7.1.1 Point

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/*

A 2D point class templated on the coordinate type `T`. Algebraic operations preserve `T`, while
metric operations use `fp_t`, which is `T` for floating-point coordinates and `double` otherwise.

Non-floating-point coordinate types such as `Modular` or `Rational` compose for exact operations
(arithmetic, `dot`, `cross`, `sqnorm`, comparisons, and `EQ`) using the coordinate type's own
operators. Their metric operations convert coordinates explicitly to `double`; for `Modular`, this
uses the stored representative and is not finite-field geometry.

- `TPoint<T>()` constructs the origin, while `TPoint<T>(x, y)` constructs point $(x, y)$. A pair of
  values can also be converted explicitly to a point.
- Operators `+`, `-`, `*`, `/`, and their compound forms act element-wise on points or apply a
  scalar. Comparisons are exact and lexicographic, as required by standard algorithms and
  containers. `EQ(p, q)` instead uses `EPS` for floating-point coordinates and remains exact for
  other coordinate types.
- `sqnorm()`, `dot(p)`, and `cross(p)` return the exact algebraic results in coordinate type `T`.
- `norm()`, `arg()`, `proj(p)`, and `normalize()` return metric results in `fp_t`; division also
  promotes to `fp_t` to avoid integer truncation.
- `rotate90()`, `rotate180()`, and `rotate270()` rotate by the corresponding counter-clockwise
  cardinal angle; overloads taking `p` rotate about point `p`.
- `rotate_cw(t)` and `rotate_ccw(t)` rotate by an arbitrary angle `t` in radians; overloads taking
  (`p`, `t`) rotate about point `p`.
- `reflect(p)` reflects across point `p`, while `reflect(p, q)` reflects across the line through
  points `p` and `q`.
- `to_double()` and `to_ldouble()` explicitly convert the coordinate type. Integral points also
  convert implicitly to floating-point point types, so `PointI` can be passed where `PointD` is
  expected.
- `PointI`, `PointL`, `PointD`, and `PointLD` use `int`, `long long`, `double`, and `long double`
  coordinates, respectively. `Point` aliases `PointD`.

Overflow warning: the exact products `dot()`, `cross()`, and `sqnorm()` grow like the squared
coordinate magnitude. With `PointI` these overflow a 32-bit `int` once coordinates exceed roughly a
few tens of thousands, so use `PointL` for larger integer coordinates.

Time Complexity:
- O(1) per call to the constructor and all other operations.

Space Complexity:
- O(1) for storage of the point.
- O(1) auxiliary for all operations.

*/

#include <cmath>
#include <ostream>
#include <tuple>
#include <type_traits>
#include <utility>

const double EPS = 1e-9;

// Epsilon-aware for floating-point coordinates; exact for int and for coordinate types like Modular
// or Rational, which therefore compose for all of the predicates below.
template<typename T, typename U, typename C = std::common_type_t<T, U>>
bool EQ(T a, U b) {
  if constexpr (std::is_floating_point_v<C>) return C(a) == C(b) || std::fabs(C(a) - C(b)) <= EPS;
  return C(a) == C(b);
}

template<typename T, typename U, typename C = std::common_type_t<T, U>>
bool LT(T a, U b) {
  if constexpr (std::is_floating_point_v<C>) return C(a) < C(b) - EPS;
  return C(a) < C(b);
}

template<typename T>
struct TPoint {
  // Metric operations preserve floating-point coordinates and otherwise promote to double.
  using fp_t = std::conditional_t<std::is_floating_point<T>::value, T, double>;

  T x, y;

  TPoint() : x(0), y(0) {}
  TPoint(T x, T y) : x(x), y(y) {}
  explicit TPoint(const std::pair<T, T> &p) : x(p.first), y(p.second) {}

  // Implicit conversion from integral to floating-point point (optional, can just use to_double()).
  template<
      typename U,
      typename = std::enable_if_t<std::is_integral<U>::value && std::is_floating_point<T>::value>>
  TPoint(const TPoint<U> &p) : x(static_cast<T>(p.x)), y(static_cast<T>(p.y)) {}

  friend bool EQ(const TPoint &a, const TPoint &b) { return EQ(a.x, b.x) && EQ(a.y, b.y); }

  bool operator==(const TPoint &p) const { return x == p.x && y == p.y; }
  bool operator!=(const TPoint &p) const { return !(*this == p); }
  bool operator<(const TPoint &p) const { return std::tie(x, y) < std::tie(p.x, p.y); }
  bool operator>(const TPoint &p) const { return p < *this; }
  bool operator<=(const TPoint &p) const { return !(*this > p); }
  bool operator>=(const TPoint &p) const { return !(*this < p); }
  TPoint operator+(const TPoint &p) const { return {x + p.x, y + p.y}; }
  TPoint operator-(const TPoint &p) const { return {x - p.x, y - p.y}; }
  TPoint operator+(T v) const { return {x + v, y + v}; }
  TPoint operator-(T v) const { return {x - v, y - v}; }
  TPoint operator*(T v) const { return {x * v, y * v}; }

  // Division always promotes to fp_t to avoid integer truncation.
  TPoint<fp_t> operator/(fp_t v) const { return {fp_t(x) / v, fp_t(y) / v}; }

  // clang-format off
  TPoint &operator+=(const TPoint &p) { x += p.x; y += p.y; return *this; }
  TPoint &operator-=(const TPoint &p) { x -= p.x; y -= p.y; return *this; }
  TPoint &operator+=(T v) { x += v; y += v; return *this; }
  TPoint &operator-=(T v) { x -= v; y -= v; return *this; }
  TPoint &operator*=(T v) { x *= v; y *= v; return *this; }
  friend TPoint operator+(T v, const TPoint &p) { return p + v; }
  friend TPoint operator*(T v, const TPoint &p) { return p * v; }
  // clang-format on

  // --- Exact operations: return T or TPoint<T>, work for any coordinate type ---

  T sqnorm() const { return x * x + y * y; }                    // Overflow warning.
  T dot(const TPoint &p) const { return x * p.x + y * p.y; }    // Overflow warning.
  T cross(const TPoint &p) const { return x * p.y - y * p.x; }  // Overflow warning.

  // --- Floating-point operations: return fp_t or TPoint<fp_t> ---

  fp_t norm() const { return std::hypot(fp_t(x), fp_t(y)); }
  fp_t arg() const { return std::atan2(fp_t(y), fp_t(x)); }
  fp_t proj(const TPoint &p) const { return (fp_t(x) * p.x + fp_t(y) * p.y) / p.norm(); }

  TPoint<fp_t> normalize() const {
    fp_t n = norm();
    if (n < 1e-30) {              // guard against dividing by a near-zero norm, not a
      return {fp_t{0}, fp_t{0}};  // geometric tolerance (EPS is too coarse here)
    }
    return {fp_t(x) / n, fp_t(y) / n};
  }

  // --- Cardinal rotations: exact for all coordinate types including int ---

  // Returns (x, y) rotated 90/180/270 degrees counter-clockwise about the origin.
  TPoint rotate90() const { return {-y, x}; }
  TPoint rotate180() const { return {-x, -y}; }
  TPoint rotate270() const { return {y, -x}; }

  // Returns (x, y) rotated 90/180/270 degrees counter-clockwise about point p.
  TPoint rotate90(const TPoint &p) const { return (*this - p).rotate90() + p; }
  TPoint rotate180(const TPoint &p) const { return (*this - p).rotate180() + p; }
  TPoint rotate270(const TPoint &p) const { return (*this - p).rotate270() + p; }

  // --- Arbitrary-angle rotations: always return floating-point ---

  // Returns (x, y) rotated t radians clockwise about the origin.
  TPoint<fp_t> rotate_cw(fp_t t) const {
    fp_t fx(x), fy(y);
    return {fx * std::cos(t) + fy * std::sin(t), fy * std::cos(t) - fx * std::sin(t)};
  }

  // Returns (x, y) rotated t radians counter-clockwise about the origin.
  TPoint<fp_t> rotate_ccw(fp_t t) const {
    fp_t fx(x), fy(y);
    return {fx * std::cos(t) - fy * std::sin(t), fx * std::sin(t) + fy * std::cos(t)};
  }

  // Returns (x, y) rotated t radians clockwise about point p.
  TPoint<fp_t> rotate_cw(const TPoint &p, fp_t t) const {
    return TPoint<fp_t>{fp_t(x) - p.x, fp_t(y) - p.y}.rotate_cw(t) +
           TPoint<fp_t>{fp_t(p.x), fp_t(p.y)};
  }

  // Returns (x, y) rotated t radians counter-clockwise about point p.
  TPoint<fp_t> rotate_ccw(const TPoint &p, fp_t t) const {
    return TPoint<fp_t>{fp_t(x) - p.x, fp_t(y) - p.y}.rotate_ccw(t) +
           TPoint<fp_t>{fp_t(p.x), fp_t(p.y)};
  }

  // --- Reflections ---

  // Returns (x, y) reflected across point p. Exact for any coordinate type.
  TPoint reflect(const TPoint &p) const { return {2 * p.x - x, 2 * p.y - y}; }

  // Returns (x, y) reflected across the line containing points p and q.
  // Always returns floating-point coordinates.
  TPoint<fp_t> reflect(const TPoint &p, const TPoint &q) const {
    TPoint<fp_t> fp{fp_t(p.x), fp_t(p.y)};
    if (EQ(p, q)) {
      return TPoint<fp_t>{fp_t(x), fp_t(y)}.reflect(fp);
    }
    TPoint<fp_t> r{fp_t(x) - p.x, fp_t(y) - p.y};
    TPoint<fp_t> s{fp_t(q.x) - p.x, fp_t(q.y) - p.y};
    fp_t ssq = s.sqnorm();
    r = TPoint<fp_t>{(r.x * s.x + r.y * s.y) / ssq, (r.x * s.y - r.y * s.x) / ssq};
    return TPoint<fp_t>{r.x * s.x - r.y * s.y + fp.x, r.x * s.y + r.y * s.x + fp.y};
  }

  // --- Explicit type conversions ---

  TPoint<double> to_double() const { return {static_cast<double>(x), static_cast<double>(y)}; }

  TPoint<long double> to_ldouble() const {
    return {static_cast<long double>(x), static_cast<long double>(y)};
  }

  // --- Friend free-function versions ---

  // clang-format off
  friend T sqnorm(const TPoint &p) { return p.sqnorm(); }
  friend fp_t norm(const TPoint &p) { return p.norm(); }
  friend fp_t arg(const TPoint &p) { return p.arg(); }
  friend T dot(const TPoint &p, const TPoint &q) { return p.dot(q); }
  friend T cross(const TPoint &p, const TPoint &q) { return p.cross(q); }
  friend fp_t proj(const TPoint &p, const TPoint &q) { return p.proj(q); }
  friend TPoint<fp_t> normalize(const TPoint &p) { return p.normalize(); }
  friend TPoint rotate90(const TPoint &p) { return p.rotate90(); }
  friend TPoint rotate180(const TPoint &p) { return p.rotate180(); }
  friend TPoint rotate270(const TPoint &p) { return p.rotate270(); }
  friend TPoint rotate90(const TPoint &p, const TPoint &q) { return p.rotate90(q); }
  friend TPoint rotate180(const TPoint &p, const TPoint &q) { return p.rotate180(q); }
  friend TPoint rotate270(const TPoint &p, const TPoint &q) { return p.rotate270(q); }
  friend TPoint<fp_t> rotate_cw(const TPoint &p, fp_t t) { return p.rotate_cw(t); }
  friend TPoint<fp_t> rotate_ccw(const TPoint &p, fp_t t) { return p.rotate_ccw(t); }
  friend TPoint<fp_t> rotate_cw(const TPoint &p, const TPoint &q, fp_t t) { return p.rotate_cw(q, t); }
  friend TPoint<fp_t> rotate_ccw(const TPoint &p, const TPoint &q, fp_t t) { return p.rotate_ccw(q, t); }
  friend TPoint reflect(const TPoint &p, const TPoint &q) { return p.reflect(q); }
  friend TPoint<fp_t> reflect(const TPoint &p, const TPoint &a, const TPoint &b) { return p.reflect(a, b); }
  // clang-format on

  friend std::ostream &operator<<(std::ostream &out, const TPoint &p) {
    if constexpr (std::is_floating_point<T>::value) {
      return out << "(" << (std::fabs(p.x) < EPS ? 0 : p.x) << ","
                 << (std::fabs(p.y) < EPS ? 0 : p.y) << ")";
    }
    return out << "(" << p.x << "," << p.y << ")";
  }
};

using PointI = TPoint<int>;
using PointL = TPoint<long long>;
using PointD = TPoint<double>;
using PointLD = TPoint<long double>;
using Point = PointD;  // Default point type is double.

// Can compose with numerical types from chapter 6:
// using PointB = TPoint<BigInt>;
// using PointR = TPoint<Rational<int64_t>>;
// using PointM = TPoint<Modular<1000000007>>;

/*** Example Usage ***/

#include <cassert>
using namespace std;

const double PI = acos(-1.0);

int main() {
  Point p(-10, 3);
  assert(EQ(Point(-18, 29), p + Point(-3, 9) * 6.0 / 2.0 - Point(-1, 1)));
  assert(EQ(109, p.sqnorm()));
  assert(EQ(10.44030650891, p.norm()));
  assert(EQ(2.850135859112, p.arg()));
  assert(EQ(0, p.dot(Point(3, 10))));
  assert(EQ(0, p.cross(Point(10, -3))));
  assert(EQ(10, p.proj(Point(-10, 0))));
  assert(EQ(1, p.normalize().norm()));
  assert(EQ(Point(-3, -10), p.rotate90()));
  assert(EQ(Point(10, -3), p.rotate180()));
  assert(EQ(Point(3, 10), p.rotate270()));
  assert(EQ(Point(3, 12), p.rotate_cw(Point(1, 1), PI / 2)));
  assert(EQ(Point(1, -10), p.rotate_ccw(Point(2, 2), PI / 2)));
  assert(EQ(Point(10, -3), p.reflect(Point(0, 0))));
  assert(EQ(Point(-10, -3), p.reflect(Point(-2, 0), Point(5, 0))));

  // Integer point - exact arithmetic, float-only ops return PointD.
  PointI a(3, 4), b(1, 0);
  assert(a + b == PointI(4, 4));
  assert(a - b == PointI(2, 4));
  assert(a * 2 == PointI(6, 8));
  assert(a.sqnorm() == 25);
  assert(a.dot(b) == 3);
  assert(a.cross(b) == -4);
  assert(EQ(a.norm(), 5.0));
  assert(a.rotate90() == PointI(-4, 3));
  assert(a.rotate180() == PointI(-3, -4));
  assert(a.rotate270() == PointI(4, -3));
  PointD anorm = a.normalize();  // returns PointD
  assert(EQ(anorm.norm(), 1.0));
  PointD adiv = a / 2.0;  // returns PointD (division always promotes)
  assert(EQ(adiv.x, 1.5) && EQ(adiv.y, 2.0));
  auto rotated = PointI(1, 0).rotate_ccw(PI / 4);  // returns PointD (arbitrary rotation promotes)
  double root2over2 = sqrt(2.0) / 2.0;
  assert(EQ(rotated.x, root2over2) && EQ(rotated.y, root2over2));

  // Implicit conversion PointI -> PointD.
  PointD promoted = PointI(1, 2);
  assert(promoted == PointD(1, 2));
  return 0;
}