Геометрия

Вектор - направленныйй отрезок

Полярные кординаты - представление вектора с длиноной и углом

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\[ \begin{align}\begin{aligned}\overrightarrow{A} * \overrightarrow{B} = (a_x * b_x, a_y * b_y)\\\overrightarrow{A} * \overrightarrow{A} = \overrightarrow{A} ^ 2\\\overrightarrow{A} * 0 = 0\\\overrightarrow{A} * \overrightarrow{B} = \overrightarrow{B} * \overrightarrow{A}\\\overrightarrow{A} * (\overrightarrow{B} + C) = \overrightarrow{A} * \overrightarrow{B} + \overrightarrow{A} * C\\\overrightarrow{A} * (\overrightarrow{B} * C) = (\overrightarrow{A} * \overrightarrow{B}) * C\end{aligned}\end{align} \]

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Перевод из Декартовой в Полярную систему и обратно

\[ \begin{align}\begin{aligned}(x, y) \Longrightarrow (a, \varphi)\\\varphi = \arctan \frac{y}{x}\\a = \sqrt{x ^ 2 + y ^ 2}\end{aligned}\end{align} \]

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\[ \begin{align}\begin{aligned}(a, \varphi) \Longrightarrow (x, y)\\x = a * \cos \varphi\\y = a * \sin \varphi\end{aligned}\end{align} \]

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Нормирование вектора

\[ \begin{align}\begin{aligned}a = \sqrt{x ^ 2 + y ^ 2}\\(\frac{x}{a}; \frac{y}{a})\end{aligned}\end{align} \]

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\[ \begin{align}\begin{aligned}\overrightarrow{A} + \overrightarrow{B} = (a_x + b_x, a_y + b_y)\\\overrightarrow{A} - \overrightarrow{B} = (a_x - b_x, a_y - b_y)\end{aligned}\end{align} \]

#include <bits/stdc++.h>

using namespace std;

typedef long long ll;
typedef long double ld;

struct point {
    ld x, y;

    friend bool operator==(const point &f, const point &s) {
        return f.x == s.x && f.y == s.y;
    }

    friend bool operator!=(const point &f, const point &s) {
        return !(f.x == s.x && f.y == s.y);
    }
};

struct Coefficient {
    ld a, b, c;

    Coefficient(ld a_, ld b_, ld c_) {
        a = a_;
        b = b_;
        c = c_;
    }

    Coefficient(point p1, point p2) {
        a = p2.y - p1.y;
        b = -(p2.x - p1.x);
        c = -(a * p1.x + b * p1.y);
    }

    friend bool operator==(const Coefficient &f, const Coefficient &s) {
        return f.a == s.a && f.b == s.b && f.c == f.c;
    }
};

struct Vector {
    point p;

    Vector(point f, point s) { // f -> s
        p = {s.x - f.x, s.y - f.y};
    }

    Vector(point t) {
        p = t;
    }

    Vector(ld x, ld y) {
        p = {x, y};
    }

    friend Vector operator+(const Vector &f, const Vector &s) {
        return Vector(f.p.x + s.p.x, f.p.y + s.p.y);
    }

    friend Vector operator-(const Vector &f, const Vector &s) {
        return Vector(f.p.x - s.p.x, f.p.y - s.p.y);
    }

    friend ld operator*(const Vector &f, const Vector &s) {
        return f.p.x * s.p.y - f.p.y * s.p.x;
    }

    friend ld operator^(const Vector &f, const Vector &s) {
        return f.p.x * s.p.x + f.p.y * s.p.y;
    }

};

ld rast(point a, point b) {
    return sqrt((a.x - b.x) * (a.x - b.x) + (a.y - b.y) * (a.y - b.y));
}


ld angle(point a, point b, point c) {
    auto Vec1 = Vector(a, c);
    auto Vec2 = Vector(a, b);
    return atan2(Vec1 * Vec2, Vec1 ^ Vec2);
}

ld point_to_vector(point p1, point p2, point p3) {
    point p; // point vector
    p.x = p2.x - p1.x;
    p.y = p2.y - p1.y;
    point pn; // point normal
    pn.x = -p.y;
    pn.y = p.x;
    point p4; // point p3 p4
    p4.x = p3.x + pn.x;
    p4.y = p3.y + pn.y;
    auto line = Coefficient(p3, p4);
    auto vector = Coefficient(p1, p2);
    ld DET = vector.a * line.b - line.a * vector.b;
    ld DETx = -vector.c * line.b - -line.c * vector.b;
    ld DETy = vector.a * -line.c - line.a * -vector.c;
    point P;
    P.x = DETx / DET;
    P.y = DETy / DET;
    return rast(P, p3);
}

ld Point_to_vector(point c, point a, point b) {
    if (angle(a, b, c) < 0 || angle(b, a, c) < 0) {
        return min(rast(c, a), rast(c, b));
    } else {
        return point_to_vector(a, b, c);
    }
}

bool p(point a, point b, point c, point d) {
    // c-d ? a-b
    auto p = Vector(d, c);
    auto v1 = Vector(d, a);
    auto v2 = Vector(d, b);
    auto vec1 = v1 * p;
    auto vec2 = v2 * p;
    if (!((vec1 <= 0 && vec2 <= 0) || (vec1 >= 0 && vec2 >= 0))) {
        return true;
    } else {
        return false;
    }
}

ld sq(vector<point> &fig) {
    ld res = 0;
    for (ll i = 0; i < fig.size(); i++) {
        point a = i ? fig[i - 1] : fig.back(), b = fig[i];
        res += Vector(a) * Vector(b);
    }
    return abs(res) / 2;
}

int main() {
    return 0;
}