Counter clockwise function (tools/ccw.hpp)

template <typename T>
int ccw(tools::vector2<T> a, tools::vector2<T> b, tools::vector2<T> c);

It returns

\[\begin{align*} \left\{\begin{array}{ll} +1 & \text{(if $a$, $b$ and $c$ make a counterclockwise turn on the regular orthogonal coordinate system ($(\infty, 0)$ is on the right side and $(0, \infty)$ is on the upper side))}\\ -1 & \text{(if $a$, $b$ and $c$ make a clockwise turn on the regular orthogonal coordinate system)}\\ +2 & \text{(if $c$ - $a$ - $b$ are on a line in this order ($c \neq a$))}\\ -2 & \text{(if $a$ - $b$ - $c$ are on a line in this order ($c \neq b$))}\\ 0 & \text{(otherwise; i.e., $a$ - $c$ - $b$ are on a line in this order, $c = a$, $c = b$ or $a = b$)} \end{array}\right.& \end{align*}\]

Constraints

None

Time Complexity

$O(1)$

References

Spaghetti Source (ccw) - 点の進行方向

License

Author

anqooqie

Depends on

Required by

Verified with

Code

#ifndef TOOLS_CCW_HPP
#define TOOLS_CCW_HPP

#include "tools/vector2.hpp"

namespace tools {
  template <typename T>
  int ccw(const ::tools::vector2<T>& a, ::tools::vector2<T> b, ::tools::vector2<T> c) {
    b -= a;
    c -= a;
    if (b.outer_product(c) > T(0)) return +1;
    if (b.outer_product(c) < T(0)) return -1;
    if (b.inner_product(c) < T(0)) return +2;
    if (b.squared_l2_norm() < c.squared_l2_norm()) return -2;
    return 0;
  }
}

#endif

#line 1 "tools/ccw.hpp"



#line 1 "tools/vector2.hpp"



#line 1 "tools/vector.hpp"



#include <cstddef>
#include <array>
#include <initializer_list>
#include <cassert>
#include <limits>
#include <vector>
#include <type_traits>
#include <iterator>
#include <algorithm>
#include <utility>
#include <cmath>
#include <iostream>
#include <string>
#include <functional>
#include <tuple>
#line 1 "tools/abs.hpp"



namespace tools {
  constexpr float abs(const float x) {
    return x < 0 ? -x : x;
  }
  constexpr double abs(const double x) {
    return x < 0 ? -x : x;
  }
  constexpr long double abs(const long double x) {
    return x < 0 ? -x : x;
  }
  constexpr int abs(const int x) {
    return x < 0 ? -x : x;
  }
  constexpr long abs(const long x) {
    return x < 0 ? -x : x;
  }
  constexpr long long abs(const long long x) {
    return x < 0 ? -x : x;
  }
  constexpr unsigned int abs(const unsigned int x) {
    return x;
  }
  constexpr unsigned long abs(const unsigned long x) {
    return x;
  }
  constexpr unsigned long long abs(const unsigned long long x) {
    return x;
  }
}


#line 1 "tools/tuple_hash.hpp"



#line 1 "tools/now.hpp"



#include <chrono>

namespace tools {
  inline long long now() {
    return ::std::chrono::duration_cast<::std::chrono::nanoseconds>(::std::chrono::high_resolution_clock::now().time_since_epoch()).count();
  }
}


#line 1 "tools/hash_combine.hpp"



#line 6 "tools/hash_combine.hpp"

// Source: https://github.com/google/cityhash/blob/f5dc54147fcce12cefd16548c8e760d68ac04226/src/city.h
// License: MIT
// Author: Google Inc.

// Copyright (c) 2011 Google, Inc.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.

namespace tools {
  template <typename T>
  void hash_combine(::std::size_t& seed, const T& v) {
    static const ::std::hash<T> hasher;
    static constexpr ::std::size_t k_mul = 0x9ddfea08eb382d69ULL;
    ::std::size_t a = (hasher(v) ^ seed) * k_mul;
    a ^= (a >> 47);
    ::std::size_t b = (seed ^ a) * k_mul;
    b ^= (b >> 47);
    seed = b * k_mul;
  }
}


#line 11 "tools/tuple_hash.hpp"

namespace tools {
  template <typename... Ts>
  struct tuple_hash {
    template <::std::size_t I = sizeof...(Ts) - 1>
    ::std::size_t operator()(const ::std::tuple<Ts...>& key) const {
      if constexpr (I == ::std::numeric_limits<::std::size_t>::max()) {
        static const ::std::size_t seed = ::tools::now();
        return seed;
      } else {
        ::std::size_t seed = this->operator()<I - 1>(key);
        ::tools::hash_combine(seed, ::std::get<I>(key));
        return seed;
      }
    }
  };
}


#line 21 "tools/vector.hpp"

namespace tools {
  namespace detail {
    namespace vector {
      template <typename T, ::std::size_t N>
      class members {
      protected:
        constexpr static bool variable_sized = false;
        constexpr static bool has_aliases = false;
        ::std::array<T, N> m_values;
        members() : m_values() {}
        members(const ::std::initializer_list<T> il) : m_values(il) {
          assert(il.size() == N);
        }
      };

      template <typename T>
      class members<T, 2> {
      protected:
        constexpr static bool variable_sized = false;
        constexpr static bool has_aliases = true;
        ::std::array<T*, 2> m_values;
        members() : m_values{&this->x, &this->y} {}
        members(const T& x, const T& y) : m_values{&this->x, &this->y}, x(x), y(y) {}
        members(const ::std::initializer_list<T> il) : m_values{&this->x, &this->y}, x(il.begin()[0]), y(il.begin()[1]) {
          assert(il.size() == 2);
        }

      public:
        T x;
        T y;
      };

      template <typename T>
      class members<T, 3> {
      protected:
        constexpr static bool variable_sized = false;
        constexpr static bool has_aliases = true;
        ::std::array<T*, 3> m_values;
        members() : m_values{&this->x, &this->y, &this->z} {}
        members(const T& x, const T& y, const T& z) : m_values{&this->x, &this->y, &this->z}, x(x), y(y), z(z) {}
        members(const ::std::initializer_list<T> il) : m_values{&this->x, &this->y, &this->z}, x(il.begin()[0]), y(il.begin()[1]), z(il.begin()[2]) {
          assert(il.size() == 3);
        }

      public:
        T x;
        T y;
        T z;
      };

      template <typename T>
      class members<T, 4> {
      protected:
        constexpr static bool variable_sized = false;
        constexpr static bool has_aliases = true;
        ::std::array<T*, 4> m_values;
        members() : m_values{&this->x, &this->y, &this->z, &this->w} {}
        members(const T& x, const T& y, const T& z, const T& w) : m_values{&this->x, &this->y, &this->z, &this->w}, x(x), y(y), z(z), w(w) {}
        members(const ::std::initializer_list<T> il) : m_values{&this->x, &this->y, &this->z, &this->w}, x(il.begin()[0]), y(il.begin()[1]), z(il.begin()[2]), w(il.begin()[3]) {
          assert(il.size() == 4);
        }

      public:
        T x;
        T y;
        T z;
        T w;
      };

      template <typename T>
      class members<T, ::std::numeric_limits<::std::size_t>::max()> {
      protected:
        constexpr static bool variable_sized = true;
        constexpr static bool has_aliases = false;
        ::std::vector<T> m_values;
        members() = default;
        members(const ::std::size_t n) : m_values(n) {}
        members(const ::std::size_t n, const T& value) : m_values(n, value) {}
        template <typename InputIter>
        members(const InputIter first, const InputIter last) : m_values(first, last) {}
        members(const ::std::initializer_list<T> il) : m_values(il) {}
      };
    }
  }

  template <typename T, ::std::size_t N = ::std::numeric_limits<::std::size_t>::max()>
  class vector : public ::tools::detail::vector::members<T, N> {
  private:
    using Base = ::tools::detail::vector::members<T, N>;
    using F = ::std::conditional_t<::std::is_floating_point_v<T>, T, double>;
    using V = ::tools::vector<T, N>;
    constexpr static bool variable_sized = Base::variable_sized;
    constexpr static bool has_aliases = Base::has_aliases;

  public:
    using reference = T&;
    using const_reference = const T&;
    using size_type = ::std::size_t;
    using difference_type = ::std::ptrdiff_t;
    using pointer = T*;
    using const_pointer = const T*;
    using value_type = T;
    class iterator {
    private:
      V* m_parent;
      size_type m_i;

    public:
      using difference_type = ::std::ptrdiff_t;
      using value_type = T;
      using reference = T&;
      using pointer = T*;
      using iterator_category = ::std::random_access_iterator_tag;

      iterator() = default;
      iterator(const iterator&) = default;
      iterator(iterator&&) = default;
      ~iterator() = default;
      iterator& operator=(const iterator&) = default;
      iterator& operator=(iterator&&) = default;

      iterator(V * const parent, const size_type i) : m_parent(parent), m_i(i) {}

      reference operator*() const {
        return (*this->m_parent)[this->m_i];
      }
      pointer operator->() const {
        return &(*(*this));
      }

      iterator& operator++() {
        ++this->m_i;
        return *this;
      }
      iterator operator++(int) {
        const auto self = *this;
        ++*this;
        return self;
      }
      iterator& operator--() {
        --this->m_i;
        return *this;
      }
      iterator operator--(int) {
        const auto self = *this;
        --*this;
        return self;
      }
      iterator& operator+=(const difference_type n) {
        this->m_i += n;
        return *this;
      }
      iterator& operator-=(const difference_type n) {
        this->m_i -= n;
        return *this;
      }
      friend iterator operator+(const iterator& self, const difference_type n) {
        return iterator(self) += n;
      }
      friend iterator operator+(const difference_type n, const iterator& self) {
        return iterator(self) += n;
      }
      friend iterator operator-(const iterator& self, const difference_type n) {
        return iterator(self) -= n;
      }
      friend difference_type operator-(const iterator& lhs, const iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return static_cast<difference_type>(lhs.m_i) - static_cast<difference_type>(rhs.m_i);
      }
      reference operator[](const difference_type n) const {
        return *(*this + n);
      }

      friend bool operator==(const iterator& lhs, const iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i == rhs.m_i;
      }
      friend bool operator!=(const iterator& lhs, const iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i != rhs.m_i;
      }
      friend bool operator<(const iterator& lhs, const iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i < rhs.m_i;
      }
      friend bool operator<=(const iterator& lhs, const iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i <= rhs.m_i;
      }
      friend bool operator>(const iterator& lhs, const iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i > rhs.m_i;
      }
      friend bool operator>=(const iterator& lhs, const iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i >= rhs.m_i;
      }
    };
    class const_iterator {
    private:
      const V *m_parent;
      size_type m_i;

    public:
      using difference_type = ::std::ptrdiff_t;
      using value_type = T;
      using reference = const T&;
      using pointer = const T*;
      using iterator_category = ::std::random_access_iterator_tag;

      const_iterator() = default;
      const_iterator(const const_iterator&) = default;
      const_iterator(const_iterator&&) = default;
      ~const_iterator() = default;
      const_iterator& operator=(const const_iterator&) = default;
      const_iterator& operator=(const_iterator&&) = default;

      const_iterator(const V * const parent, const size_type i) : m_parent(parent), m_i(i) {}

      reference operator*() const {
        return (*this->m_parent)[this->m_i];
      }
      pointer operator->() const {
        return &(*(*this));
      }

      const_iterator& operator++() {
        ++this->m_i;
        return *this;
      }
      const_iterator operator++(int) {
        const auto self = *this;
        ++*this;
        return self;
      }
      const_iterator& operator--() {
        --this->m_i;
        return *this;
      }
      const_iterator operator--(int) {
        const auto self = *this;
        --*this;
        return self;
      }
      const_iterator& operator+=(const difference_type n) {
        this->m_i += n;
        return *this;
      }
      const_iterator& operator-=(const difference_type n) {
        this->m_i -= n;
        return *this;
      }
      friend const_iterator operator+(const const_iterator& self, const difference_type n) {
        return const_iterator(self) += n;
      }
      friend const_iterator operator+(const difference_type n, const const_iterator& self) {
        return const_iterator(self) += n;
      }
      friend const_iterator operator-(const const_iterator& self, const difference_type n) {
        return const_iterator(self) -= n;
      }
      friend difference_type operator-(const const_iterator& lhs, const const_iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return static_cast<difference_type>(lhs.m_i) - static_cast<difference_type>(rhs.m_i);
      }
      reference operator[](const difference_type n) const {
        return *(*this + n);
      }

      friend bool operator==(const const_iterator& lhs, const const_iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i == rhs.m_i;
      }
      friend bool operator!=(const const_iterator& lhs, const const_iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i != rhs.m_i;
      }
      friend bool operator<(const const_iterator& lhs, const const_iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i < rhs.m_i;
      }
      friend bool operator<=(const const_iterator& lhs, const const_iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i <= rhs.m_i;
      }
      friend bool operator>(const const_iterator& lhs, const const_iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i > rhs.m_i;
      }
      friend bool operator>=(const const_iterator& lhs, const const_iterator& rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i >= rhs.m_i;
      }
    };
    using reverse_iterator = ::std::reverse_iterator<iterator>;
    using const_reverse_iterator = ::std::reverse_iterator<const_iterator>;

    vector() = default;
    vector(const V& other) : Base() {
      if constexpr (has_aliases) {
        ::std::copy(other.begin(), other.end(), this->begin());
      } else {
        this->m_values = other.m_values;
      }
    }
    vector(V&& other) noexcept {
      if constexpr (has_aliases) {
        ::std::copy(other.begin(), other.end(), this->begin());
      } else {
        this->m_values = ::std::move(other.m_values);
      }
    }
    ~vector() = default;
    V& operator=(const V& other) {
      if constexpr (has_aliases) {
        ::std::copy(other.begin(), other.end(), this->begin());
      } else {
        this->m_values = other.m_values;
      }
      return *this;
    }
    V& operator=(V&& other) noexcept {
      if constexpr (has_aliases) {
        ::std::copy(other.begin(), other.end(), this->begin());
      } else {
        this->m_values = ::std::move(other.m_values);
      }
      return *this;
    }

    template <bool SFINAE = variable_sized, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    explicit vector(size_type n) : Base(n) {}
    template <bool SFINAE = variable_sized, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    vector(size_type n, const_reference value) : Base(n, value) {}
    template <typename InputIter, bool SFINAE = variable_sized, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    vector(const InputIter first, const InputIter last) : Base(first, last) {}
    template <bool SFINAE = N == 2, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    vector(const T& x, const T& y) : Base(x, y) {}
    template <bool SFINAE = N == 3, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    vector(const T& x, const T& y, const T& z) : Base(x, y, z) {}
    template <bool SFINAE = N == 4, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    vector(const T& x, const T& y, const T& z, const T& w) : Base(x, y, z, w) {}
    vector(const ::std::initializer_list<T> il) : Base(il) {}

    iterator begin() noexcept { return iterator(this, 0); }
    const_iterator begin() const noexcept { return const_iterator(this, 0); }
    const_iterator cbegin() const noexcept { return const_iterator(this, 0); }
    iterator end() noexcept { return iterator(this, this->size()); }
    const_iterator end() const noexcept { return const_iterator(this, this->size()); }
    const_iterator cend() const noexcept { return const_iterator(this, this->size()); }
    reverse_iterator rbegin() noexcept { return ::std::make_reverse_iterator(this->end()); }
    const_reverse_iterator rbegin() const noexcept { return ::std::make_reverse_iterator(this->end()); }
    const_reverse_iterator crbegin() const noexcept { return ::std::make_reverse_iterator(this->cend()); }
    reverse_iterator rend() noexcept { return ::std::make_reverse_iterator(this->begin()); }
    const_reverse_iterator rend() const noexcept { return ::std::make_reverse_iterator(this->begin()); }
    const_reverse_iterator crend() const noexcept { return ::std::make_reverse_iterator(this->cbegin()); }

    size_type size() const noexcept { return this->m_values.size(); }
    bool empty() const noexcept { return this->m_values.empty(); }

    reference operator[](const size_type n) {
      if constexpr (has_aliases) {
        return *this->m_values[n];
      } else {
        return this->m_values[n];
      }
    }
    const_reference operator[](const size_type n) const {
      if constexpr (has_aliases) {
        return *this->m_values[n];
      } else {
        return this->m_values[n];
      }
    }
    reference front() { return *this->begin(); }
    const_reference front() const { return *this->begin(); }
    reference back() { return *this->rbegin(); }
    const_reference back() const { return *this->rbegin(); }

    V operator+() const {
      return *this;
    }
    V operator-() const {
      V res = *this;
      for (auto& v : res) v = -v;
      return res;
    }
    V& operator+=(const V& other) {
      assert(this->size() == other.size());
      for (::std::size_t i = 0; i < this->size(); ++i) {
        (*this)[i] += other[i];
      }
      return *this;
    }
    friend V operator+(const V& lhs, const V& rhs) {
      return V(lhs) += rhs;
    }
    V& operator-=(const V& other) {
      assert(this->size() == other.size());
      for (::std::size_t i = 0; i < this->size(); ++i) {
        (*this)[i] -= other[i];
      }
      return *this;
    }
    friend V operator-(const V& lhs, const V& rhs) {
      return V(lhs) -= rhs;
    }
    V& operator*=(const T& c) {
      for (auto& v : *this) v *= c;
      return *this;
    }
    friend V operator*(const T& lhs, const V& rhs) {
      return V(rhs) *= lhs;
    }
    friend V operator*(const V& lhs, const T& rhs) {
      return V(lhs) *= rhs;
    }
    V& operator/=(const T& c) {
      for (auto& v : *this) v /= c;
      return *this;
    }
    friend V operator/(const V& lhs, const T& rhs) {
      return V(lhs) /= rhs;
    }

    friend bool operator==(const V& lhs, const V& rhs) {
      return ::std::equal(lhs.begin(), lhs.end(), rhs.begin(), rhs.end());
    }
    friend bool operator!=(const V& lhs, const V& rhs) {
      return !(lhs == rhs);
    }

    T inner_product(const V& other) const {
      assert(this->size() == other.size());
      T res{};
      for (::std::size_t i = 0; i < this->size(); ++i) {
        res += (*this)[i] * other[i];
      }
      return res;
    }
    T l1_norm() const {
      T res{};
      for (const auto& v : *this) {
        res += ::tools::abs(v);
      }
      return res;
    }
    T squared_l2_norm() const {
      return this->inner_product(*this);
    }
    F l2_norm() const {
      return ::std::sqrt(static_cast<F>(this->squared_l2_norm()));
    }
    template <bool SFINAE = ::std::is_floating_point_v<T>, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    V normalized() const {
      return *this / this->l2_norm();
    }

    friend ::std::ostream& operator<<(::std::ostream& os, const V& self) {
      os << '(';
      ::std::string delimiter = "";
      for (const auto& v : self) {
        os << delimiter << v;
        delimiter = ", ";
      }
      return os << ')';
    }
    friend ::std::istream& operator>>(::std::istream& is, V& self) {
      for (auto& v : self) {
        is >> v;
      }
      return is;
    }

    template <bool SFINAE = N == 2, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    T outer_product(const V& other) const {
      return this->x * other.y - this->y * other.x;
    }
    template <bool SFINAE = N == 2, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    V turned90() const {
      return V{-this->y, this->x};
    }
    template <bool SFINAE = N == 2, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    V turned270() const {
      return V{this->y, -this->x};
    }

    template <bool SFINAE = N == 2, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    static const ::std::array<V, 4>& four_directions() {
      static const ::std::array<V, 4> res = {
        V{T(1), T(0)},
        V{T(0), T(1)},
        V{T(-1), T(0)},
        V{T(0), T(-1)}
      };
      return res;
    }
    template <bool SFINAE = N == 2, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    static const ::std::array<V, 8>& eight_directions() {
      static const ::std::array<V, 8> res = {
        V{T(1), T(0)},
        V{T(1), T(1)},
        V{T(0), T(1)},
        V{T(-1), T(1)},
        V{T(-1), T(0)},
        V{T(-1), T(-1)},
        V{T(0), T(-1)},
        V{T(1), T(-1)}
      };
      return res;
    }

    template <bool SFINAE = N == 3, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    V outer_product(const V& other) const {
      return V{this->y * other.z - this->z * other.y, this->z * other.x - this->x * other.z, this->x * other.y - this->y * other.x};
    }
    template <bool SFINAE = N == 3 && ::std::is_floating_point_v<T>, ::std::enable_if_t<SFINAE, ::std::nullptr_t> = nullptr>
    ::std::array<V, 3> orthonormal_basis() const {
      assert((*this != V{0, 0, 0}));

      ::std::array<V, 3> v;
      v[0] = *this;
      v[1] = V{0, this->z, -this->y};
      if (v[1] == V{0, 0, 0}) {
        v[1] = V{-this->z, 0, this->x};
      }
      v[1] -= v[0].inner_product(v[1]) / v[0].inner_product(v[0]) * v[0];

      v[0] = v[0].normalized();
      v[1] = v[1].normalized();
      v[2] = v[0].outer_product(v[1]);

      return v;
    }
  };
}

namespace std {
  template <typename T>
  struct hash<::tools::vector<T, 2>> {
    using result_type = ::std::size_t;
    using argument_type = ::tools::vector<T, 2>;
    result_type operator()(const argument_type& key) const {
      static const ::tools::tuple_hash<T, T> hasher;
      return hasher(::std::make_tuple(key.x, key.y));
    }
  };
  template <typename T>
  struct hash<::tools::vector<T, 3>> {
    using result_type = ::std::size_t;
    using argument_type = ::tools::vector<T, 3>;
    result_type operator()(const argument_type& key) const {
      static const ::tools::tuple_hash<T, T, T> hasher;
      return hasher(::std::make_tuple(key.x, key.y, key.z));
    }
  };
  template <typename T>
  struct hash<::tools::vector<T, 4>> {
    using result_type = ::std::size_t;
    using argument_type = ::tools::vector<T, 4>;
    result_type operator()(const argument_type& key) const {
      static const ::tools::tuple_hash<T, T, T, T> hasher;
      return hasher(::std::make_tuple(key.x, key.y, key.z, key.w));
    }
  };
}


#line 5 "tools/vector2.hpp"

namespace tools {
  template <typename T>
  using vector2 = ::tools::vector<T, 2>;
}


#line 5 "tools/ccw.hpp"

namespace tools {
  template <typename T>
  int ccw(const ::tools::vector2<T>& a, ::tools::vector2<T> b, ::tools::vector2<T> c) {
    b -= a;
    c -= a;
    if (b.outer_product(c) > T(0)) return +1;
    if (b.outer_product(c) < T(0)) return -1;
    if (b.inner_product(c) < T(0)) return +2;
    if (b.squared_l2_norm() < c.squared_l2_norm()) return -2;
    return 0;
  }
}