumber twiceArea() const {
        if (points_.size() < 3) {
            return ResultNumber(0);
        }
        return pgl::detail::abs(signedTwiceArea<ResultNumber>());
    }

    template <class ResultNumber = division_result_t<NumberType>>
    constexpr auto area() const {
        ResultNumber result = static_cast<ResultNumber>(twiceArea());
        return result / ResultNumber(2);
    }

    [[nodiscard]] constexpr bool empty() const {
        return points_.empty();
    }

    constexpr bool isDegenerate() const {
        return empty() || isPoint() || isSegment() || hasNoArea();
    }

    [[nodiscard]] constexpr bool isPoint() const {
        return detail::allPointsEqual(points_);
    }

    [[nodiscard]] constexpr std::optional<PointType> getIfPoint() const {
        if (!isPoint()) {
            return std::nullopt;
        }
        return points_.front() + translation_;
    }

    [[nodiscard]] constexpr bool isSegment() const {
        return detail::pointsSpanSegment(points_);
    }

    [[nodiscard]] constexpr std::optional<BoundaryType<false>> getIfSegment() const {
        if (!isSegment()) {
            return std::nullopt;
        }
        return detail::spannedSegment<BoundaryType<false>>(points_) + translation_;
    }

    [[nodiscard]] constexpr bool isUndefined() const {
        // Ordered so the cheap emptiness and point/segment scans reject the
        // common cases before paying for the full area sum.
        return !empty() && !isPoint() && !isSegment() && hasNoArea();
    }

    template <class Rational = pgl::Rational<pgl::BigInt>>
    [[nodiscard]] bool isSimple() const;

    [[nodiscard]] constexpr bool isConvex() const {
        const std::ptrdiff_t n = static_cast<std::ptrdiff_t>(size());
        if (n < 3) {
            return false;
        }
        bool sawPositive = false;
        bool sawNegative = false;
        for (std::ptrdiff_t i = 0; i < n; ++i) {
            const auto turn = orientationSign(get(i), get(i + 1), get(i + 2));
            if (turn > 0) {
                sawPositive = true;
            } else if (turn < 0) {
                sawNegative = true;
            }
            if (sawPositive && sawNegative) {
                return false;
            }
        }
        return true;
    }

    [[nodiscard]] constexpr std::size_t chainCount() const {
        const std::ptrdiff_t n = static_cast<std::ptrdiff_t>(size());
        if (n < 2) {
            return 0;
        }
        // Matches BoundaryChains exactly, down to how it classifies a repeated
        // vertex: `ascends` is a strict lexicographic test, so a level edge
        // counts as descending in both places.
        std::size_t breaks = 0;
        bool previous = get(n - 1) < get(0);
        for (std::ptrdiff_t i = 0; i < n; ++i) {
            const bool ascends = get(i) < get(i + 1);
            if (ascends != previous) {
                ++breaks;
            }
            previous = ascends;
        }
        return breaks;
    }

    [[nodiscard]] constexpr std::optional<HalfplaneIntersection<PointType>> getStarShapedKernel() const;

    [[nodiscard]] constexpr bool isStarShaped() const {
        return getStarShapedKernel().has_value();
    }

    constexpr Segment<PointType> diameter() const {
        return Convex<PointType>(vertices()).diameter();
    }

    constexpr Convex<PointType> convexHull() const {
        return Convex<PointType>(vertices());
    }

    constexpr const Rectangle<PointType>& bbox() const;

    template <class ResultNumber = grid_number_t<typename PointType_::NumberType>>
        requires(detail::extended_integral<ResultNumber> || std::same_as<ResultNumber, BigInt>)
    [[nodiscard]] std::vector<Point<ResultNumber, typename PointType::LabelType>>
    latticePoints() const;

    template <std::floating_point ResultNumber = double>
    constexpr Rectangle<Point<ResultNumber>> fbox() const;

    constexpr std::vector<PointType> vertices() const {
        auto ret = points_;
        for (auto& vertex : ret) {
            vertex += translation_;
        }
        return ret;
    }

    constexpr std::vector<Segment<PointType>> edges() const {
        std::vector<Segment<PointType>> result;
        const auto translatedVertices = vertices();
        for (std::size_t i = 0; i < translatedVertices.size(); ++i) {
            const auto& p1 = translatedVertices[i];
            const auto& p2 = translatedVertices[(i + 1) % translatedVertices.size()];
            result.emplace_back(p1, p2);
        }
        return result;
    }

    constexpr std::vector<OrientedSegment<PointType>> orientedEdges() const {
        std::vector<OrientedSegment<PointType>> result;
        const auto translatedVertices = vertices();
        for (std::size_t i = 0; i < translatedVertices.size(); ++i) {
            const auto& p1 = translatedVertices[i];
            const auto& p2 = translatedVertices[(i + 1) % translatedVertices.size()];
            result.emplace_back(p1, p2);
        }
        return result;
    }

    [[nodiscard]] Graph<PointType> visibilityGraph() const;

    [[nodiscard]] Graph<PointType> clearVisibilityGraph() const;

    [[nodiscard]] Graph<PointType> reducedVisibilityGraph() const;

    [[nodiscard]] std::vector<PointType> visibleVertices(const PointType& query) const;

    [[nodiscard]] std::vector<PointType> clearlyVisibleVertices(const PointType& query) const;

    template <class ResultNumber = division_result_t<NumberType>>
    [[nodiscard]] Polygon<Point<ResultNumber>> regularizedVisiblePolygon(
        const PointType& query) const;

    constexpr auto verticesView() const {
        return std::ranges::subrange(begin(), end());
    }

    constexpr auto edgesView() const {
        return std::ranges::subrange(edgesBegin(), edgesEnd());
    }

    constexpr auto orientedEdgesView() const {
        return std::ranges::subrange(orientedEdgesBegin(), orientedEdgesEnd());
    }

    constexpr EdgeIterator edgesBegin() const {
        return EdgeIterator(this, 0);
    }

    constexpr EdgeIterator edgesEnd() const {
        return EdgeIterator(this, size());
    }

    constexpr OrientedEdgeIterator orientedEdgesBegin() const {
        return OrientedEdgeIterator(this, 0);
    }

    constexpr OrientedEdgeIterator orientedEdgesEnd() const {
        return OrientedEdgeIterator(this, size());
    }

    [[nodiscard]] constexpr PolygonWithHoles<PointType> asPolygonWithHoles() const {
        return PolygonWithHoles<PointType>(*this);
    }

    [[nodiscard]] constexpr PolygonSet<PointType> asPolygonSet() const {
        return PolygonSet<PointType>(asPolygonWithHoles());
    }

    template <class ResultNumber = division_result_t<NumberType>>
    constexpr Point<ResultNumber> centroid() const {
        if (points_.size() < 3) {
            return verticesCentroid<ResultNumber>();
        }
        // The weights and the area they are divided by are summed in the same
        // pass and the same type: taken from signedTwiceArea() the divisor
        // would come back narrowed to NumberType, where an area past the
        // coordinate range wraps and either sends a perfectly ordinary polygon
        // down the no-area path or scales the answer by a wrapped divisor.
        ResultNumber cx = 0;
        ResultNumber cy = 0;
        ResultNumber areaTwice = 0;
        const std::size_t n = points_.size();
        for (std::size_t i = 0; i < n; ++i) {
            const auto& p1 = points_[i];
            const auto& p2 = points_[(i + 1) % n];
            const ResultNumber cross = detail::asNumber<ResultNumber>(p1.x()) * detail::asNumber<ResultNumber>(p2.y())
                                     - detail::asNumber<ResultNumber>(p2.x()) * detail::asNumber<ResultNumber>(p1.y());
            areaTwice += cross;
            cx += (detail::asNumber<ResultNumber>(p1.x()) + detail::asNumber<ResultNumber>(p2.x())) * cross;
            cy += (detail::asNumber<ResultNumber>(p1.y()) + detail::asNumber<ResultNumber>(p2.y())) * cross;
        }
        if (areaTwice == ResultNumber(0)) {
            return verticesCentroid<ResultNumber>();
        }
        const ResultNumber denom = ResultNumber(3) * areaTwice;
        return Point<ResultNumber>(cx / denom, cy / denom) + static_cast<Point<ResultNumber>>(translation_);
    }

    template <class ResultNumber = division_result_t<NumberType>>
    constexpr Point<ResultNumber> verticesCentroid() const {
        if (points_.empty()) {
            return Point<ResultNumber>();
        }
        ResultNumber cx = 0;
        ResultNumber cy = 0;
        for (const auto& vertex : points_) {
            cx += detail::asNumber<ResultNumber>(vertex.x());
            cy += detail::asNumber<ResultNumber>(vertex.y());
        }
        return Point<ResultNumber>(cx / static_cast<ResultNumber>(points_.size()),
                                   cy / static_cast<ResultNumber>(points_.size()))
               + static_cast<Point<ResultNumber>>(translation_);
    }

    template <class ResultNumber = division_result_t<NumberType>>
    [[nodiscard]] constexpr Point<ResultNumber> pointInside() const;

    template <class OtherShape>
    [[nodiscard]] constexpr bool pointInsideInteriorContainedIn(const OtherShape& shape) const;

    auto triangulation() const;

    template <class SegmentRange>
    auto triangulation(const SegmentRange& segments) const;

    [[nodiscard]] std::vector<Convex<PointType>> convexPartition() const;

    [[nodiscard]] std::vector<Convex<PointType>> convexCovering() const;

    template <class ResultNumber = grid_number_t<typename PointType_::NumberType>>
        requires(std::signed_integral<ResultNumber>)
    [[nodiscard]] auto asBitMatrix() const;

    template <class PointRange, class SegmentRange>
    auto triangulation(const PointRange& points, const SegmentRange& segments) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherRegion& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonSetConcept OtherSet>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherSet& other) const;

    template <class ResultNumber = division_result_t<NumberType>, HalfplaneIntersectionConcept OtherIntersection>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherIntersection& other) const;

    template <class ResultNumber = division_result_t<NumberType>, HalfplaneConcept OtherHalfplane>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherHalfplane& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    regularizedUnion(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    regularizedUnion(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    regularizedUnion(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    regularizedUnion(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    regularizedUnion(const OtherRegion& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonSetConcept OtherSet>
    [[nodiscard]] auto regularizedUnion(const OtherSet& other) const {
        return other.template regularizedUnion<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    symmetricDifference(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    symmetricDifference(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    symmetricDifference(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    symmetricDifference(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    symmetricDifference(const OtherRegion& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonSetConcept OtherSet>
    [[nodiscard]] auto symmetricDifference(const OtherSet& other) const {
        return other.template symmetricDifference<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherRegion& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolylineConcept OtherPolyline>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherPolyline& other) const;

    template <class ResultNumber = division_result_t<NumberType>, MonotoneChainConcept OtherChain>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherChain& other) const;

    template <class ResultNumber = division_result_t<NumberType>, SegmentConcept OtherSegment>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedSegmentConcept OtherSegment>
    [[nodiscard]] PolygonWithHoles<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonSetConcept OtherSet>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    minkowskiSum(const OtherSet& other) const;

    template<PointConcept OtherPoint>
    constexpr bool contains(const OtherPoint& point) const;

    template<SegmentConcept OtherSegment>
    constexpr bool contains(const OtherSegment& other) const;

    template<OrientedSegmentConcept OtherOrientedSegment>
    constexpr bool contains(const OtherOrientedSegment& other) const;

    template<LineConcept OtherLine>
    constexpr bool contains(const OtherLine& other) const;

    template<OrientedLineConcept OtherOrientedLine>
    constexpr bool contains(const OtherOrientedLine& other) const;

    template<RayConcept OtherRay>
    constexpr bool contains(const OtherRay& other) const;

    template<HalfplaneConcept OtherHalfplane>
    constexpr bool contains(const OtherHalfplane& other) const;

    template<RectangleConcept OtherRectangle>
    constexpr bool contains(const OtherRectangle& other) const;

    template<TriangleConcept OtherTriangle>
    constexpr bool contains(const OtherTriangle& other) const;

    template<ConvexConcept OtherConvex>
    constexpr bool contains(const OtherConvex& other) const;

    template<PolygonConcept OtherPolygon>
    constexpr bool contains(const OtherPolygon& other) const;

    template<PolygonConcept OtherPolygon>
    constexpr bool containsChainBased(const OtherPolygon& other) const;

    template<DiskConcept OtherDisk>
    constexpr bool contains(const OtherDisk& other) const;

    constexpr bool contains(const Shape<PointType>& other) const;
 
    // The empty set is a subset of every shape, so its containment relations are
    // true; symmetric crossing reaches the empty set through the generic
    // OtherShape fallback declared below.
    template <class EmptyPoint>
    [[nodiscard]] constexpr bool contains(const EmptyShape<EmptyPoint>&) const {
        return true;
    }

    template <class EmptyPoint>
    [[nodiscard]] constexpr bool boundaryContains(const EmptyShape<EmptyPoint>&) const {
        return true;
    }

    template <class EmptyPoint>
    [[nodiscard]] constexpr bool interiorContains(const EmptyShape<EmptyPoint>&) const {
        return true;
    }

    template<PointConcept OtherPoint>
    constexpr bool interiorContains(const OtherPoint& point) const;

    template<SegmentConcept OtherSegment>
    constexpr bool interiorContains(const OtherSegment& other) const;

    template<SegmentConcept OtherSegment>
    constexpr bool interiorContainsInterior(const OtherSegment& other) const;

    template<OrientedSegmentConcept OtherOrientedSegment>
    constexpr bool interiorContains(const OtherOrientedSegment& other) const;

    template<LineConcept OtherLine>
    constexpr bool interiorContains(const OtherLine& other) const;

    template<OrientedLineConcept OtherOrientedLine>
    constexpr bool interiorContains(const OtherOrientedLine& other) const;

    template<RayConcept OtherRay>
    constexpr bool interiorContains(const OtherRay& other) const;

    template<HalfplaneConcept OtherHalfplane>
    constexpr bool interiorContains(const OtherHalfplane& other) const;

    template<RectangleConcept OtherRectangle>
    constexpr bool interiorContains(const OtherRectangle& other) const;

    template<TriangleConcept OtherTriangle>
    constexpr bool interiorContains(const OtherTriangle& other) const;

    template<ConvexConcept OtherConvex>
    constexpr bool interiorContains(const OtherConvex& other) const;

    template<PolygonConcept OtherPolygon>
    constexpr bool interiorContains(const OtherPolygon& other) const;

    template<PointConcept OtherPoint>
    constexpr bool boundaryContains(const OtherPoint& point) const;

    template<SegmentConcept OtherSegment>
    constexpr bool boundaryContains(const OtherSegment& other) const;

    template<OrientedSegmentConcept OtherOrientedSegment>
    constexpr bool boundaryContains(const OtherOrientedSegment& other) const;

    template<LineConcept OtherLine>
    constexpr bool boundaryContains(const OtherLine& other) const;

    template<OrientedLineConcept OtherOrientedLine>
    constexpr bool boundaryContains(const OtherOrientedLine& other) const;

    template<RayConcept OtherRay>
    constexpr bool boundaryContains(const OtherRay& other) const;

    template<HalfplaneConcept OtherHalfplane>
    constexpr bool boundaryContains(const OtherHalfplane& other) const;

    template<RectangleConcept OtherRectangle>
    constexpr bool boundaryContains(const OtherRectangle& other) const;

    template<TriangleConcept OtherTriangle>
    constexpr bool boundaryContains(const OtherTriangle& other) const;

    template<ConvexConcept OtherConvex>
    constexpr bool boundaryContains(const OtherConvex& other) const;

    template<PolygonConcept OtherPolygon>
    constexpr bool boundaryContains(const OtherPolygon& other) const;

    template<DiskConcept OtherDisk>
    constexpr bool boundaryContains(const OtherDisk& other) const;

    template<PointConcept OtherPoint>
    constexpr bool boundaryContains(const Shape<OtherPoint>& other) const;
 
    // --- not-yet-implemented predicate pairs (throw); see implementation ---
    template<DiskConcept OtherDisk>
    [[nodiscard]] constexpr bool interiorContains(const OtherDisk& other) const;

    template<PointConcept OtherPoint>
    [[nodiscard]] constexpr bool separates(const OtherPoint& other) const;

    template<HalfplaneConcept OtherHalfplane>
    [[nodiscard]] constexpr bool separates(const OtherHalfplane& other) const;

    template<RectangleConcept OtherRectangle>
    [[nodiscard]] constexpr bool separates(const OtherRectangle& other) const;

    template<TriangleConcept OtherTriangle>
    [[nodiscard]] constexpr bool separates(const OtherTriangle& other) const;

    template<DiskConcept OtherDisk>
    [[nodiscard]] constexpr bool separates(const OtherDisk& other) const;

    template<ConvexConcept OtherConvex>
    [[nodiscard]] constexpr bool separates(const OtherConvex& other) const;

    template<PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr bool separates(const OtherPolygon& other) const;

    template<MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr bool contains(const OtherChain& other) const;

    template<MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr bool boundaryContains(const OtherChain& other) const;

    template<MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr bool interiorContains(const OtherChain& other) const;

    template<MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr bool intersects(const OtherChain& other) const;

    template<MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr bool interiorsIntersect(const OtherChain& other) const;

    template<MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr bool separates(const OtherChain& other) const;

    template<MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr bool crosses(const OtherChain& other) const;

    template <class ResultNumber = division_result_t<NumberType>, MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr auto squaredDistance(const OtherChain& other) const;

    template <class ResultNumber = division_result_t<NumberType>, MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr auto distanceL1(const OtherChain& other) const;

    template <class ResultNumber = division_result_t<NumberType>, MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr auto distanceLInf(const OtherChain& other) const;

    template<PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr bool contains(const OtherPolyline& other) const;

    template<PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr bool boundaryContains(const OtherPolyline& other) const;

    template<PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr bool interiorContains(const OtherPolyline& other) const;

    template<PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr bool intersects(const OtherPolyline& other) const;

    template<PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr bool interiorsIntersect(const OtherPolyline& other) const;

    template<PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr bool separates(const OtherPolyline& other) const;

    template<HalfplaneIntersectionConcept OtherRegion>
    [[nodiscard]] constexpr bool contains(const OtherRegion& other) const;

    template<HalfplaneIntersectionConcept OtherRegion>
    [[nodiscard]] constexpr bool boundaryContains(const OtherRegion& other) const;

    template<HalfplaneIntersectionConcept OtherRegion>
    [[nodiscard]] constexpr bool interiorContains(const OtherRegion& other) const;

    template<HalfplaneIntersectionConcept OtherRegion>
    [[nodiscard]] constexpr bool separates(const OtherRegion& other) const;

    template<PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] constexpr bool contains(const OtherRegion& other) const;

    template<PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] constexpr bool boundaryContains(const OtherRegion& other) const;

    template<PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] constexpr bool interiorContains(const OtherRegion& other) const;

    template<PolygonWithHolesConcept OtherRegion>
    [[nodiscard]] bool separates(const OtherRegion& other) const;
 
    // -------------------------------------------------------------------------
    // A set of regions
    //
    // It outranks every other shape, so the symmetric relations reach it through
    // the rank-based forwarders and only the asymmetric ones are answered here.
    // A set is the union of its components, so it is contained exactly when
    // every component is — no matter what this shape is.

    template<PolygonSetConcept OtherSet>
    [[nodiscard]] constexpr bool contains(const OtherSet& other) const {
        for (const auto& component : other) {
            if (!contains(component)) {
                return false;
            }
        }
        return true;
    }

    template<PolygonSetConcept OtherSet>
    [[nodiscard]] constexpr bool boundaryContains(const OtherSet& other) const {
        for (const auto& component : other) {
            if (!boundaryContains(component)) {
                return false;
            }
        }
        return true;
    }

    template<PolygonSetConcept OtherSet>
    [[nodiscard]] constexpr bool interiorContains(const OtherSet& other) const {
        for (const auto& component : other) {
            if (!interiorContains(component)) {
                return false;
            }
        }
        return true;
    }

    template<PolygonSetConcept OtherSet>
    [[nodiscard]] bool separates(const OtherSet& other) const;

    template<PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr bool crosses(const OtherPolyline& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr auto squaredDistance(const OtherPolyline& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr auto distanceL1(const OtherPolyline& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr auto distanceLInf(const OtherPolyline& other) const;
 

    template<PointConcept OtherPoint>
    constexpr bool intersects(const OtherPoint& other) const;

    template<SegmentConcept OtherSegment>
    constexpr bool intersects(const OtherSegment& other) const;

    template<OrientedSegmentConcept OtherOrientedSegment>
    constexpr bool intersects(const OtherOrientedSegment& other) const;

    template<LineConcept OtherLine>
    constexpr bool intersects(const OtherLine& other) const;

    template<OrientedLineConcept OtherOrientedLine>
    constexpr bool intersects(const OtherOrientedLine& other) const;

    template<RayConcept OtherRay>
    constexpr bool intersects(const OtherRay& other) const;

    template<HalfplaneConcept OtherHalfplane>
    constexpr bool intersects(const OtherHalfplane& other) const;

    template<RectangleConcept OtherRectangle>
    constexpr bool intersects(const OtherRectangle& other) const;

    template<TriangleConcept OtherTriangle>
    constexpr bool intersects(const OtherTriangle& other) const;

    template<ConvexConcept OtherConvex>
    constexpr bool intersects(const OtherConvex& other) const;

    template<PolygonConcept OtherPolygon>
    constexpr bool intersects(const OtherPolygon& other) const;

    template<PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr bool boundariesIntersect(const OtherPolygon& other) const;

    template<PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr bool boundariesStrongCross(const OtherPolygon& other) const;
 

    template<DiskConcept OtherDisk>
    constexpr bool intersects(const OtherDisk& other) const;

    template<PointConcept OtherPoint>
    constexpr bool interiorsIntersect(const OtherPoint& other) const;

    template<LineConcept OtherLine>
    constexpr bool interiorsIntersect(const OtherLine& other) const;

    template<OrientedLineConcept OtherOrientedLine>
    constexpr bool interiorsIntersect(const OtherOrientedLine& other) const;

    template<SegmentConcept OtherSegment>
    constexpr bool interiorsIntersect(const OtherSegment& other) const;

    template<OrientedSegmentConcept OtherOrientedSegment>
    constexpr bool interiorsIntersect(const OtherOrientedSegment& other) const;

    template<RayConcept OtherRay>
    constexpr bool interiorsIntersect(const OtherRay& other) const;

    template<HalfplaneConcept OtherHalfplane>
    constexpr bool interiorsIntersect(const OtherHalfplane& other) const;

    template<RectangleConcept OtherRectangle>
    constexpr bool interiorsIntersect(const OtherRectangle& other) const;

    template<TriangleConcept OtherTriangle>
    constexpr bool interiorsIntersect(const OtherTriangle& other) const;

    template<ConvexConcept OtherConvex>
    constexpr bool interiorsIntersect(const OtherConvex& other) const;

    template<PolygonConcept OtherPolygon>
    constexpr bool interiorsIntersect(const OtherPolygon& other) const;

    template<DiskConcept OtherDisk>
    constexpr bool interiorsIntersect(const OtherDisk& other) const;

    template<SegmentConcept OtherSegment>
    constexpr bool separates(const OtherSegment& other) const;

    template<OrientedSegmentConcept OtherOrientedSegment>
    constexpr bool separates(const OtherOrientedSegment& other) const;

    template<RayConcept OtherRay>
    constexpr bool separates(const OtherRay& other) const;

    template<LineConcept OtherLine>
    constexpr bool separates(const OtherLine& other) const;

    template<OrientedLineConcept OtherOrientedLine>
    constexpr bool separates(const OtherOrientedLine& other) const;

    template<PointConcept OtherPoint>
    [[nodiscard]] constexpr bool crosses(const OtherPoint&) const;

    template<SegmentConcept OtherSegment>
    [[nodiscard]] constexpr bool crosses(const OtherSegment& other) const;

    template<OrientedSegmentConcept OtherOrientedSegment>
    [[nodiscard]] constexpr bool crosses(const OtherOrientedSegment& other) const;

    template<RayConcept OtherRay>
    [[nodiscard]] constexpr bool crosses(const OtherRay& other) const;

    template<LineConcept OtherLine>
    [[nodiscard]] constexpr bool crosses(const OtherLine& other) const;

    template<OrientedLineConcept OtherOrientedLine>
    [[nodiscard]] constexpr bool crosses(const OtherOrientedLine& other) const;

    template<HalfplaneConcept OtherHalfplane>
    [[nodiscard]] constexpr bool crosses(const OtherHalfplane&) const;

    template<RectangleConcept OtherRectangle>
    [[nodiscard]] constexpr bool crosses(const OtherRectangle&) const;

    template<TriangleConcept OtherTriangle>
    [[nodiscard]] constexpr bool crosses(const OtherTriangle&) const;

    template<ConvexConcept OtherConvex>
    [[nodiscard]] constexpr bool crosses(const OtherConvex&) const;

    template<DiskConcept OtherDisk>
    [[nodiscard]] constexpr bool crosses(const OtherDisk&) const;

    template<PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr bool crosses(const OtherPolygon&) const;

    template<PointConcept OtherPoint>
    [[nodiscard]] constexpr bool crosses(const Shape<OtherPoint>& other) const;

    template<PointConcept OtherPoint>
    [[nodiscard]] constexpr bool intersects(const Shape<OtherPoint>& other) const;

    template<PointConcept OtherPoint>
    [[nodiscard]] constexpr bool interiorsIntersect(const Shape<OtherPoint>& other) const;

    template<typename OtherShape>
        requires (!PointConcept<OtherShape> && detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
    [[nodiscard]] constexpr bool crosses(const OtherShape& other) const {
        return other.crosses(*this);
    }

    template<typename OtherShape>
        requires (!PointConcept<OtherShape> && detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
    [[nodiscard]] constexpr bool intersects(const OtherShape& other) const {
        return other.intersects(*this);
    }

    template<typename OtherShape>
        requires (!PointConcept<OtherShape> && detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
    [[nodiscard]] constexpr bool interiorsIntersect(const OtherShape& other) const {
        return other.interiorsIntersect(*this);
    }

    template <class EmptyPoint>
    [[nodiscard]] constexpr bool crosses(const EmptyShape<EmptyPoint>&) const {
        return false;
    }

    template <class EmptyPoint>
    [[nodiscard]] constexpr bool intersects(const EmptyShape<EmptyPoint>&) const {
        return false;
    }

    template <class EmptyPoint>
    [[nodiscard]] constexpr bool interiorsIntersect(const EmptyShape<EmptyPoint>&) const {
        return false;
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] constexpr auto squaredDistance(const OtherPoint& point) const;

    template <class ResultNumber = division_result_t<NumberType>, SegmentConcept OtherSegment>
    [[nodiscard]] constexpr auto squaredDistance(const OtherSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedSegmentConcept OtherOrientedSegment>
    [[nodiscard]] constexpr auto squaredDistance(const OtherOrientedSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, LineConcept OtherLine>
    [[nodiscard]] constexpr auto squaredDistance(const OtherLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedLineConcept OtherOrientedLine>
    [[nodiscard]] constexpr auto squaredDistance(const OtherOrientedLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RayConcept OtherRay>
    [[nodiscard]] constexpr auto squaredDistance(const OtherRay& other) const;

    template <class ResultNumber = division_result_t<NumberType>, HalfplaneConcept OtherHalfplane>
    [[nodiscard]] constexpr auto squaredDistance(const OtherHalfplane& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] constexpr auto squaredDistance(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] constexpr auto squaredDistance(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] constexpr auto squaredDistance(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr auto squaredDistance(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
                  && requires(const OtherShape& o, const Polygon& self) {
                         o.template squaredDistance<ResultNumber>(self);
                     })
    [[nodiscard]] constexpr auto squaredDistance(const OtherShape& other) const {
        return other.template squaredDistance<ResultNumber>(*this);
    }

    template <class ResultNumber = double, class DiskPointType, class DiskLabel>
    [[nodiscard]] detail::floating_result_t<ResultNumber> squaredDistance(
        const Disk<DiskPointType, DiskLabel>& disk) const;

    template <class ResultNumber = NumberType, BoundedPolygonalConcept OtherShape>
        requires detail::ClosestPairConcept<Polygon<PointType_, TLabel>, OtherShape>
    [[nodiscard]] constexpr auto closestSegments(const OtherShape& other) const;

    template <class ResultNumber = division_result_t<NumberType>, class OtherShape>
        requires detail::ClosestPointsPairConcept<Polygon<PointType_, TLabel>, OtherShape>
    [[nodiscard]] constexpr auto closestPoints(const OtherShape& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] constexpr auto distanceL1(const OtherPoint& point) const;

    template <class ResultNumber = division_result_t<NumberType>, SegmentConcept OtherSegment>
    [[nodiscard]] constexpr auto distanceL1(const OtherSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedSegmentConcept OtherOrientedSegment>
    [[nodiscard]] constexpr auto distanceL1(const OtherOrientedSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, LineConcept OtherLine>
    [[nodiscard]] constexpr auto distanceL1(const OtherLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedLineConcept OtherOrientedLine>
    [[nodiscard]] constexpr auto distanceL1(const OtherOrientedLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RayConcept OtherRay>
    [[nodiscard]] constexpr auto distanceL1(const OtherRay& other) const;

    template <class ResultNumber = division_result_t<NumberType>, HalfplaneConcept OtherHalfplane>
    [[nodiscard]] constexpr auto distanceL1(const OtherHalfplane& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] constexpr auto distanceL1(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] constexpr auto distanceL1(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] constexpr auto distanceL1(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr auto distanceL1(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
                  && requires(const OtherShape& o, const Polygon& self) {
                         o.template distanceL1<ResultNumber>(self);
                     })
    [[nodiscard]] constexpr auto distanceL1(const OtherShape& other) const {
        return other.template distanceL1<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] constexpr auto intersection(const Shape<OtherPoint>& other) const {
        return other.template intersection<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] auto regularizedIntersection(const Shape<OtherPoint>& other) const {
        return other.template regularizedIntersection<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] auto regularizedUnion(const Shape<OtherPoint>& other) const {
        return other.template regularizedUnion<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] auto difference(const Shape<OtherPoint>& other) const {
        return Shape<OtherPoint>(*this).template difference<ResultNumber>(other);
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] auto symmetricDifference(const Shape<OtherPoint>& other) const {
        return other.template symmetricDifference<ResultNumber>(*this);
    }

    template <class ResultNumber = double, PointConcept OtherPoint>
    [[nodiscard]] constexpr auto distanceL1(const Shape<OtherPoint>& other) const {
        return other.template distanceL1<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] constexpr auto distanceLInf(const OtherPoint& point) const;

    template <class ResultNumber = division_result_t<NumberType>, SegmentConcept OtherSegment>
    [[nodiscard]] constexpr auto distanceLInf(const OtherSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedSegmentConcept OtherOrientedSegment>
    [[nodiscard]] constexpr auto distanceLInf(const OtherOrientedSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, LineConcept OtherLine>
    [[nodiscard]] constexpr auto distanceLInf(const OtherLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedLineConcept OtherOrientedLine>
    [[nodiscard]] constexpr auto distanceLInf(const OtherOrientedLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RayConcept OtherRay>
    [[nodiscard]] constexpr auto distanceLInf(const OtherRay& other) const;

    template <class ResultNumber = division_result_t<NumberType>, HalfplaneConcept OtherHalfplane>
    [[nodiscard]] constexpr auto distanceLInf(const OtherHalfplane& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] constexpr auto distanceLInf(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] constexpr auto distanceLInf(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] constexpr auto distanceLInf(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr auto distanceLInf(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
                  && requires(const OtherShape& o, const Polygon& self) {
                         o.template distanceLInf<ResultNumber>(self);
                     })
    [[nodiscard]] constexpr auto distanceLInf(const OtherShape& other) const {
        return other.template distanceLInf<ResultNumber>(*this);
    }

    template <class ResultNumber = double, PointConcept OtherPoint>
    [[nodiscard]] constexpr auto distanceLInf(const Shape<OtherPoint>& other) const {
        return other.template distanceLInf<ResultNumber>(*this);
    }

    template <class ResultNumber = NumberType, PointConcept OtherPoint>
    [[nodiscard]] constexpr std::optional<Point<ResultNumber, typename PointType::LabelType>>
    intersection(const OtherPoint& other) const;

    template <class ResultNumber = division_result_t<NumberType>, SegmentConcept OtherSegment>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Segment<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedSegmentConcept OtherOrientedSegment>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Segment<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherOrientedSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, LineConcept OtherLine>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Segment<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedLineConcept OtherOrientedLine>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Segment<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherOrientedLine& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RayConcept OtherRay>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Segment<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherRay& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolygonConcept OtherPolygon>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Polyline<Point<ResultNumber, typename PointType::LabelType>>, Polygon<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherPolygon& other) const;

    template <class ResultNumber = division_result_t<NumberType>, ConvexConcept OtherConvex>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Polyline<Point<ResultNumber, typename PointType::LabelType>>, Polygon<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherConvex& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Polyline<Point<ResultNumber, typename PointType::LabelType>>, Polygon<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Polyline<Point<ResultNumber, typename PointType::LabelType>>, Polygon<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, typename OtherShape>
        requires (!PointConcept<OtherShape>
                  && (detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
                  && requires(const OtherShape& o, const Polygon& self) {
                         o.template intersection<ResultNumber>(self);
                     })
    [[nodiscard]] auto intersection(const OtherShape& other) const {
        return other.template intersection<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, typename OtherShape>
        requires (!PointConcept<OtherShape>
                  && (detail::shapeRank<OtherShape> > detail::shapeRank<Polygon>)
                  && requires(const OtherShape& o, const Polygon& self) {
                         o.template regularizedIntersection<ResultNumber>(self);
                     })
    [[nodiscard]] auto regularizedIntersection(const OtherShape& other) const {
        return other.template regularizedIntersection<ResultNumber>(*this);
    }

    template <class ResultNumber = NumberType, class EmptyPoint>
    [[nodiscard]] constexpr EmptyShape<EmptyPoint> intersection(const EmptyShape<EmptyPoint>&) const {
        return {};
    }

    template <class ResultNumber = division_result_t<NumberType>, HalfplaneConcept OtherHalfplane>
    [[nodiscard]] constexpr std::vector<std::variant<Point<ResultNumber, typename PointType::LabelType>, Segment<Point<ResultNumber, typename PointType::LabelType>>, Polygon<Point<ResultNumber, typename PointType::LabelType>>>>
    intersection(const OtherHalfplane& other) const;

    template <class ResultNumber = division_result_t<NumberType>, PolylineConcept OtherPolyline>
    [[nodiscard]] constexpr auto intersection(const OtherPolyline& other) const;

    template <class ResultNumber = division_result_t<NumberType>, MonotoneChainConcept OtherChain>
    [[nodiscard]] constexpr auto intersection(const OtherChain& other) const;

    [[nodiscard]] constexpr Polygon rotated90(int k = 1) const;

    constexpr void rotate90(int k = 1);

    constexpr void untangle();

    template <class OtherNumber>
    [[nodiscard]] constexpr Polygon scaledUpX(const OtherNumber scalar) const;

    template <class OtherNumber>
    constexpr void scaleUpX(const OtherNumber scalar);

    template <class OtherNumber>
    [[nodiscard]] constexpr Polygon scaledUpY(const OtherNumber scalar) const;

    template <class OtherNumber>
    constexpr void scaleUpY(const OtherNumber scalar);

    template <class OtherNumber>
    [[nodiscard]] constexpr Polygon scaledDownX(const OtherNumber scalar) const;

    template <class OtherNumber>
    constexpr void scaleDownX(const OtherNumber scalar);

    template <class OtherNumber>
    [[nodiscard]] constexpr Polygon scaledDownY(const OtherNumber scalar) const;

    template <class OtherNumber>
    constexpr void scaleDownY(const OtherNumber scalar);

    template <class OtherShape>
        requires MinkowskiSummableConcept<Polygon<PointType_, TLabel>, OtherShape>
    [[nodiscard]] constexpr auto minkowskiSum(const OtherShape& other) const;

    template <class OtherShape>
        requires MinkowskiSummableConcept<Polygon<PointType_, TLabel>, OtherShape>
    [[nodiscard]] constexpr auto minkowskiErosion(const OtherShape& other) const;

    template <class ResultNumber = division_result_t<NumberType>, class OtherShape>
        requires (!MinkowskiSummableConcept<Polygon<PointType_, TLabel>, OtherShape> &&
                  BoundedPolygonalConcept<OtherShape>)
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType_::LabelType>>
    minkowskiErosion(const OtherShape& other) const;

    template<PointConcept OtherPoint>
    constexpr Polygon& operator+=(const OtherPoint& translation) {
        translation_ += translation;
        // A pure translation merely shifts the bounding box, so update the
        // cached bbox in place rather than discarding it. The hash, however,
        // depends on the absolute vertex positions, so it must be invalidated.
        if (!bbox_.empty()) {
            bbox_ += translation;
        }
        hash_ = hashUnset_;
        return *this;
    }

    template<PointConcept OtherPoint>
    constexpr Polygon& operator-=(const OtherPoint& translation) {
        translation_ -= translation;
        if (!bbox_.empty()) {
            bbox_ -= translation;
        }
        hash_ = hashUnset_;
        return *this;
    }

    template <class Scalar>
        requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
    constexpr Polygon& operator*=(const Scalar& scalar) {
        for (auto& vertex : points_) {
            vertex *= scalar;
        }
        translation_ *= scalar;
        normalize();
        resetCache();
        return *this;
    }

    template <class Scalar>
        requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
    constexpr Polygon& operator/=(const Scalar& scalar) {
        for (auto& vertex : points_) {
            vertex /= scalar;
        }
        translation_ /= scalar;
        normalize();
        resetCache();
        return *this;
    }

    template <bool Oriented>
    class BoundaryIterator {
      public:
        using iterator_category = std::forward_iterator_tag;
        using iterator_concept = std::forward_iterator_tag;
        using value_type = BoundaryType<Oriented>;
        using difference_type = std::ptrdiff_t;
        using reference = value_type;
 
        constexpr BoundaryIterator() = default;
 
        constexpr value_type operator*() const {
            assert(polygon != nullptr);
            return polygon->template boundaryAt<Oriented>(index);
        }
 
        constexpr BoundaryIterator& operator++() {
            ++index;
            return *this;
        }
 
        constexpr BoundaryIterator operator++(int) {
            BoundaryIterator copy(*this);
            ++(*this);
            return copy;
        }
 
        constexpr bool operator==(const BoundaryIterator& other) const = default;
 
      private:
        friend struct Polygon;
 
        constexpr BoundaryIterator(const Polygon* polygon_arg, std::size_t index_arg)
            : polygon(polygon_arg), index(index_arg) {}
 
        const Polygon* polygon = nullptr;
        std::size_t index = 0;
    };
 
  private:
    std::vector<PointType> points_{};
    [[no_unique_address]] mutable LabelType label_{};
    PointType translation_{};
    // Lazily computed bounding box, invalidated by resetCache() on every
    // mutation. The empty rectangle doubles as "not computed yet": a shape
    // whose box is genuinely empty has no vertices, so bbox() re-derives it
    // with one size check rather than any real work.
    mutable Rectangle<PointType> bbox_{};
 
    // Memoized hash, computed lazily by std::hash<Polygon>. hashUnset_ means "not
    // yet computed"; SIZE_MAX is chosen as the sentinel because it is a rare hash
    // output, and the one true hash that would collide with it is remapped to
    // hashUnset_ - 1 so the sentinel is never stored as a real value. Unlike the
    // bbox, the hash is not translation-invariant, so operator+=/-= reset it.
    static constexpr std::size_t hashUnset_ = pgl::detail::numeric_limits<std::size_t>::max();
    mutable std::size_t hash_ = hashUnset_;
    friend struct std::hash<Polygon>;
 
    // Drops the memoized caches; call after any operation that mutates the
    // polygon's vertices. A pure translation does not need to drop bbox_ (it
    // shifts in place, see operator+=), but it must still reset hash_, which
    // depends on the absolute vertex positions.
    constexpr void resetCache() const {
        bbox_ = {};
        hash_ = hashUnset_;
    }
 
    // Runtime implementation of untangle(), defined after IntervalTree is
    // available. The constexpr front-end keeps the allocation-free pairwise
    // implementation for constant evaluation.
    void untangleRuntime();
 
    template <bool Oriented>
    constexpr BoundaryType<Oriented> boundaryAt(std::size_t index) const {
        const auto i = static_cast<std::ptrdiff_t>(index);
        return BoundaryType<Oriented>(get(i), get(i + 1));
    }

    template <class ResultNumber, class OtherShape>
    constexpr ResultNumber edgeMinSquaredDistance(const OtherShape& other) const;

    template <class ResultNumber, class OtherShape>
    constexpr ResultNumber edgeMinDistanceL1(const OtherShape& other) const;

    template <class ResultNumber, class OtherShape>
    constexpr ResultNumber edgeMinDistanceLInf(const OtherShape& other) const;

    template <class ResultNumber = NumberType>
    constexpr ResultNumber signedTwiceArea() const {
        ResultNumber sum = 0;
        const std::size_t n = points_.size();
        for (std::size_t i = 0; i < n; ++i) {
            const auto& p1 = points_[i];
            const auto& p2 = points_[(i + 1) % n];
            sum += detail::asNumber<ResultNumber>(p1.x()) * detail::asNumber<ResultNumber>(p2.y())
                 - detail::asNumber<ResultNumber>(p2.x()) * detail::asNumber<ResultNumber>(p1.y());
        }
        return sum;
    }

    constexpr bool hasNoArea() const {
        using Exact = detail::promoted_number_t<NumberType>;
        return signedTwiceArea<Exact>() == Exact(0);
    }

    constexpr bool windsClockwise(std::size_t pivot) const {
        const std::size_t n = points_.size();
        const PointType& vertex = points_[pivot];
        std::size_t before = (pivot + n - 1) % n;
        while (before != pivot && points_[before] == vertex) {
            before = (before + n - 1) % n;
        }
        std::size_t after = (pivot + 1) % n;
        while (after != pivot && points_[after] == vertex) {
            after = (after + 1) % n;
        }
        return orientationSign(points_[before], vertex, points_[after]) < 0;
    }

    constexpr void normalize() {
        if (points_.empty()) {
            return;
        }
        auto minIt = std::min_element(points_.begin(), points_.end());
        if (points_.size() >= 3 &&
            windsClockwise(static_cast<std::size_t>(minIt - points_.begin()))) {
            // Reversing relocates the smallest vertex, so it is found again
            // rather than assumed to still sit where it did.
            std::reverse(points_.begin(), points_.end());
            minIt = std::min_element(points_.begin(), points_.end());
        }
        std::rotate(points_.begin(), minIt, points_.end());
    }

    template <class Poly>
    class BoundaryChains {
      public:
        using PT = typename Poly::PointType;
        using ChainView = MonotoneChainView<PT>;
 
        explicit BoundaryChains(const Poly& poly) : verts_(poly.vertices()) {
            n_ = verts_.size();
            buffer_.reserve(2 * n_);
 
            // Edge i (verts_[i] -> verts_[i+1]) ascends lexicographically.
            const auto ascends = [&](std::size_t i) { return verts_[i] < verts_[(i + 1) % n_]; };
 
            // Anchor at a break vertex (its incoming edge reverses) so no run
            // straddles index 0 ambiguously; record each run's first vertex and
            // direction. The run's last vertex is the next run's first.
            std::size_t start = 0;
            bool broke = false;
            for (std::size_t j = 0; j < n_; ++j) {
                if (ascends((j + n_ - 1) % n_) != ascends(j)) {
                    start = j;
                    broke = true;
                    break;
                }
            }
            if (!broke) {
                // Every edge is level, so all vertices coincide: the boundary
                // is a single point and decomposes into no monotone chain. A
                // polygon with two or more distinct vertices always reverses
                // direction somewhere, so this is the only way to get here.
                return;
            }
            std::size_t i = start;
            do {
                const bool up = ascends(i);
                runs_.push_back({i, up});
                std::size_t k = i;
                while (ascends(k) == up) {
                    k = (k + 1) % n_;
                }
                i = k;
            } while (i != start);
        }
 
        bool exhausted() const { return produced_ == runs_.size(); }
        const std::vector<ChainView>& produced() const { return chains_; }
 
        // Unrolls the next run into the shared buffer and returns its view.
        const ChainView& produceNext() {
            const auto [begin, up] = runs_[produced_];
            const std::size_t end = runs_[(produced_ + 1) % runs_.size()].first;  // inclusive
            const std::size_t bufStart = buffer_.size();
            std::size_t idx = begin;
            buffer_.push_back(verts_[idx]);
            while (idx != end) {
                idx = (idx + 1) % n_;
                buffer_.push_back(verts_[idx]);
            }
            const std::size_t len = buffer_.size() - bufStart;
            if (!up) {
                std::reverse(buffer_.begin() + static_cast<std::ptrdiff_t>(bufStart), buffer_.end());
            }
            chains_.emplace_back(std::span<const PT>(buffer_.data() + bufStart, len), /*trusted=*/true);
            ++produced_;
            return chains_.back();
        }
 
      private:
        std::vector<PT> verts_;                            // translated boundary vertices
        std::vector<PT> buffer_;                            // runs unrolled ascending, contiguous
        std::vector<std::pair<std::size_t, bool>> runs_;   // (first-vertex index, ascending?)
        std::vector<ChainView> chains_;                    // materialized views into buffer_
        std::size_t n_ = 0;
        std::size_t produced_ = 0;
    };
 
    class Iterator {
    private:
        std::vector<PointType>::const_iterator it;
        PointType x;
 
    public:
        using iterator_category = std::random_access_iterator_tag;
        using difference_type = std::ptrdiff_t;
        using value_type = PointType;
        using pointer = PointType*;
        using reference = PointType&;
 
        Iterator() = default;
        Iterator(std::vector<PointType>::const_iterator it, PointType x) : it(it), x(x) {}
 
        // Dereference returns value + x
        PointType operator*() const {
            return *it + x;
        }
 
        // Pre-increment
        Iterator& operator++() {
            ++it;
            return *this;
        }
 
        // Post-increment
        Iterator operator++(int) {
            Iterator tmp = *this;
            ++it;
            return tmp;
        }
 
        // Pre-decrement
        Iterator& operator--() {
            --it;
            return *this;
        }
 
        // Post-decrement
        Iterator operator--(int) {
            Iterator tmp = *this;
            --it;
            return tmp;
        }
 
        // Equality comparison
        bool operator==(const Iterator& other) const {
            return it == other.it;
        }
 
        // Other comparisons
        auto operator<=>(const Iterator& other) const {
            return it <=> other.it;
        }
 
        // Addition
        Iterator operator+(difference_type n) const {
            return Iterator(it + n, x);
        }
 
        // Subtraction
        Iterator operator-(difference_type n) const {
            return Iterator(it - n, x);
        }
 
        // Difference
        difference_type operator-(const Iterator& other) const {
            return it - other.it;
        }
 
        // Array subscript operator
        PointType operator[](difference_type n) const {
            return *(it + n) + x;
        }
    };
}; // struct Polygon
 
template <class PointType, class LabelType, class TranslationNumber, class TranslationLabel>
constexpr auto operator-(const Polygon<PointType, LabelType>& polygon, const Point<TranslationNumber, TranslationLabel>& translation) {
    return polygon + (-translation);
}
 
template <class PointType, class LabelType, class Scalar>
    requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
constexpr auto operator*(const Polygon<PointType, LabelType>& polygon, const Scalar& scalar) {
    using ResultPointType = Point<decltype(std::declval<PointType>().x() * scalar), typename PointType::LabelType>;
    Polygon<ResultPointType, LabelType> result(polygon);
    result *= scalar;
    if constexpr (detail::has_label_v<LabelType>) {
        result.label() = LabelType{};
    }
    return result;
}
 
template <class Scalar, class PointType, class LabelType>
    requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
constexpr auto operator*(const Scalar& scalar, const Polygon<PointType, LabelType>& polygon) {
    return polygon * scalar;
}
 
template <class PointType, class LabelType, class Scalar>
    requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
constexpr auto operator/(const Polygon<PointType, LabelType>& polygon, const Scalar& scalar) {
    using ResultPointType = Point<decltype(std::declval<PointType>().x() / scalar), typename PointType::LabelType>;
    Polygon<ResultPointType, LabelType> result(polygon);
    result /= scalar;
    if constexpr (detail::has_label_v<LabelType>) {
        result.label() = LabelType{};
    }
    return result;
}
 
template <class PointType, class LabelType>
std::ostream& operator<<(std::ostream& stream, const Polygon<PointType, LabelType>& polygon);
 
}  // namespace pgl