Geometry/basic.h

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• Last update: 2022-07-20 01:28:51+08:00
• Link: https://cses.fi/problemset/task/2190/

Verified with

• Geometry/tests/aizu_cgl_1_a_basic_projection.test.cpp
• Geometry/tests/aizu_cgl_1_b_basic_reflection.test.cpp
• Geometry/tests/aizu_cgl_1_c_basic_ccw.test.cpp
• Geometry/tests/aizu_cgl_2_a_basic_line.test.cpp
• Geometry/tests/aizu_cgl_2_b_basic_segment_intersect.test.cpp
• Geometry/tests/aizu_cgl_2_c_basic_line_intersection.test.cpp
• Geometry/tests/aizu_cgl_2_d_basic_segment_distance.test.cpp
• Geometry/tests/aizu_cgl_3_a_polygon_area.test.cpp
• Geometry/tests/aizu_cgl_3_b_polygon_is_convex.test.cpp
• Geometry/tests/aizu_cgl_3_c_polygon_in_polygon.test.cpp
• Geometry/tests/aizu_cgl_4_a_polygon_convex_hull.test.cpp
• Geometry/tests/aizu_cgl_4_b_polygon_convex_diameter.test.cpp
• Geometry/tests/aizu_cgl_4_c_polygon_convex_cut.test.cpp
• Geometry/tests/aizu_cgl_5_a_closest_pair.test.cpp
• Geometry/tests/aizu_cgl_7_a_cicle_tangents.test.cpp
• Geometry/tests/aizu_cgl_7_d_circle_line_intersection.test.cpp
• Geometry/tests/aizu_cgl_7_e_circle_circle_intersection.test.cpp
• Geometry/tests/aizu_cgl_7_f_circle_tangent_points.test.cpp
• Geometry/tests/aizu_cgl_7_g_circle_circle_tangent_points.test.cpp
• Geometry/tests/aizu_cgl_7_i_circle_common_area.test.cpp
• Geometry/tests/polygon_in_convex.test.cpp
• Geometry/tests/z_basic_ccw.test.cpp
• Geometry/tests/z_basic_segment_intersect.test.cpp
• Geometry/tests/z_polygon_area.test.cpp
• Geometry/tests/z_polygon_convexhull.test.cpp
• Geometry/tests/z_polygon_is_convex.test.cpp

Code

#pragma once

// Basic geometry objects: Point, Line, Segment
// Works with both integers and floating points
// Unless the problem has precision issue, can use Point, which uses double
// and has more functionalities.
// For integers, can use P<long long>

#ifndef EPS  // allow test files to overwrite EPS
#define EPS 1e-6
#endif

const double PI = acos(-1.0l);

double DEG_to_RAD(double d) { return d * PI / 180.0; }
double RAD_to_DEG(double r) { return r * 180.0 / PI; }

inline int cmp(double a, double b) {
    return (a < b - EPS) ? -1 : ((a > b + EPS) ? 1 : 0);
}

// for int types
template<typename T, typename std::enable_if<!std::is_floating_point<T>::value>::type * = nullptr>
inline int cmp(T a, T b) {
    return (a == b) ? 0 : (a < b) ? -1 : 1;
}

template<typename T>
struct P {
    T x, y;
    P() { x = y = T(0); }
    P(T _x, T _y) : x(_x), y(_y) {}

    P operator + (const P& a) const { return P(x+a.x, y+a.y); }
    P operator - (const P& a) const { return P(x-a.x, y-a.y); }
    P operator * (T k) const { return P(x*k, y*k); }
    P<double> operator / (double k) const { return P(x/k, y/k); }

    T operator * (const P& a) const { return x*a.x + y*a.y; } // dot product
    T operator % (const P& a) const { return x*a.y - y*a.x; } // cross product

    int cmp(const P<T>& q) const { if (int t = ::cmp(x,q.x)) return t; return ::cmp(y,q.y); }

    #define Comp(x) bool operator x (const P& q) const { return cmp(q) x 0; }
    Comp(>) Comp(<) Comp(==) Comp(>=) Comp(<=) Comp(!=)
    #undef Comp

    T norm() { return x*x + y*y; }

    // Note: There are 2 ways for implementing len():
    // 1. sqrt(norm()) --> fast, but inaccurate (produce some values that are of order X^2)
    // 2. hypot(x, y) --> slow, but much more accurate
    double len() { return hypot(x, y); }

    P<double> rotate(double alpha) {
        double cosa = cos(alpha), sina = sin(alpha);
        return P(x * cosa - y * sina, x * sina + y * cosa);
    }
};
using Point = P<double>;

// Compare points by (y, x)
template<typename T = double>
bool cmpy(const P<T>& a, const P<T>& b) {
    if (cmp(a.y, b.y)) return a.y < b.y;
    return a.x < b.x;
};

template<typename T>
int ccw(P<T> a, P<T> b, P<T> c) {
    return cmp((b-a)%(c-a), T(0));
}

int RE_TRAI = ccw(P<int>(0, 0), P<int>(0, 1), P<int>(-1, 1));
int RE_PHAI = ccw(P<int>(0, 0), P<int>(0, 1), P<int>(1, 1));

template<typename T>
istream& operator >> (istream& cin, P<T>& p) {
    cin >> p.x >> p.y;
    return cin;
}
template<typename T>
ostream& operator << (ostream& cout, const P<T>& p) {
    cout << p.x << ' ' << p.y;
    return cout;
}

double angle(Point a, Point o, Point b) { // min of directed angle AOB & BOA
    a = a - o; b = b - o;
    return acos((a * b) / sqrt(a.norm()) / sqrt(b.norm()));
}

double directed_angle(Point a, Point o, Point b) { // angle AOB, in range [0, 2*PI)
    double t = -atan2(a.y - o.y, a.x - o.x)
            + atan2(b.y - o.y, b.x - o.x);
    while (t < 0) t += 2*PI;
    return t;
}

// Distance from p to Line ab (closest Point --> c)
// i.e. c is projection of p on AB
double distToLine(Point p, Point a, Point b, Point &c) {
    Point ap = p - a, ab = b - a;
    double u = (ap * ab) / ab.norm();
    c = a + (ab * u);
    return (p-c).len();
}

// Distance from p to segment ab (closest Point --> c)
double distToLineSegment(Point p, Point a, Point b, Point &c) {
    Point ap = p - a, ab = b - a;
    double u = (ap * ab) / ab.norm();
    if (u < 0.0) {
        c = Point(a.x, a.y);
        return (p - a).len();
    }
    if (u > 1.0) {
        c = Point(b.x, b.y);
        return (p - b).len();
    }
    return distToLine(p, a, b, c);
}

// NOTE: WILL NOT WORK WHEN a = b = 0.
struct Line {
    double a, b, c;  // ax + by + c = 0
    Point A, B;  // Added for polygon intersect line. Do not rely on assumption that these are valid

    Line(double _a, double _b, double _c) : a(_a), b(_b), c(_c) {} 

    Line(Point _A, Point _B) : A(_A), B(_B) {
        a = B.y - A.y;
        b = A.x - B.x;
        c = - (a * A.x + b * A.y);
    }
    Line(Point P, double m) {
        a = -m; b = 1;
        c = -((a * P.x) + (b * P.y));
    }
    double f(Point p) {
        return a*p.x + b*p.y + c;
    }
};
ostream& operator << (ostream& cout, const Line& l) {
    cout << l.a << "*x + " << l.b << "*y + " << l.c;
    return cout;
}

bool areParallel(Line l1, Line l2) {
    return cmp(l1.a*l2.b, l1.b*l2.a) == 0;
}

bool areSame(Line l1, Line l2) {
    return areParallel(l1 ,l2) && cmp(l1.c*l2.a, l2.c*l1.a) == 0
                && cmp(l1.c*l2.b, l1.b*l2.c) == 0;
}

bool areIntersect(Line l1, Line l2, Point &p) {
    if (areParallel(l1, l2)) return false;
    double dx = l1.b*l2.c - l2.b*l1.c;
    double dy = l1.c*l2.a - l2.c*l1.a;
    double d  = l1.a*l2.b - l2.a*l1.b;
    p = Point(dx/d, dy/d);
    return true;
}

// closest point from p in line l.
void closestPoint(Line l, Point p, Point &ans) {
    if (fabs(l.b) < EPS) {
        ans.x = -(l.c) / l.a; ans.y = p.y;
        return;
    }
    if (fabs(l.a) < EPS) {
        ans.x = p.x; ans.y = -(l.c) / l.b;
        return;
    }
    Line perp(l.b, -l.a, - (l.b*p.x - l.a*p.y));
    areIntersect(l, perp, ans);
}

// Segment intersect
// Tested:
// - https://cses.fi/problemset/task/2190/
// returns true if p is on segment [a, b]
template<typename T>
bool onSegment(const P<T>& a, const P<T>& b, const P<T>& p) {
    return ccw(a, b, p) == 0
        && min(a.x, b.x) <= p.x && p.x <= max(a.x, b.x)
        && min(a.y, b.y) <= p.y && p.y <= max(a.y, b.y);
}

// Returns true if segment [a, b] and [c, d] intersects
// End point also returns true
template<typename T>
bool segmentIntersect(const P<T>& a, const P<T>& b, const P<T>& c, const P<T>& d) {
    if (onSegment(a, b, c)
            || onSegment(a, b, d)
            || onSegment(c, d, a)
            || onSegment(c, d, b)) {
        return true;
    }

    return ccw(a, b, c) * ccw(a, b, d) < 0
        && ccw(c, d, a) * ccw(c, d, b) < 0;
}

#line 2 "Geometry/basic.h"

// Basic geometry objects: Point, Line, Segment
// Works with both integers and floating points
// Unless the problem has precision issue, can use Point, which uses double
// and has more functionalities.
// For integers, can use P<long long>

#ifndef EPS  // allow test files to overwrite EPS
#define EPS 1e-6
#endif

const double PI = acos(-1.0l);

double DEG_to_RAD(double d) { return d * PI / 180.0; }
double RAD_to_DEG(double r) { return r * 180.0 / PI; }

inline int cmp(double a, double b) {
    return (a < b - EPS) ? -1 : ((a > b + EPS) ? 1 : 0);
}

// for int types
template<typename T, typename std::enable_if<!std::is_floating_point<T>::value>::type * = nullptr>
inline int cmp(T a, T b) {
    return (a == b) ? 0 : (a < b) ? -1 : 1;
}

template<typename T>
struct P {
    T x, y;
    P() { x = y = T(0); }
    P(T _x, T _y) : x(_x), y(_y) {}

    P operator + (const P& a) const { return P(x+a.x, y+a.y); }
    P operator - (const P& a) const { return P(x-a.x, y-a.y); }
    P operator * (T k) const { return P(x*k, y*k); }
    P<double> operator / (double k) const { return P(x/k, y/k); }

    T operator * (const P& a) const { return x*a.x + y*a.y; } // dot product
    T operator % (const P& a) const { return x*a.y - y*a.x; } // cross product

    int cmp(const P<T>& q) const { if (int t = ::cmp(x,q.x)) return t; return ::cmp(y,q.y); }

    #define Comp(x) bool operator x (const P& q) const { return cmp(q) x 0; }
    Comp(>) Comp(<) Comp(==) Comp(>=) Comp(<=) Comp(!=)
    #undef Comp

    T norm() { return x*x + y*y; }

    // Note: There are 2 ways for implementing len():
    // 1. sqrt(norm()) --> fast, but inaccurate (produce some values that are of order X^2)
    // 2. hypot(x, y) --> slow, but much more accurate
    double len() { return hypot(x, y); }

    P<double> rotate(double alpha) {
        double cosa = cos(alpha), sina = sin(alpha);
        return P(x * cosa - y * sina, x * sina + y * cosa);
    }
};
using Point = P<double>;

// Compare points by (y, x)
template<typename T = double>
bool cmpy(const P<T>& a, const P<T>& b) {
    if (cmp(a.y, b.y)) return a.y < b.y;
    return a.x < b.x;
};

template<typename T>
int ccw(P<T> a, P<T> b, P<T> c) {
    return cmp((b-a)%(c-a), T(0));
}

int RE_TRAI = ccw(P<int>(0, 0), P<int>(0, 1), P<int>(-1, 1));
int RE_PHAI = ccw(P<int>(0, 0), P<int>(0, 1), P<int>(1, 1));

template<typename T>
istream& operator >> (istream& cin, P<T>& p) {
    cin >> p.x >> p.y;
    return cin;
}
template<typename T>
ostream& operator << (ostream& cout, const P<T>& p) {
    cout << p.x << ' ' << p.y;
    return cout;
}

double angle(Point a, Point o, Point b) { // min of directed angle AOB & BOA
    a = a - o; b = b - o;
    return acos((a * b) / sqrt(a.norm()) / sqrt(b.norm()));
}

double directed_angle(Point a, Point o, Point b) { // angle AOB, in range [0, 2*PI)
    double t = -atan2(a.y - o.y, a.x - o.x)
            + atan2(b.y - o.y, b.x - o.x);
    while (t < 0) t += 2*PI;
    return t;
}

// Distance from p to Line ab (closest Point --> c)
// i.e. c is projection of p on AB
double distToLine(Point p, Point a, Point b, Point &c) {
    Point ap = p - a, ab = b - a;
    double u = (ap * ab) / ab.norm();
    c = a + (ab * u);
    return (p-c).len();
}

// Distance from p to segment ab (closest Point --> c)
double distToLineSegment(Point p, Point a, Point b, Point &c) {
    Point ap = p - a, ab = b - a;
    double u = (ap * ab) / ab.norm();
    if (u < 0.0) {
        c = Point(a.x, a.y);
        return (p - a).len();
    }
    if (u > 1.0) {
        c = Point(b.x, b.y);
        return (p - b).len();
    }
    return distToLine(p, a, b, c);
}

// NOTE: WILL NOT WORK WHEN a = b = 0.
struct Line {
    double a, b, c;  // ax + by + c = 0
    Point A, B;  // Added for polygon intersect line. Do not rely on assumption that these are valid

    Line(double _a, double _b, double _c) : a(_a), b(_b), c(_c) {} 

    Line(Point _A, Point _B) : A(_A), B(_B) {
        a = B.y - A.y;
        b = A.x - B.x;
        c = - (a * A.x + b * A.y);
    }
    Line(Point P, double m) {
        a = -m; b = 1;
        c = -((a * P.x) + (b * P.y));
    }
    double f(Point p) {
        return a*p.x + b*p.y + c;
    }
};
ostream& operator << (ostream& cout, const Line& l) {
    cout << l.a << "*x + " << l.b << "*y + " << l.c;
    return cout;
}

bool areParallel(Line l1, Line l2) {
    return cmp(l1.a*l2.b, l1.b*l2.a) == 0;
}

bool areSame(Line l1, Line l2) {
    return areParallel(l1 ,l2) && cmp(l1.c*l2.a, l2.c*l1.a) == 0
                && cmp(l1.c*l2.b, l1.b*l2.c) == 0;
}

bool areIntersect(Line l1, Line l2, Point &p) {
    if (areParallel(l1, l2)) return false;
    double dx = l1.b*l2.c - l2.b*l1.c;
    double dy = l1.c*l2.a - l2.c*l1.a;
    double d  = l1.a*l2.b - l2.a*l1.b;
    p = Point(dx/d, dy/d);
    return true;
}

// closest point from p in line l.
void closestPoint(Line l, Point p, Point &ans) {
    if (fabs(l.b) < EPS) {
        ans.x = -(l.c) / l.a; ans.y = p.y;
        return;
    }
    if (fabs(l.a) < EPS) {
        ans.x = p.x; ans.y = -(l.c) / l.b;
        return;
    }
    Line perp(l.b, -l.a, - (l.b*p.x - l.a*p.y));
    areIntersect(l, perp, ans);
}

// Segment intersect
// Tested:
// - https://cses.fi/problemset/task/2190/
// returns true if p is on segment [a, b]
template<typename T>
bool onSegment(const P<T>& a, const P<T>& b, const P<T>& p) {
    return ccw(a, b, p) == 0
        && min(a.x, b.x) <= p.x && p.x <= max(a.x, b.x)
        && min(a.y, b.y) <= p.y && p.y <= max(a.y, b.y);
}

// Returns true if segment [a, b] and [c, d] intersects
// End point also returns true
template<typename T>
bool segmentIntersect(const P<T>& a, const P<T>& b, const P<T>& c, const P<T>& d) {
    if (onSegment(a, b, c)
            || onSegment(a, b, d)
            || onSegment(c, d, a)
            || onSegment(c, d, b)) {
        return true;
    }

    return ccw(a, b, c) * ccw(a, b, d) < 0
        && ccw(c, d, a) * ccw(c, d, b) < 0;
}

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