triangle.hpp

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#pragma once
 
#include "shape/rectangle.hpp"

 
#include <array>
#include <cassert>
#include <compare>
#include <concepts>
#include <cstddef>
#include <iterator>
#include <optional>
#include <vector>
#include <ostream>
#include <type_traits>
#include <utility>
 
 
namespace pgl {
 
template <class PointType = Point<>, class Label>
struct Triangle;
 
Triangle() -> Triangle<Point<>, NoLabel>;
 
template <class PointType>
Triangle(PointType, PointType, PointType) -> Triangle<PointType, NoLabel>;
 
template <class PointType, class A>
Triangle(PointType, PointType, PointType, A) -> Triangle<PointType, std::decay_t<A>>;
 
template <class Number>
Triangle(Number, Number, Number, Number, Number, Number) -> Triangle<Point<Number>, NoLabel>;

template <class PointType_, class TLabel>
struct Triangle {
    using PointType = PointType_;
    using NumberType = PointType::NumberType;
    using LabelType = TLabel;

    using value_type = PointType;
    using size_type = std::size_t;
    using difference_type = std::ptrdiff_t;
    using reference = const PointType&;
    using const_reference = const PointType&;
    using iterator = typename std::array<PointType, 3>::const_iterator;
    using const_iterator = iterator;
 
    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>;
 
    static_assert(detail::is_point_v<PointType>, "Triangle requires pgl::Point vertices");

    constexpr Triangle() = default;

    constexpr Triangle(PointType first, PointType second, PointType third)
        : points_(canonicalizeVertices(std::move(first), std::move(second), std::move(third))) {}

    constexpr Triangle(NumberType x1, NumberType y1, NumberType x2, NumberType y2, NumberType x3, NumberType y3)
        : Triangle(PointType(x1, y1), PointType(x2, y2), PointType(x3, y3)) {}

    template <class A>
        requires(detail::has_label_v<LabelType> && std::constructible_from<LabelType, A&&>)
    constexpr Triangle(PointType first, PointType second, PointType third, A&& label)
        : points_(canonicalizeVertices(std::move(first), std::move(second), std::move(third))),
          label_(std::forward<A>(label)) {}

    template <class A>
        requires(detail::has_label_v<LabelType> && std::constructible_from<LabelType, A&&>)
    constexpr Triangle(NumberType x1, NumberType y1, NumberType x2, NumberType y2,
                       NumberType x3, NumberType y3, A&& label)
        : Triangle(PointType(x1, y1), PointType(x2, y2), PointType(x3, y3), std::forward<A>(label)) {}

    template<PointConcept OtherPointType, class OtherLabelType>
        requires(std::constructible_from<PointType, const OtherPointType&>)
    constexpr Triangle(const Triangle<OtherPointType, OtherLabelType>& other)
        : Triangle(PointType(other.a()), PointType(other.b()), PointType(other.c())) {
        label_ = detail::copyLabel<LabelType>(other);
    }

    template<PointConcept OtherPointType, class OtherLabelType>
        requires(std::constructible_from<PointType, const OtherPointType&>)
    constexpr Triangle& operator=(const Triangle<OtherPointType, OtherLabelType>& other) {
        points_ = canonicalizeVertices(
            PointType(other.a()),
            PointType(other.b()),
            PointType(other.c()));
        label_ = detail::copyLabel<LabelType>(other);
        return *this;
    }

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

    static constexpr std::size_t size() {
        return 3;
    }

    constexpr const 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 const PointType& a() const {
        return points_[0];
    }

    constexpr const PointType& b() const {
        return points_[1];
    }

    constexpr const PointType& c() const {
        return points_[2];
    }

    constexpr auto begin() const {
        return points_.cbegin();
    }

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

    constexpr auto end() const {
        return points_.cend();
    }

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

    constexpr bool operator==(const Triangle& other) const {
        return points_ == other.points_;
    }

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

    constexpr auto operator<=>(const Triangle& other) const {
        return points_ <=> other.points_;
    }

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

    [[nodiscard]] constexpr NumberType twiceArea() const;

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

    [[nodiscard]] constexpr bool isDegenerate() const;

    [[nodiscard]] constexpr bool isPoint() const;

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

    [[nodiscard]] constexpr bool isSegment() const;

    [[nodiscard]] constexpr std::optional<BoundaryType<false>> getIfSegment() const;

    [[nodiscard]] constexpr bool isUndefined() const;

    [[nodiscard]] constexpr 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>
    [[nodiscard]] constexpr Rectangle<Point<ResultNumber>> fbox() const;

    [[nodiscard]] constexpr std::array<PointType, 3> vertices() const;

    [[nodiscard]] constexpr std::array<Segment<PointType>, 3> edges() const;

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

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

    [[nodiscard]] constexpr std::array<OrientedSegment<PointType>, 3> orientedEdges() const;

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

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

    [[nodiscard]] constexpr explicit operator Convex<PointType>() const {
        return Convex<PointType>(*this, !isDegenerate());
    }

    [[nodiscard]] constexpr Convex<PointType> asConvex() const {
        return static_cast<Convex<PointType>>(*this);
    }

    [[nodiscard]] constexpr Convex<PointType> convexHull() const {
        return asConvex();
    }

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

    [[nodiscard]] constexpr explicit operator Polygon<PointType>() const {
        return Polygon<PointType>(*this, !isDegenerate());
    }

    [[nodiscard]] constexpr Polygon<PointType> asPolygon() const {
        return static_cast<Polygon<PointType>>(*this);
    }

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

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

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

    constexpr void rotate90(int k = 1);

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

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

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

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

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

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

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

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

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

    [[nodiscard]] constexpr Disk<PointType, NoLabel> circumcircle() const;

    [[nodiscard]] constexpr Segment<PointType> diameter() const;

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

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

    [[nodiscard]] constexpr bool isRectangle() const;

    [[nodiscard]] constexpr bool isObtuse() const;

    [[nodiscard]] constexpr bool isIsosceles() const;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    [[nodiscard]] constexpr bool contains(const Shape<PointType>& other) const;

    [[nodiscard]] constexpr bool boundaryContains(const Shape<PointType>& other) const;
 
    // The empty set is a subset of every shape (contained in all of them) and
    // disjoint from all of them, so containment is true while separation is
    // false. These overloads let an EmptyShape flow through Shape's variant
    // dispatch without special-casing.
    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 <class EmptyPoint>
    [[nodiscard]] constexpr bool separates(const EmptyShape<EmptyPoint>&) const {
        return false;
    }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    [[nodiscard]] constexpr bool intersects(const Shape<PointType>& other) const;

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

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

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

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

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

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

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

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

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

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

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

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

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

    [[nodiscard]] constexpr bool interiorsIntersect(const Shape<PointType>& other) const;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    [[nodiscard]] constexpr bool crosses(const Shape<PointType>& other) const;

    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>, LineConcept OtherLine>
    [[nodiscard]] constexpr std::optional<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::optional<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>, SegmentConcept OtherSegment>
    [[nodiscard]] constexpr std::optional<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::optional<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>, RayConcept OtherRay>
    [[nodiscard]] constexpr std::optional<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>, HalfplaneConcept OtherHalfplane>
    [[nodiscard]] constexpr auto intersection(const OtherHalfplane& other) const;

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

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

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

    template <class ResultNumber = division_result_t<NumberType>, typename OtherShape>
        requires (!PointConcept<OtherShape>
                  && !HalfplaneIntersectionConcept<OtherShape>
                  && (detail::shapeRank<OtherShape> > detail::shapeRank<Triangle>)
                  && requires(const OtherShape& o, const Triangle& self) {
                         o.template intersection<ResultNumber>(self);
                     })
    [[nodiscard]] constexpr 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<Triangle>)
                  && requires(const OtherShape& o, const Triangle& self) {
                         o.template regularizedIntersection<ResultNumber>(self);
                     })
    [[nodiscard]] constexpr 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>, 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>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Triangle>)
                  && requires(const OtherShape& o, const Triangle& 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 {
        return disk.template squaredDistance<ResultNumber>(*this);
    }

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

    template <class ResultNumber = division_result_t<NumberType>, class OtherShape>
        requires detail::ClosestPointsPairConcept<Triangle<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>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Triangle>)
                  && requires(const OtherShape& o, const Triangle& 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>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Triangle>)
                  && requires(const OtherShape& o, const Triangle& 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 auto hausdorffDistanceL1(const OtherPoint& point) const;

    template <class ResultNumber = division_result_t<NumberType>, SegmentConcept OtherSegment>
    [[nodiscard]] constexpr auto hausdorffDistanceL1(const OtherSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, OrientedSegmentConcept OtherOrientedSegment>
    [[nodiscard]] constexpr auto hausdorffDistanceL1(const OtherOrientedSegment& other) const;

    template <class ResultNumber = division_result_t<NumberType>, RectangleConcept OtherRectangle>
    [[nodiscard]] constexpr auto hausdorffDistanceL1(const OtherRectangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, TriangleConcept OtherTriangle>
    [[nodiscard]] constexpr auto hausdorffDistanceL1(const OtherTriangle& other) const;

    template <class ResultNumber = division_result_t<NumberType>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Triangle>)
                  && requires(const OtherShape& o, const Triangle& self) {
                         o.template hausdorffDistanceL1<ResultNumber>(self);
                     })
    [[nodiscard]] constexpr auto hausdorffDistanceL1(const OtherShape& other) const {
        return other.template hausdorffDistanceL1<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PointConcept OtherPoint>
    [[nodiscard]] constexpr auto hausdorffDistanceL1(const Shape<OtherPoint>& other) const {
        return other.template hausdorffDistanceL1<ResultNumber>(*this);
    }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    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>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Triangle>)
                  && requires(const OtherShape& o, const Triangle& self) {
                         o.template regularizedUnion<ResultNumber>(self);
                     })
    [[nodiscard]] auto regularizedUnion(const OtherShape& other) const {
        return other.template regularizedUnion<ResultNumber>(*this);
    }

    template <class ResultNumber = division_result_t<NumberType>, PolygonalRegionConcept OtherRegion>
    [[nodiscard]] PolygonSet<Point<ResultNumber, typename PointType::LabelType>>
    difference(const OtherRegion& 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>, 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>, typename OtherShape>
        requires ((detail::shapeRank<OtherShape> > detail::shapeRank<Triangle>)
                  && requires(const OtherShape& o, const Triangle& self) {
                         o.template symmetricDifference<ResultNumber>(self);
                     })
    [[nodiscard]] auto symmetricDifference(const OtherShape& other) const {
        return other.template symmetricDifference<ResultNumber>(*this);
    }

    template<PointConcept OtherPoint>
    constexpr Triangle& operator+=(const OtherPoint& translation);

    template<PointConcept OtherPoint>
    constexpr Triangle& operator-=(const OtherPoint& translation);

    template <class Scalar>
        requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
    constexpr Triangle& operator*=(const Scalar& scalar);

    template <class Scalar>
        requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
    constexpr Triangle& operator/=(const Scalar& scalar);
 
    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(triangle != nullptr);
            return triangle->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 Triangle;
 
        constexpr BoundaryIterator(const Triangle* triangle_arg, std::size_t index_arg)
            : triangle(triangle_arg), index(index_arg) {}
 
        const Triangle* triangle = nullptr;
        std::size_t index = 0;
    };
 
  private:
    static constexpr std::size_t edgeCount = 3;

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

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

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

    template <class ResultNumber, class OtherShape>
    constexpr ResultNumber vertexMinDistanceL1(const OtherShape& other) const;

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

    template <class ResultNumber, class OtherShape>
    constexpr ResultNumber vertexMinDistanceLInf(const OtherShape& other) const;
 
    static constexpr std::array<PointType, 3> canonicalizeVertices(PointType first, PointType second, PointType third) {
        std::array<PointType, 3> vertices{
            std::move(first),
            std::move(second),
            std::move(third),
        };
 
        if (vertices[1] < vertices[0]) {
            std::swap(vertices[0], vertices[1]);
        }
        if (vertices[2] < vertices[1]) {
            std::swap(vertices[1], vertices[2]);
        }
        if (vertices[1] < vertices[0]) {
            std::swap(vertices[0], vertices[1]);
        }
 
        if (orientationDeterminant(vertices[0], vertices[1], vertices[2]) < 0) {
            std::swap(vertices[1], vertices[2]);
        }
 
        return vertices;
    }
 
    constexpr void normalize() {
        points_ = canonicalizeVertices(points_[0], points_[1], points_[2]);
    }
 
    template <bool Oriented>
    constexpr BoundaryType<Oriented> boundaryAt(std::size_t index) const {
        assert(index < edgeCount);
 
        switch (index) {
        case 0:
            return BoundaryType<Oriented>(a(), b());
        case 1:
            return BoundaryType<Oriented>(b(), c());
        default:
            return BoundaryType<Oriented>(c(), a());
        }
    }
 
    std::array<PointType, 3> points_{};
    [[no_unique_address]] mutable LabelType label_{};
};

template <class PointType, class LabelType, class TranslationNumber, class TranslationLabel>
constexpr auto operator-(const Triangle<PointType, LabelType>& triangle, const Point<TranslationNumber, TranslationLabel>& translation);

template <class PointType, class LabelType, class Scalar>
    requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
constexpr auto operator*(const Triangle<PointType, LabelType>& triangle, const Scalar& scalar);

template <class Scalar, class PointType, class LabelType>
    requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
constexpr auto operator*(const Scalar& scalar, const Triangle<PointType, LabelType>& triangle);

template <class PointType, class LabelType, class Scalar>
    requires(!detail::is_point_v<Scalar> && !TransformationConcept<Scalar>)
constexpr auto operator/(const Triangle<PointType, LabelType>& triangle, const Scalar& scalar);

template <class PointType, class LabelType>
std::ostream& operator<<(std::ostream& stream, const Triangle<PointType, LabelType>& triangle);
 
}  // namespace pgl