Task: Line circle intersection
C: #include <math.h>
#include <stdbool.h>
#include <stdio.h>

const double eps = 1e-14;

typedef struct point_t {
    double x, y;
} point;

point make_point(double x, double y) {
    point p = { x, y };
    return p;
}

void print_point(point p) {
    double x = p.x;
    double y = p.y;
    if (x == 0) {
        x = 0;
    }
    if (y == 0) {
        y = 0;
    }
    printf("(%g, %g)", x, y);
}

double sq(double x) {
    return x * x;
}

bool within(double x1, double y1, double x2, double y2, double x, double y) {
    double d1 = sqrt(sq(x2 - x1) + sq(y2 - y1));    
    double d2 = sqrt(sq(x - x1) + sq(y - y1));      
    double d3 = sqrt(sq(x2 - x) + sq(y2 - y));      
    double delta = d1 - d2 - d3;
    return fabs(delta) < eps;   
}

int rxy(double x1, double y1, double x2, double y2, double x, double y, bool segment) {
    if (!segment || within(x1, y1, x2, y2, x, y)) {
        print_point(make_point(x, y));
        return 1;
    } else {
        return 0;
    }
}

double fx(double A, double B, double C, double x) {
    return -(A * x + C) / B;
}

double fy(double A, double B, double C, double y) {
    return -(B * y + C) / A;
}




void intersects(point p1, point p2, point cp, double r, bool segment) {
    double x0 = cp.x, y0 = cp.y;
    double x1 = p1.x, y1 = p1.y;
    double x2 = p2.x, y2 = p2.y;
    double A = y2 - y1;
    double B = x1 - x2;
    double C = x2 * y1 - x1 * y2;
    double a = sq(A) + sq(B);
    double b, c, d;
    bool bnz = true;
    int cnt = 0;

    if (fabs(B) >= eps) {
        
        b = 2 * (A * C + A * B * y0 - sq(B) * x0);
        c = sq(C) + 2 * B * C * y0 - sq(B) * (sq(r) - sq(x0) - sq(y0));
    } else {
        b = 2 * (B * C + A * B * x0 - sq(A) * y0);
        c = sq(C) + 2 * A * C * x0 - sq(A) * (sq(r) - sq(x0) - sq(y0));
        bnz = false;
    }
    d = sq(b) - 4 * a * c; 
    if (d < 0) {
        
        printf("[]\n");
        return;
    }

    if (d == 0) {
        
        if (bnz) {
            double x = -b / (2 * a);
            double y = fx(A, B, C, x);
            cnt = rxy(x1, y1, x2, y2, x, y, segment);
        } else {
            double y = -b / (2 * a);
            double x = fy(A, B, C, y);
            cnt = rxy(x1, y1, x2, y2, x, y, segment);
        }
    } else {
        
        d = sqrt(d);
        if (bnz) {
            double x = (-b + d) / (2 * a);
            double y = fx(A, B, C, x);
            cnt = rxy(x1, y1, x2, y2, x, y, segment);

            x = (-b - d) / (2 * a);
            y = fx(A, B, C, x);
            cnt += rxy(x1, y1, x2, y2, x, y, segment);
        } else {
            double y = (-b + d) / (2 * a);
            double x = fy(A, B, C, y);
            cnt = rxy(x1, y1, x2, y2, x, y, segment);

            y = (-b - d) / (2 * a);
            x = fy(A, B, C, y);
            cnt += rxy(x1, y1, x2, y2, x, y, segment);
        }
    }

    if (cnt <= 0) {
        printf("[]");
    }
}

int main() {
    point cp = make_point(3, -5);
    double r = 3.0;
    printf("The intersection points (if any) between:\n");
    printf("  A circle, center (3, -5) with radius 3, and:\n");
    printf("    a line containing the points (-10, 11) and (10, -9) is/are:\n");
    printf("      ");
    intersects(make_point(-10, 11), make_point(10, -9), cp, r, false);
    printf("\n    a segment starting at (-10, 11) and ending at (-11, 12) is/are\n");
    printf("      ");
    intersects(make_point(-10, 11), make_point(-11, 12), cp, r, true);
    printf("\n    a horizontal line containing the points (3, -2) and (7, -2) is/are:\n");
    printf("      ");
    intersects(make_point(3, -2), make_point(7, -2), cp, r, false);
    printf("\n");

    cp = make_point(0, 0);
    r = 4.0;
    printf("  A circle, center (0, 0) with radius 4, and:\n");
    printf("    a vertical line containing the points (0, -3) and (0, 6) is/are:\n");
    printf("      ");
    intersects(make_point(0, -3), make_point(0, 6), cp, r, false);
    printf("\n    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:\n");
    printf("      ");
    intersects(make_point(0, -3), make_point(0, 6), cp, r, true);
    printf("\n");

    cp = make_point(4,2);
    r = 5.0;
    printf("  A circle, center (4, 2) with radius 5, and:\n");
    printf("    a line containing the points (6, 3) and (10, 7) is/are:\n");
    printf("      ");
    intersects(make_point(6, 3), make_point(10, 7), cp, r, false);
    printf("\n    a segment starting at (7, 4) and ending at (11, 8) is/are:\n");
    printf("      ");
    intersects(make_point(7, 4), make_point(11, 8), cp, r, true);
    printf("\n");

    return 0;
}

C++: #include <iostream>
#include <utility>
#include <vector>

using Point = std::pair<double, double>;
constexpr auto eps = 1e-14;

std::ostream &operator<<(std::ostream &os, const Point &p) {
    auto x = p.first;
    if (x == 0.0) {
        x = 0.0;
    }
    auto y = p.second;
    if (y == 0.0) {
        y = 0.0;
    }
    return os << '(' << x << ", " << y << ')';
}

template <typename T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &v) {
    auto itr = v.cbegin();
    auto end = v.cend();

    os << '[';
    if (itr != end) {
        os << *itr;
        itr = std::next(itr);
    }
    while (itr != end) {
        os << ", " << *itr;
        itr = std::next(itr);
    }
    return os << ']';
}

double sq(double x) {
    return x * x;
}

std::vector<Point> intersects(const Point &p1, const Point &p2, const Point &cp, double r, bool segment) {
    std::vector<Point> res;
    auto x0 = cp.first;
    auto y0 = cp.second;
    auto x1 = p1.first;
    auto y1 = p1.second;
    auto x2 = p2.first;
    auto y2 = p2.second;
    auto A = y2 - y1;
    auto B = x1 - x2;
    auto C = x2 * y1 - x1 * y2;
    auto a = sq(A) + sq(B);
    double b, c;
    bool bnz = true;
    if (abs(B) >= eps) {
        b = 2 * (A * C + A * B * y0 - sq(B) * x0);
        c = sq(C) + 2 * B * C * y0 - sq(B) * (sq(r) - sq(x0) - sq(y0));
    } else {
        b = 2 * (B * C + A * B * x0 - sq(A) * y0);
        c = sq(C) + 2 * A * C * x0 - sq(A) * (sq(r) - sq(x0) - sq(y0));
        bnz = false;
    }
    auto d = sq(b) - 4 * a * c; 
    if (d < 0) {
        return res;
    }

    
    auto within = [x1, y1, x2, y2](double x, double y) {
        auto d1 = sqrt(sq(x2 - x1) + sq(y2 - y1));  
        auto d2 = sqrt(sq(x - x1) + sq(y - y1));    
        auto d3 = sqrt(sq(x2 - x) + sq(y2 - y));    
        auto delta = d1 - d2 - d3;
        return abs(delta) < eps;                    
    };

    auto fx = [A, B, C](double x) {
        return -(A * x + C) / B;
    };

    auto fy = [A, B, C](double y) {
        return -(B * y + C) / A;
    };

    auto rxy = [segment, &res, within](double x, double y) {
        if (!segment || within(x, y)) {
            res.push_back(std::make_pair(x, y));
        }
    };

    double x, y;
    if (d == 0.0) {
        
        if (bnz) {
            x = -b / (2 * a);
            y = fx(x);
            rxy(x, y);
        } else {
            y = -b / (2 * a);
            x = fy(y);
            rxy(x, y);
        }
    } else {
        
        d = sqrt(d);
        if (bnz) {
            x = (-b + d) / (2 * a);
            y = fx(x);
            rxy(x, y);
            x = (-b - d) / (2 * a);
            y = fx(x);
            rxy(x, y);
        } else {
            y = (-b + d) / (2 * a);
            x = fy(y);
            rxy(x, y);
            y = (-b - d) / (2 * a);
            x = fy(y);
            rxy(x, y);
        }
    }

    return res;
}

int main() {
    std::cout << "The intersection points (if any) between:\n";

    auto cp = std::make_pair(3.0, -5.0);
    auto r = 3.0;
    std::cout << "  A circle, center " << cp << " with radius " << r << ", and:\n";

    auto p1 = std::make_pair(-10.0, 11.0);
    auto p2 = std::make_pair(10.0, -9.0);
    std::cout << "    a line containing the points " << p1 << " and " << p2 << " is/are:\n";
    std::cout << "     " << intersects(p1, p2, cp, r, false) << '\n';

    p2 = std::make_pair(-10.0, 12.0);
    std::cout << "    a segment starting at " << p1 << " and ending at " << p2 << " is/are:\n";
    std::cout << "     " << intersects(p1, p2, cp, r, true) << '\n';

    p1 = std::make_pair(3.0, -2.0);
    p2 = std::make_pair(7.0, -2.0);
    std::cout << "    a horizontal line containing the points " << p1 << " and " << p2 << " is/are:\n";
    std::cout << "     " << intersects(p1, p2, cp, r, false) << '\n';

    cp = std::make_pair(0.0, 0.0);
    r = 4.0;
    std::cout << "  A circle, center " << cp << " with radius " << r << ", and:\n";

    p1 = std::make_pair(0.0, -3.0);
    p2 = std::make_pair(0.0, 6.0);
    std::cout << "    a vertical line containing the points " << p1 << " and " << p2 << " is/are:\n";
    std::cout << "     " << intersects(p1, p2, cp, r, false) << '\n';
    std::cout << "    a vertical segment containing the points " << p1 << " and " << p2 << " is/are:\n";
    std::cout << "     " << intersects(p1, p2, cp, r, true) << '\n';

    cp = std::make_pair(4.0, 2.0);
    r = 5.0;
    std::cout << "  A circle, center " << cp << " with radius " << r << ", and:\n";

    p1 = std::make_pair(6.0, 3.0);
    p2 = std::make_pair(10.0, 7.0);
    std::cout << "    a line containing the points " << p1 << " and " << p2 << " is/are:\n";
    std::cout << "     " << intersects(p1, p2, cp, r, false) << '\n';

    p1 = std::make_pair(7.0, 4.0);
    p2 = std::make_pair(11.0, 8.0);
    std::cout << "    a segment starting at " << p1 << " and ending at " << p2 << " is/are:\n";
    std::cout << "     " << intersects(p1, p2, cp, r, true) << '\n';

    return 0;
}