Line circle intersection

In plane geometry, a line (or segment) may intersect a circle at 0, 1 or 2 points.

Line circle intersection is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Task

Implement a method (function, procedure etc.) in your language which takes as parameters:

  • the starting point of a line;
  • the point where the line ends;
  • the center point of a circle;
  • the circle's radius; and
  • whether the line is a segment or extends to infinity beyond the above points.


The method should return the intersection points (if any) of the circle and the line.

Illustrate your method with some examples (or use the Go examples below).

Stretch Task

Include results for a circle with center at (10, 10) and radius 5 combined with a line through points (5, 0) and (5, 20) and that same circle combined with a line segment from (-5, 10) to (5, 10).


References

AutoHotkeyEdit

data :=  [[[3, -5], 3,    [-10, 11],    [10, -9],    0]
        , [[3, -5], 3,    [-10, 11],    [-11, 12],    1]
        , [[3, -5], 3,    [3, -2],    [7, -2],    1]
        , [[0, 0],  4,    [0, -3],    [0, 6],     0]
        , [[0, 0],  4,    [0, -3],    [0, 6],        1]
        , [[4, 2],  5,    [6, 3],        [10, 7],    0]
        , [[4, 2],  5,    [7, 4],        [11, 8],    1]]

Result := "Center`tRad`tP1`tP2`tSegment`tintersect 1`tIntersect 2`n"
for i, obj in data
{
    x := Line_circle_intersection(center := obj.1, radius := obj.2, P1 := obj.3, P2 := obj.4, Segment := obj.5)
    Result .= "[" center.1 "," center.2 "]`t" radius "`t[" p1.1 "," p1.2 "]`t[" p2.1 "," p2.2 "]`t" Segment
    for i, v in x
        Result .=  "`t[" i "]"
    Result .= "`n"
}
MsgBox % Result
return

Line_circle_intersection(c, r, p1, p2, segment:=0){
    global result
    p1.1 -= c.1,    p2.1 -= c.1,    p1.2 -= c.2,    p2.2 -= c.2
    
    dx := p2.1 - p1.1,    dy := p2.2 - p1.2
    dr := Sqrt(dx**2 + dy**2)
    D  := p1.1*p2.2 - p2.1*p1.2
    x1 := (D * dy + sgn(dy) * dx * Sqrt(r**2 * dr**2 - D**2)) / dr**2
    x2 := (D * dy - sgn(dy) * dx * Sqrt(r**2 * dr**2 - D**2)) / dr**2
    y1  := (0-D * dx + Abs(dy) * Sqrt(r**2 * dr**2 - D**2)) / dr**2
    y2  := (0-D * dx - Abs(dy) * Sqrt(r**2 * dr**2 - D**2)) / dr**2

    p1.1 += c.1,    p2.1 += c.1,    p1.2 += c.2,    p2.2 += c.2
    x1 += c.1,    x2 += c.1,    y1 += c.2,    y2 += c.2

    res := []
    if segment
    {
        if !((x1 < p1.1 && x1 < p2.1) || (x1 > p1.1 && x1 > p2.1)
          || (y1 < p1.2 && y1 < p2.2) || (y1 > p1.2 && y1 > p2.2))
            res[x1 ", " y1] := true
        if !((x2 < p1.1 && x2 < p2.1) || (x2 > p1.1 && x2 > p2.1)
          || (y2 < p1.2 && y2 < p2.2) || (y2 > p1.2 && y2 > p2.2))
            res[x2 ", " y2] := true
    }    
    else
        res[x1 ", " y1] := true, res[x2 ", " y2] := true
    return res
}
sgn(x){
    return x<0?-1:1
}
Output:
Center	Rad	P1		P2		Segment	intersect 1		Intersect 2
[3,-5]	3	[-10,11]	[10,-9]		0	[3.000000, -2.000000]	[6.000000, -5.000000]
[3,-5]	3	[-10,11]	[-11,12]	1
[3,-5]	3	[3,-2]		[7,-2]		1	[3.000000, -2.000000]
[0,0]	4	[0,-3]		[0,6]		0	[0.000000, -4.000000]	[0.000000, 4.000000]
[0,0]	4	[0,-3]		[0,6]		1	[0.000000, 4.000000]
[4,2]	5	[6,3]		[10,7]		0	[1.000000, -2.000000]	[8.000000, 5.000000]
[4,2]	5	[7,4]		[11,8]		1	[8.000000, 5.000000]

CEdit

Translation of: Go
#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));    // distance between end-points
    double d2 = sqrt(sq(x - x1) + sq(y - y1));      // distance from point to one end
    double d3 = sqrt(sq(x2 - x) + sq(y2 - y));      // distance from point to other end
    double delta = d1 - d2 - d3;
    return fabs(delta) < eps;   // true if delta is less than a small tolerance
}

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;
}

// Prints the intersection points (if any) of a circle, center 'cp' with radius 'r',
// and either an infinite line containing the points 'p1' and 'p2'
// or a segment drawn between those points.
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) {
        // if B isn't zero or close to it
        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; // discriminant
    if (d < 0) {
        // line & circle don't intersect
        printf("[]\n");
        return;
    }

    if (d == 0) {
        // line is tangent to circle, so just one intersect at most
        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 {
        // two intersects at most
        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;
}
Output:
The intersection points (if any) between:
  A circle, center (3, -5) with radius 3, and:
    a line containing the points (-10, 11) and (10, -9) is/are:
      (6, -5)(3, -2)
    a segment starting at (-10, 11) and ending at (-11, 12) is/are
      []
    a horizontal line containing the points (3, -2) and (7, -2) is/are:
      (3, -2)
  A circle, center (0, 0) with radius 4, and:
    a vertical line containing the points (0, -3) and (0, 6) is/are:
      (0, 4)(0, -4)
    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:
      (0, 4)
  A circle, center (4, 2) with radius 5, and:
    a line containing the points (6, 3) and (10, 7) is/are:
      (8, 5)(1, -2)
    a segment starting at (7, 4) and ending at (11, 8) is/are:
      (8, 5)

C++Edit

Translation of: Kotlin
#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; // discriminant
    if (d < 0) {
        return res;
    }

    // checks whether a point is within a segment
    auto within = [x1, y1, x2, y2](double x, double y) {
        auto d1 = sqrt(sq(x2 - x1) + sq(y2 - y1));  // distance between end-points
        auto d2 = sqrt(sq(x - x1) + sq(y - y1));    // distance from point to one end
        auto d3 = sqrt(sq(x2 - x) + sq(y2 - y));    // distance from point to other end
        auto delta = d1 - d2 - d3;
        return abs(delta) < eps;                    // true if delta is less than a small tolerance
    };

    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) {
        // line is tangent to circle, so just one intersect at most
        if (bnz) {
            x = -b / (2 * a);
            y = fx(x);
            rxy(x, y);
        } else {
            y = -b / (2 * a);
            x = fy(y);
            rxy(x, y);
        }
    } else {
        // two intersects at most
        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;
}
Output:
The intersection points (if any) between:
  A circle, center (3, -5) with radius 3, and:
    a line containing the points (-10, 11) and (10, -9) is/are:
     [(6, -5), (3, -2)]
    a segment starting at (-10, 11) and ending at (-10, 12) is/are:
     []
    a horizontal line containing the points (3, -2) and (7, -2) is/are:
     [(3, -2)]
  A circle, center (0, 0) with radius 4, and:
    a vertical line containing the points (0, -3) and (0, 6) is/are:
     [(0, 4), (0, -4)]
    a vertical segment containing the points (0, -3) and (0, 6) is/are:
     [(0, 4)]
  A circle, center (4, 2) with radius 5, and:
    a line containing the points (6, 3) and (10, 7) is/are:
     [(8, 5), (1, -2)]
    a segment starting at (7, 4) and ending at (11, 8) is/are:
     [(8, 5)]

C#Edit

Translation of: C++
using System;
using System.Collections.Generic;
using System.Linq;

public class Program
{
    public static void Main()
    {
        Circle circle = ((3, -5), 3);
        Line[] lines = {
            ((-10, 11), (10, -9)),
            ((-10, 11), (-11, 12), true),
            ((3, -2), (7, -2))
        };
        Print(circle, lines);
        
        circle = ((0, 0), 4);
        lines = new Line[] {
            ((0, -3), (0, 6)),
            ((0, -3), (0, 6), true)
        };
        Print(circle, lines);
        
        circle = ((4, 2), 5);
        lines = new Line[] {
            ((6, 3), (10, 7)),
            ((7, 4), (11, 8), true)
        };
        Print(circle, lines);
    }
    
    static void Print(Circle circle, Line[] lines)
    {
        Console.WriteLine($"Circle: {circle}");
        foreach (var line in lines) {
            Console.WriteLine($"\t{(line.IsSegment ? "Segment:" : "Line:")} {line}");
            var points = Intersection(circle, line).ToList();
            Console.WriteLine(points.Count == 0 ? "\t\tdo not intersect" : "\t\tintersect at " + string.Join(" and ", points));
        }
        Console.WriteLine();
    }
    
    static IEnumerable<Point> Intersection(Circle circle, Line line)
    {
        var intersection = LineIntersection(circle, line);
        return line.IsSegment
            ? intersection.Where(p => p.CompareTo(line.P1) >= 0 && p.CompareTo(line.P2) <= 0)
            : intersection;

        static IEnumerable<Point> LineIntersection(Circle circle, Line line)
        {
            double x, y, A, B, C, D;
            var (m, c) = (line.Slope, line.YIntercept);
            var (p, q, r) = (circle.X, circle.Y, circle.Radius);

            if (line.IsVertical) {
                x = line.P1.X;
                B = -2 * q;
                C = p * p + q * q - r * r + x * x - 2 * p * x;
                D = B * B - 4 * C;
                if (D == 0) yield return (x, -q);
                else if (D > 0) {
                    D = Math.Sqrt(D);
                    yield return (x, (-B - D) / 2);
                    yield return (x, (-B + D) / 2);
                }
            } else {
                A = m * m + 1;
                B = 2 * (m * c - m * q - p);
                C = p * p + q * q - r * r + c * c - 2 * c * q;
                D = B * B - 4 * A * C;
                if (D == 0) {
                    x = -B / (2 * A);
                    y = m * x + c;
                    yield return (x, y);
                } else if (D > 0) {
                    D = Math.Sqrt(D);
                    x = (-B - D) / (2 * A);
                    y = m * x + c;
                    yield return (x, y);
                    x = (-B + D) / (2 * A);
                    y = m * x + c;
                    yield return (x, y);
                }
            }
        }

    }
    
    readonly struct Point : IComparable<Point>
    {
        public Point(double x, double y) => (X, Y) = (x, y);
        
        public static implicit operator Point((double x, double y) p) => new Point(p.x, p.y);
        
        public double X { get; }
        public double Y { get; }
        
        public int CompareTo(Point other)
        {
            int c = X.CompareTo(other.X);
            if (c != 0) return c;
            return Y.CompareTo(other.Y);
        }
        
        public override string ToString() => $"({X}, {Y})";
    }
    
    readonly struct Line
    {
        public Line(Point p1, Point p2, bool isSegment = false)
        {
            (P1, P2) = p2.CompareTo(p1) < 0 ? (p2, p1) : (p1, p2);
            IsSegment = isSegment;
            if (p1.X == p2.X) (Slope, YIntercept) = (double.PositiveInfinity, double.NaN);
            else {
                Slope = (P2.Y - P1.Y) / (P2.X - P1.X);
                YIntercept = P2.Y - Slope * P2.X;
            }
        }
        
        public static implicit operator Line((Point p1, Point p2) l) => new Line(l.p1, l.p2);
        public static implicit operator Line((Point p1, Point p2, bool isSegment) l) => new Line(l.p1, l.p2, l.isSegment);
        
        public Point P1 { get; }
        public Point P2 { get; }
        public double Slope { get; }
        public double YIntercept { get; }
        public bool IsSegment { get; }
        public bool IsVertical => P1.X == P2.X;
        
        public override string ToString() => $"[{P1}, {P2}]";
    }
    
    readonly struct Circle
    {
        public Circle(Point center, double radius) => (Center, Radius) = (center, radius);
        
        public static implicit operator Circle((Point center, double radius) c) => new Circle(c.center, c.radius);
        
        public Point Center { get; }
        public double Radius { get; }
        public double X => Center.X;
        public double Y => Center.Y;
        
        public override string ToString() => $"{{ C:{Center}, R:{Radius} }}";
    }   
}
Output:
Circle: { C:(3, -5), R:3 }
    Line: [(-10, 11), (10, -9)]
        intersect at (3, -2) and (6, -5)
    Segment: [(-11, 12), (-10, 11)]
        do not intersect
    Line: [(3, -2), (7, -2)]
        intersect at (3, -2)

Circle: { C:(0, 0), R:4 }
    Line: [(0, -3), (0, 6)]
        intersect at (0, -4) and (0, 4)
    Segment: [(0, -3), (0, 6)]
        intersect at (0, 4)

Circle: { C:(4, 2), R:5 }
    Line: [(6, 3), (10, 7)]
        intersect at (1, -2) and (8, 5)
    Segment: [(7, 4), (11, 8)]
        intersect at (8, 5)

DEdit

Translation of: C++
import std.format;
import std.math;
import std.stdio;

immutable EPS = 1e-14;

struct Point {
    private double x;
    private double y;

    public this(double x, double y) {
        this.x = x;
        this.y = y;
    }

    public double getX() {
        return x;
    }

    public double getY() {
        return y;
    }

    void toString(scope void delegate(const(char)[]) sink, FormatSpec!char fmt) const {
        double mx = x;
        double my = y;

        // eliminate negative zero
        if (mx == 0.0) {
            mx = 0.0;
        }

        // eliminate negative zero
        if (my == 0.0) {
            my = 0.0;
        }

        sink("(");
        formatValue(sink, mx, fmt);
        sink(", ");
        formatValue(sink, my, fmt);
        sink(")");
    }
}

auto sq(T)(T x) {
    return x * x;
}

auto intersects(const Point p1, const Point p2, const Point cp, double r, bool segment) {
    auto x0 = cp.x;
    auto y0 = cp.y;
    auto x1 = p1.x;
    auto y1 = p1.y;
    auto x2 = p2.x;
    auto y2 = p2.y;

    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;

    Point[] res;

    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; // discriminant
    if (d < 0) {
        return res;
    }

    auto within(double x, double y) {
        auto d1 = sqrt(sq(x2 - x1) + sq(y2 - y1));  // distance between end-points
        auto d2 = sqrt(sq(x - x1) + sq(y - y1));    // distance from point to one end
        auto d3 = sqrt(sq(x2 - x) + sq(y2 - y));    // distance from point to other end
        auto delta = d1 - d2 - d3;
        return abs(delta) < EPS;                    // true if delta is less than a small tolerance
    }

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

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

    auto rxy(double x, double y) {
        if (!segment || within(x, y)) {
            res ~= Point(x, y);
        }
    }

    double x, y;
    if (d == 0.0) {
        // line is tangent to circle, so just one intersect at most
        if (bnz) {
            x = -b / (2 * a);
            y = fx(x);
            rxy(x, y);
        } else {
            y = -b / (2 * a);
            x = fy(y);
            rxy(x, y);
        }
    } else {
        // two intersects at most
        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;
}

void main() {
    writeln("The intersection points (if any) between:");

    auto cp = Point(3.0, -5.0);
    auto r = 3.0;
    writeln("  A circle, center ", cp, " with radius ", r, ", and:");

    auto p1 = Point(-10.0, 11.0);
    auto p2 = Point(10.0, -9.0);
    writeln("    a line containing the points ", p1, " and ", p2, " is/are:");
    writeln("     ", intersects(p1, p2, cp, r, false));

    p2 = Point(-10.0, 12.0);
    writeln("    a segment starting at ", p1, " and ending at ", p2, " is/are:");
    writeln("     ", intersects(p1, p2, cp, r, true));

    p1 = Point(3.0, -2.0);
    p2 = Point(7.0, -2.0);
    writeln("    a horizontal line containing the points ", p1, " and ", p2, " is/are:");
    writeln("     ", intersects(p1, p2, cp, r, false));

    cp = Point(0.0, 0.0);
    r = 4.0;
    writeln("  A circle, center ", cp, " with radius ", r, ", and:");

    p1 = Point(0.0, -3.0);
    p2 = Point(0.0, 6.0);
    writeln("    a vertical line containing the points ", p1, " and ", p2, " is/are:");
    writeln("     ", intersects(p1, p2, cp, r, false));
    writeln("    a vertical segment containing the points ", p1, " and ", p2, " is/are:");
    writeln("     ", intersects(p1, p2, cp, r, true));

    cp = Point(4.0, 2.0);
    r = 5.0;
    writeln("  A circle, center ", cp, " with radius ", r, ", and:");

    p1 = Point(6.0, 3.0);
    p2 = Point(10.0, 7.0);
    writeln("    a line containing the points ", p1, " and ", p2, " is/are:");
    writeln("     ", intersects(p1, p2, cp, r, false));

    p1 = Point(7.0, 4.0);
    p2 = Point(11.0, 8.0);
    writeln("    a segment starting at ", p1, " and ending at ", p2, " is/are:");
    writeln("     ", intersects(p1, p2, cp, r, true));
}
Output:
The intersection points (if any) between:
  A circle, center (3, -5) with radius 3, and:
    a line containing the points (-10, 11) and (10, -9) is/are:
     [(6, -5), (3, -2)]
    a segment starting at (-10, 11) and ending at (-10, 12) is/are:
     []
    a horizontal line containing the points (3, -2) and (7, -2) is/are:
     [(3, -2)]
  A circle, center (0, 0) with radius 4, and:
    a vertical line containing the points (0, -3) and (0, 6) is/are:
     [(0, 4), (0, -4)]
    a vertical segment containing the points (0, -3) and (0, 6) is/are:
     [(0, 4)]
  A circle, center (4, 2) with radius 5, and:
    a line containing the points (6, 3) and (10, 7) is/are:
     [(8, 5), (1, -2)]
    a segment starting at (7, 4) and ending at (11, 8) is/are:
     [(8, 5)]

GoEdit

package main

import (
    "fmt"
    "math"
)

const eps = 1e-14 // say

type point struct{ x, y float64 }

func (p point) String() string {
    // hack to get rid of negative zero
    // compiler treats 0 and -0 as being same 
    if p.x == 0 {
        p.x = 0
    }
    if p.y == 0 {
        p.y = 0
    }
    return fmt.Sprintf("(%g, %g)", p.x, p.y)
}

func sq(x float64) float64 { return x * x }

// Returns the intersection points (if any) of a circle, center 'cp' with radius 'r',
// and either an infinite line containing the points 'p1' and 'p2'
// or a segment drawn between those points.
func intersects(p1, p2, cp point, r float64, segment bool) []point {
    var res []point
    x0, y0 := cp.x, cp.y
    x1, y1 := p1.x, p1.y
    x2, y2 := p2.x, p2.y
    A := y2 - y1
    B := x1 - x2
    C := x2*y1 - x1*y2
    a := sq(A) + sq(B)
    var b, c float64
    var bnz = true
    if math.Abs(B) >= eps { // if B isn't zero or close to it
        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 // discriminant
    if d < 0 {
        // line & circle don't intersect
        return res
    }

    // checks whether a point is within a segment
    within := func(x, y float64) bool {
        d1 := math.Sqrt(sq(x2-x1) + sq(y2-y1)) // distance between end-points
        d2 := math.Sqrt(sq(x-x1) + sq(y-y1))   // distance from point to one end
        d3 := math.Sqrt(sq(x2-x) + sq(y2-y))   // distance from point to other end
        delta := d1 - d2 - d3
        return math.Abs(delta) < eps // true if delta is less than a small tolerance
    }

    var x, y float64
    fx := func() float64 { return -(A*x + C) / B }
    fy := func() float64 { return -(B*y + C) / A }
    rxy := func() {
        if !segment || within(x, y) {
            res = append(res, point{x, y})
        }
    }

    if d == 0 {
        // line is tangent to circle, so just one intersect at most
        if bnz {
            x = -b / (2 * a)
            y = fx()
            rxy()
        } else {
            y = -b / (2 * a)
            x = fy()
            rxy()
        }
    } else {
        // two intersects at most
        d = math.Sqrt(d)
        if bnz {
            x = (-b + d) / (2 * a)
            y = fx()
            rxy()
            x = (-b - d) / (2 * a)
            y = fx()
            rxy()
        } else {
            y = (-b + d) / (2 * a)
            x = fy()
            rxy()
            y = (-b - d) / (2 * a)
            x = fy()
            rxy()
        }
    }
    return res
}

func main() {
    cp := point{3, -5}
    r := 3.0
    fmt.Println("The intersection points (if any) between:")
    fmt.Println("\n  A circle, center (3, -5) with radius 3, and:")
    fmt.Println("\n    a line containing the points (-10, 11) and (10, -9) is/are:")
    fmt.Println("     ", intersects(point{-10, 11}, point{10, -9}, cp, r, false))
    fmt.Println("\n    a segment starting at (-10, 11) and ending at (-11, 12) is/are")
    fmt.Println("     ", intersects(point{-10, 11}, point{-11, 12}, cp, r, true))
    fmt.Println("\n    a horizontal line containing the points (3, -2) and (7, -2) is/are:")
    fmt.Println("     ", intersects(point{3, -2}, point{7, -2}, cp, r, false))
    cp = point{0, 0}
    r = 4.0
    fmt.Println("\n  A circle, center (0, 0) with radius 4, and:")
    fmt.Println("\n    a vertical line containing the points (0, -3) and (0, 6) is/are:")
    fmt.Println("     ", intersects(point{0, -3}, point{0, 6}, cp, r, false))
    fmt.Println("\n    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:")
    fmt.Println("     ", intersects(point{0, -3}, point{0, 6}, cp, r, true))
    cp = point{4, 2}
    r = 5.0
    fmt.Println("\n  A circle, center (4, 2) with radius 5, and:")
    fmt.Println("\n    a line containing the points (6, 3) and (10, 7) is/are:")
    fmt.Println("     ", intersects(point{6, 3}, point{10, 7}, cp, r, false))
    fmt.Println("\n    a segment starting at (7, 4) and ending at (11, 8) is/are:")
    fmt.Println("     ", intersects(point{7, 4}, point{11, 8}, cp, r, true))
    cp = point{10, 10}
    r = 5.0
    fmt.Println("\n  A circle, center (10, 10) with radius 5, and:")
    fmt.Println("\n    a vertical line containing the points (5, 0) and (5, 20) is/are:")
    fmt.Println("     ", intersects(point{5, 0}, point{5, 20}, cp, r, false))
    fmt.Println("\n    a horizontal segment starting at (-5, 10) and ending at (5, 10) is/are:")
    fmt.Println("     ", intersects(point{-5, 10}, point{5, 10}, cp, r, true))
}
Output:
The intersection points (if any) between:

  A circle, center (3, -5) with radius 3, and:

    a line containing the points (-10, 11) and (10, -9) is/are:
      [(6, -5) (3, -2)]

    a segment starting at (-10, 11) and ending at (-11, 12) is/are
      []

    a horizontal line containing the points (3, -2) and (7, -2) is/are:
      [(3, -2)]

  A circle, center (0, 0) with radius 4, and:

    a vertical line containing the points (0, -3) and (0, 6) is/are:
      [(0, 4) (0, -4)]

    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:
      [(0, 4)]

  A circle, center (4, 2) with radius 5, and:

    a line containing the points (6, 3) and (10, 7) is/are:
      [(8, 5) (1, -2)]

    a segment starting at (7, 4) and ending at (11, 8) is/are:
      [(8, 5)]

  A circle, center (10, 10) with radius 5, and:

    a vertical line containing the points (5, 0) and (5, 20) is/are:
      [(5, 10)]

    a horizontal segment starting at (-5, 10) and ending at (5, 10) is/are:
      [(5, 10)]

HaskellEdit

import Data.Tuple.Curry

main :: IO ()
main =
  mapM_ putStrLn $
  concatMap
    (("" :) . uncurryN task)
    [ ((-10, 11), (10, -9), ((3, -5), 3))
    , ((-10, 11), (-11, 12), ((3, -5), 3))
    , ((3, -2), (7, -2), ((3, -5), 3))
    , ((3, -2), (7, -2), ((0, 0), 4))
    , ((0, -3), (0, 6), ((0, 0), 4))
    , ((6, 3), (10, 7), ((4, 2), 5))
    , ((7, 4), (11, 18), ((4, 2), 5))
    ]

task :: (Double, Double)
     -> (Double, Double)
     -> ((Double, Double), Double)
     -> [String]
task pt1 pt2 circle@(pt3@(a3, b3), r) = [line, segment]
  where
    xs = map fun $ lineCircleIntersection pt1 pt2 circle
    ys = map fun $ segmentCircleIntersection pt1 pt2 circle
    to x = (fromIntegral . round $ 100 * x) / 100
    fun (x, y) = (to x, to y)
    yo = show . fun
    start = "Intersection: Circle " ++ yo pt3 ++ " " ++ show (to r) ++ " and "
    end = yo pt1 ++ " " ++ yo pt2 ++ ": "
    line = start ++ "Line " ++ end ++ show xs
    segment = start ++ "Segment " ++ end ++ show ys

segmentCircleIntersection
  :: (Double, Double)
  -> (Double, Double)
  -> ((Double, Double), Double)
  -> [(Double, Double)]
segmentCircleIntersection pt1 pt2 circle =
  filter (go p1 p2) $ lineCircleIntersection pt1 pt2 circle
  where
    [p1, p2]
      | pt1 < pt2 = [pt1, pt2]
      | otherwise = [pt2, pt1]
    go (x, y) (u, v) (i, j)
      | x == u = y <= j && j <= v
      | otherwise = x <= i && i <= u

lineCircleIntersection
  :: (Double, Double)
  -> (Double, Double)
  -> ((Double, Double), Double)
  -> [(Double, Double)]
lineCircleIntersection (a1, b1) (a2, b2) ((a3, b3), r) = go delta
  where
    (x1, x2) = (a1 - a3, a2 - a3)
    (y1, y2) = (b1 - b3, b2 - b3)
    (dx, dy) = (x2 - x1, y2 - y1)
    drdr = dx * dx + dy * dy
    d = x1 * y2 - x2 * y1
    delta = r * r * drdr - d * d
    sqrtDelta = sqrt delta
    (sgnDy, absDy) = (sgn dy, abs dy)
    u1 = (d * dy + sgnDy * dx * sqrtDelta) / drdr
    u2 = (d * dy - sgnDy * dx * sqrtDelta) / drdr
    v1 = (-d * dx + absDy * sqrtDelta) / drdr
    v2 = (-d * dx - absDy * sqrtDelta) / drdr
    go x
      | 0 > x = []
      | 0 == x = [(u1 + a3, v1 + b3)]
      | otherwise = [(u1 + a3, v1 + b3), (u2 + a3, v2 + b3)]

sgn :: Double -> Double
sgn x
  | 0 > x = -1
  | otherwise = 1
Output:
Intersection: Circle (3.0,-5.0) 3.0 and Line (-10.0,11.0) (10.0,-9.0): [(3.0,-2.0),(6.0,-5.0)]
Intersection: Circle (3.0,-5.0) 3.0 and Segment (-10.0,11.0) (10.0,-9.0): [(3.0,-2.0),(6.0,-5.0)]

Intersection: Circle (3.0,-5.0) 3.0 and Line (-10.0,11.0) (-11.0,12.0): [(3.0,-2.0),(6.0,-5.0)]
Intersection: Circle (3.0,-5.0) 3.0 and Segment (-10.0,11.0) (-11.0,12.0): []

Intersection: Circle (3.0,-5.0) 3.0 and Line (3.0,-2.0) (7.0,-2.0): [(3.0,-2.0)]
Intersection: Circle (3.0,-5.0) 3.0 and Segment (3.0,-2.0) (7.0,-2.0): [(3.0,-2.0)]

Intersection: Circle (0.0,0.0) 4.0 and Line (3.0,-2.0) (7.0,-2.0): [(3.46,-2.0),(-3.46,-2.0)]
Intersection: Circle (0.0,0.0) 4.0 and Segment (3.0,-2.0) (7.0,-2.0): [(3.46,-2.0)]

Intersection: Circle (0.0,0.0) 4.0 and Line (0.0,-3.0) (0.0,6.0): [(0.0,4.0),(0.0,-4.0)]
Intersection: Circle (0.0,0.0) 4.0 and Segment (0.0,-3.0) (0.0,6.0): [(0.0,4.0)]

Intersection: Circle (4.0,2.0) 5.0 and Line (6.0,3.0) (10.0,7.0): [(8.0,5.0),(1.0,-2.0)]
Intersection: Circle (4.0,2.0) 5.0 and Segment (6.0,3.0) (10.0,7.0): [(8.0,5.0)]

Intersection: Circle (4.0,2.0) 5.0 and Line (7.0,4.0) (11.0,18.0): [(7.46,5.61),(5.03,-2.89)]
Intersection: Circle (4.0,2.0) 5.0 and Segment (7.0,4.0) (11.0,18.0): [(7.46,5.61)]

JavaEdit

import java.util.*;
import java.awt.geom.*;

public class LineCircleIntersection {
    public static void main(String[] args) {
        try {
            demo();
        } catch (Exception e) {
            e.printStackTrace();
        }
    }

    private static void demo() throws NoninvertibleTransformException {
        Point2D center = makePoint(3, -5);
        double radius = 3.0;
        System.out.println("The intersection points (if any) between:");
        System.out.println("\n  A circle, center (3, -5) with radius 3, and:");
        System.out.println("\n    a line containing the points (-10, 11) and (10, -9) is/are:");
        System.out.println("     " + toString(intersection(makePoint(-10, 11), makePoint(10, -9),
                            center, radius, false)));
        System.out.println("\n    a segment starting at (-10, 11) and ending at (-11, 12) is/are");
        System.out.println("     " + toString(intersection(makePoint(-10, 11), makePoint(-11, 12),
                            center, radius, true)));
        System.out.println("\n    a horizontal line containing the points (3, -2) and (7, -2) is/are:");
        System.out.println("     " + toString(intersection(makePoint(3, -2), makePoint(7, -2), center, radius, false)));
        center.setLocation(0, 0);
        radius = 4.0;
        System.out.println("\n  A circle, center (0, 0) with radius 4, and:");
        System.out.println("\n    a vertical line containing the points (0, -3) and (0, 6) is/are:");
        System.out.println("     " + toString(intersection(makePoint(0, -3), makePoint(0, 6),
                            center, radius, false)));
        System.out.println("\n    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:");
        System.out.println("     " + toString(intersection(makePoint(0, -3), makePoint(0, 6),
                            center, radius, true)));
        center.setLocation(4, 2);
        radius = 5.0;
        System.out.println("\n  A circle, center (4, 2) with radius 5, and:");
        System.out.println("\n    a line containing the points (6, 3) and (10, 7) is/are:");
        System.out.println("     " + toString(intersection(makePoint(6, 3), makePoint(10, 7),
                            center, radius, false)));
        System.out.println("\n    a segment starting at (7, 4) and ending at (11, 8) is/are:");
        System.out.println("     " + toString(intersection(makePoint(7, 4), makePoint(11, 8),
                            center, radius, true)));
    }

    private static Point2D makePoint(double x, double y) {
        return new Point2D.Double(x, y);
    }

    //
    // If center of the circle is at the origin and the line is horizontal,
    // it's easy to calculate the points of intersection, so to handle the
    // general case, we convert the input to a coordinate system where the
    // center of the circle is at the origin and the line is horizontal,
    // then convert the points of intersection back to the original
    // coordinate system.
    //
    public static List<Point2D> intersection(Point2D p1, Point2D p2, Point2D center,
            double radius, boolean isSegment) throws NoninvertibleTransformException {
        List<Point2D> result = new ArrayList<>();
        double dx = p2.getX() - p1.getX();
        double dy = p2.getY() - p1.getY();
        AffineTransform trans = AffineTransform.getRotateInstance(dx, dy);
        trans.invert();
        trans.translate(-center.getX(), -center.getY());
        Point2D p1a = trans.transform(p1, null);
        Point2D p2a = trans.transform(p2, null);
        double y = p1a.getY();
        double minX = Math.min(p1a.getX(), p2a.getX());
        double maxX = Math.max(p1a.getX(), p2a.getX());
        if (y == radius || y == -radius) {
            if (!isSegment || (0 <= maxX && 0 >= minX)) {
                p1a.setLocation(0, y);
                trans.inverseTransform(p1a, p1a);
                result.add(p1a);
            }
        } else if (y < radius && y > -radius) {
            double x = Math.sqrt(radius * radius - y * y);
            if (!isSegment || (-x <= maxX && -x >= minX)) {
                p1a.setLocation(-x, y);
                trans.inverseTransform(p1a, p1a);
                result.add(p1a);
            }
            if (!isSegment || (x <= maxX && x >= minX)) {
                p2a.setLocation(x, y);
                trans.inverseTransform(p2a, p2a);
                result.add(p2a);
            }
        }
        return result;
    }

    public static String toString(Point2D point) {
        return String.format("(%g, %g)", point.getX(), point.getY());
    }

    public static String toString(List<Point2D> points) {
        StringBuilder str = new StringBuilder("[");
        for (int i = 0, n = points.size(); i < n; ++i) {
            if (i > 0)
                str.append(", ");
            str.append(toString(points.get(i)));
        }
        str.append("]");
        return str.toString();
    }
}
Output:
The intersection points (if any) between:

  A circle, center (3, -5) with radius 3, and:

    a line containing the points (-10, 11) and (10, -9) is/are:
     [(3.00000, -2.00000), (6.00000, -5.00000)]

    a segment starting at (-10, 11) and ending at (-11, 12) is/are
     []

    a horizontal line containing the points (3, -2) and (7, -2) is/are:
     [(3.00000, -2.00000)]

  A circle, center (0, 0) with radius 4, and:

    a vertical line containing the points (0, -3) and (0, 6) is/are:
     [(0.00000, -4.00000), (0.00000, 4.00000)]

    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:
     [(0.00000, 4.00000)]

  A circle, center (4, 2) with radius 5, and:

    a line containing the points (6, 3) and (10, 7) is/are:
     [(1.00000, -2.00000), (8.00000, 5.00000)]

    a segment starting at (7, 4) and ending at (11, 8) is/are:
     [(8.00000, 5.00000)]

JuliaEdit

Uses the circles and points from the Go example.

using Luxor

const centers = [Point(3, -5), Point(0, 0), Point(4, 2)]
const rads = [3, 4, 5]
const lins = [
    [Point(-10, 11), Point(10, -9)], [Point(-10, 11), Point(-11, 12)], 
    [Point(3, -2), Point(7, -2)], [Point(0, -3), Point(0, 6)], 
    [Point(6, 3), Point(10, 7)], [Point(7, 4), Point(11, 8)],
]

println("Center", " "^9, "Radius", " "^4, "Line P1", " "^14, "Line P2", " "^7,
    "Segment?   Intersect 1       Intersect 2")
for (cr, l, extended) in [(1, 1, true), (1, 2, false), (1, 3, false), 
        (2, 4, true), (2, 4, false), (3, 5, true), (3, 6, false)]
    tup = intersectionlinecircle(lins[l][1], lins[l][2], centers[cr], rads[cr])
    v = [p for p in tup[2:end] if extended || ispointonline(p, lins[l][1], lins[l][2])]
    println(rpad(centers[cr], 17), rads[cr], " "^3, rpad(lins[l][1], 21),
        rpad(lins[l][2], 19), rpad(!extended, 8), isempty(v) ? "" : 
            length(v) == 2 && tup[1] == 2 ? rpad(v[1], 18) * string(v[2]) : v[1])
end
Output:
Center         Radius    Line P1              Line P2       Segment?   Intersect 1       Intersect 2
Point(3.0, -5.0) 3   Point(-10.0, 11.0)   Point(10.0, -9.0)  false   Point(6.0, -5.0)  Point(3.0, -2.0)
Point(3.0, -5.0) 3   Point(-10.0, 11.0)   Point(-11.0, 12.0) true
Point(3.0, -5.0) 3   Point(3.0, -2.0)     Point(7.0, -2.0)   true    Point(3.0, -2.0)
Point(0.0, 0.0)  4   Point(0.0, -3.0)     Point(0.0, 6.0)    false   Point(0.0, 4.0)   Point(0.0, -4.0)
Point(0.0, 0.0)  4   Point(0.0, -3.0)     Point(0.0, 6.0)    true    Point(0.0, 4.0)
Point(4.0, 2.0)  5   Point(6.0, 3.0)      Point(10.0, 7.0)   false   Point(8.0, 5.0)   Point(1.0, -2.0)
Point(4.0, 2.0)  5   Point(7.0, 4.0)      Point(11.0, 8.0)   true    Point(8.0, 5.0)

KotlinEdit

Translation of: Go
import kotlin.math.absoluteValue
import kotlin.math.sqrt

const val eps = 1e-14

class Point(val x: Double, val y: Double) {
    override fun toString(): String {
        var xv = x
        if (xv == 0.0) {
            xv = 0.0
        }
        var yv = y
        if (yv == 0.0) {
            yv = 0.0
        }
        return "($xv, $yv)"
    }
}

fun sq(x: Double): Double {
    return x * x
}

fun intersects(p1: Point, p2: Point, cp: Point, r: Double, segment: Boolean): MutableList<Point> {
    val res = mutableListOf<Point>()
    val x0 = cp.x
    val y0 = cp.y
    val x1 = p1.x
    val y1 = p1.y
    val x2 = p2.x
    val y2 = p2.y
    val A = y2 - y1
    val B = x1 - x2
    val C = x2 * y1 - x1 * y2
    val a = sq(A) + sq(B)
    val b: Double
    val c: Double
    var bnz = true
    if (B.absoluteValue >= 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
    }
    var d = sq(b) - 4 * a * c // discriminant
    if (d < 0) {
        return res
    }

    // checks whether a point is within a segment
    fun within(x: Double, y: Double): Boolean {
        val d1 = sqrt(sq(x2 - x1) + sq(y2 - y1)) // distance between end-points
        val d2 = sqrt(sq(x - x1) + sq(y - y1))   // distance from point to one end
        val d3 = sqrt(sq(x2 - x) + sq(y2 - y))   // distance from point to other end
        val delta = d1 - d2 - d3
        return delta.absoluteValue < eps // true if delta is less than a small tolerance
    }

    var x = 0.0
    fun fx(): Double {
        return -(A * x + C) / B
    }

    var y = 0.0
    fun fy(): Double {
        return -(B * y + C) / A
    }

    fun rxy() {
        if (!segment || within(x, y)) {
            res.add(Point(x, y))
        }
    }

    if (d == 0.0) {
        // line is tangent to circle, so just one intersect at most
        if (bnz) {
            x = -b / (2 * a)
            y = fx()
            rxy()
        } else {
            y = -b / (2 * a)
            x = fy()
            rxy()
        }
    } else {
        // two intersects at most
        d = sqrt(d)
        if (bnz) {
            x = (-b + d) / (2 * a)
            y = fx()
            rxy()
            x = (-b - d) / (2 * a)
            y = fx()
            rxy()
        } else {
            y = (-b + d) / (2 * a)
            x = fy()
            rxy()
            y = (-b - d) / (2 * a)
            x = fy()
            rxy()
        }
    }

    return res
}

fun main() {
    println("The intersection points (if any) between:")

    var cp = Point(3.0, -5.0)
    var r = 3.0
    println("  A circle, center $cp with radius $r, and:")

    var p1 = Point(-10.0, 11.0)
    var p2 = Point(10.0, -9.0)
    println("    a line containing the points $p1 and $p2 is/are:")
    println("     ${intersects(p1, p2, cp, r, false)}")

    p2 = Point(-10.0, 12.0)
    println("    a segment starting at $p1 and ending at $p2 is/are:")
    println("     ${intersects(p1, p2, cp, r, true)}")

    p1 = Point(3.0, -2.0)
    p2 = Point(7.0, -2.0)
    println("    a horizontal line containing the points $p1 and $p2 is/are:")
    println("     ${intersects(p1, p2, cp, r, false)}")

    cp = Point(0.0, 0.0)
    r = 4.0
    println("  A circle, center $cp with radius $r, and:")

    p1 = Point(0.0, -3.0)
    p2 = Point(0.0, 6.0)
    println("    a vertical line containing the points $p1 and $p2 is/are:")
    println("     ${intersects(p1, p2, cp, r, false)}")
    println("    a vertical segment containing the points $p1 and $p2 is/are:")
    println("     ${intersects(p1, p2, cp, r, true)}")

    cp = Point(4.0, 2.0)
    r = 5.0
    println("  A circle, center $cp with radius $r, and:")

    p1 = Point(6.0, 3.0)
    p2 = Point(10.0, 7.0)
    println("    a line containing the points $p1 and $p2 is/are:")
    println("     ${intersects(p1, p2, cp, r, false)}")

    p1 = Point(7.0, 4.0)
    p2 = Point(11.0, 8.0)
    println("    a segment starting at $p1 and ending at $p2 is/are:")
    println("     ${intersects(p1, p2, cp, r, true)}")
}
Output:
The intersection points (if any) between:
  A circle, center (3.0, -5.0) with radius 3.0, and:
    a line containing the points (-10.0, 11.0) and (10.0, -9.0) is/are:
     [(6.0, -5.0), (3.0, -2.0)]
    a segment starting at (-10.0, 11.0) and ending at (-10.0, 12.0) is/are:
     []
    a horizontal line containing the points (3.0, -2.0) and (7.0, -2.0) is/are:
     [(3.0, -2.0)]
  A circle, center (0.0, 0.0) with radius 4.0, and:
    a vertical line containing the points (0.0, -3.0) and (0.0, 6.0) is/are:
     [(0.0, 4.0), (0.0, -4.0)]
    a vertical segment containing the points (0.0, -3.0) and (0.0, 6.0) is/are:
     [(0.0, 4.0)]
  A circle, center (4.0, 2.0) with radius 5.0, and:
    a line containing the points (6.0, 3.0) and (10.0, 7.0) is/are:
     [(8.0, 5.0), (1.0, -2.0)]
    a segment starting at (7.0, 4.0) and ending at (11.0, 8.0) is/are:
     [(8.0, 5.0)]

LuaEdit

Translation of: C++
EPS = 1e-14

function pts(p)
    local x, y = p.x, p.y
    if x == 0 then
        x = 0
    end
    if y == 0 then
        y = 0
    end
    return "(" .. x .. ", " .. y .. ")"
end

function lts(pl)
    local str = "["
    for i,p in pairs(pl) do
        if i > 1 then
            str = str .. ", "
        end
        str = str .. pts(p)
    end
    return str .. "]"
end

function sq(x)
    return x * x
end

function intersects(p1, p2, cp, r, segment)
    local res = {}
    local x0, y0 = cp.x, cp.y
    local x1, y1 = p1.x, p1.y
    local x2, y2 = p2.x, p2.y
    local A = y2 - y1
    local B = x1 - x2
    local C = x2 * y1 - x1 * y2
    local a = sq(A) + sq(B)
    local b, c
    local bnz = true
    if math.abs(B) >= EPS then
        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
    end
    local d = sq(b) - 4 * a * c -- discriminant
    if d < 0 then
        return res
    end

    -- checks whether a point is within a segment
    function within(x, y)
        local d1 = math.sqrt(sq(x2 - x1) + sq(y2 - y1)) -- distance between end-points
        local d2 = math.sqrt(sq(x - x1) + sq(y - y1))   -- distance from point to one end
        local d3 = math.sqrt(sq(x2 - x) + sq(y2 - y))   -- distance from point to other end
        local delta = d1 - d2 - d3
        return math.abs(delta) < EPS
    end

    function fx(x)
        return -(A * x + C) / B
    end

    function fy(y)
        return -(B * y + C) / A
    end

    function rxy(x, y)
        if not segment or within(x, y) then
            table.insert(res, {x=x, y=y})
        end
    end

    local x, y
    if d == 0 then
        -- line is tangent to circle, so just one intersect at most
        if bnz then
            x = -b / (2 * a)
            y = fx(x)
            rxy(x, y)
        else
            y = -b / (2 * a)
            x = fy(y)
            rxy(x, y)
        end
    else
        -- two intersects at most
        d = math.sqrt(d)
        if bnz then
            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)
        end
    end

    return res
end

function main()
    print("The intersection points (if any) between:")

    local cp = {x=3, y=-5}
    local r = 3
    print("  A circle, center " .. pts(cp) .. " with radius " .. r .. ", and:")

    local p1 = {x=-10, y=11}
    local p2 = {x=10, y=-9}
    print("    a line containing the points " .. pts(p1) .. " and " .. pts(p2) .. " is/are:")
    print("      " .. lts(intersects(p1, p2, cp, r, false)))

    p2 = {x=-10, y=12}
    print("    a segment starting at " .. pts(p1) .. " and ending at " .. pts(p2) .. " is/are:")
    print("      " .. lts(intersects(p1, p2, cp, r, true)))

    p1 = {x=3, y=-2}
    p2 = {x=7, y=-2}
    print("    a horizontal line containing the points " .. pts(p1) .. " and " .. pts(p2) .. " is/are:")
    print("      " .. lts(intersects(p1, p2, cp, r, false)))

    cp = {x=0, y=0}
    r = 4
    print("  A circle, center " .. pts(cp) .. " with radius " .. r .. ", and:")

    p1 = {x=0, y=-3}
    p2 = {x=0, y=6}
    print("    a vertical line containing the points " .. pts(p1) .. " and " .. pts(p2) .. " is/are:")
    print("      " .. lts(intersects(p1, p2, cp, r, false)))
    print("    a vertical segment starting at " .. pts(p1) .. " and ending at " .. pts(p2) .. " is/are:")
    print("      " .. lts(intersects(p1, p2, cp, r, true)))

    cp = {x=4, y=2}
    r = 5
    print("  A circle, center " .. pts(cp) .. " with radius " .. r .. ", and:")

    p1 = {x=6, y=3}
    p2 = {x=10, y=7}
    print("    a line containing the points " .. pts(p1) .. " and " .. pts(p2) .. " is/are:")
    print("      " .. lts(intersects(p1, p2, cp, r, false)))

    p1 = {x=7, y=4}
    p2 = {x=11, y=8}
    print("    a segment starting at " .. pts(p1) .. " and ending at " .. pts(p2) .. " is/are:")
    print("      " .. lts(intersects(p1, p2, cp, r, true)))
end

main()
Output:
The intersection points (if any) between:
  A circle, center (3, -5) with radius 3, and:
    a line containing the points (-10, 11) and (10, -9) is/are:
      [(6, -5), (3, -2)]
    a segment starting at (-10, 11) and ending at (-10, 12) is/are:
      []
    a horizontal line containing the points (3, -2) and (7, -2) is/are:
      [(3, -2)]
  A circle, center (0, 0) with radius 4, and:
    a vertical line containing the points (0, -3) and (0, 6) is/are:
      [(0, 4), (0, -4)]
    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:
      [(0, 4)]
  A circle, center (4, 2) with radius 5, and:
    a line containing the points (6, 3) and (10, 7) is/are:
      [(8, 5), (1, -2)]
    a segment starting at (7, 4) and ending at (11, 8) is/are:
      [(8, 5)]

Mathematica/Wolfram LanguageEdit

LineCircleIntersections[p1_, p2_, c_, r_, type_] := RegionIntersection[Circle[c, r], type[{p1, p2}]]
LineCircleIntersections[{-1, 1}, {1, 1}, {0, 0}, 1, Line]
LineCircleIntersections[{-1, 0}, {2, 0.4}, {0, 0}, 1, Line]
LineCircleIntersections[{-1.5, 0}, {-2, 0.4}, {0, 0}, 1, Line]
LineCircleIntersections[{-1.5, 0}, {-2, 0.4}, {0, 0}, 1, InfiniteLine]
Output:
Point[{{0,1}}]
Point[{{-1,0},{0.965066,0.262009}}]
EmptyRegion[2]
Point[{{-0.858057,-0.513554},{-0.312675,-0.94986}}]

NimEdit

Translation of: Go
import math, strutils

const Eps = 1e-14

type Point = tuple[x, y: float]

func `$`(p: Point): string =
  let x = if p.x == 0.0: 0.0 else: p.x
  let y = if p.y == 0.0: 0.0 else: p.y
  "($1, $2)".format(x, y)


func intersects(p1, p2, cp: Point; r: float; segment: bool): seq[Point] =
  let
    (x0, y0) = cp
    (x1, y1) = p1
    (x2, y2) = p2
    A = y2 - y1
    B = x1 - x2
    C = x2 * y1 - x1 * y2
  var
    a = A^2 + B^2
    b, c: float
    bnz = true
  if abs(B) >= Eps:
    b = 2 * (A * C + A * B * y0 - B^2 * x0)
    c = C^2 + 2 * B * C * y0 - B^2 * (r^2 - x0^2 - y0^2)
  else:
    b = 2 * (B * C + A * B * x0 - A^2 * y0)
    c = C^2 + 2 * A * C * x0 - A^2 * (r^2 - x0^2 - y0^2)
    bnz = false
  let d = b^2 - 4 * a * c
  if d < 0: return  # Line & circle don't intersect.

  func within(x, y: float): bool =
    ## Checks whether a point is within a segment.
    let
      d1 = sqrt((x2 - x1)^2 + (y2 - y1)^2)    # Distance between end-points.
      d2 = sqrt((x - x1)^2 + (y - y1)^2)      # Distance from point to one end.
      d3 = sqrt((x2 - x)^2 + (y2 - y)^2)      # Distance from point to other end.
      delta = d1 - d2 - d3
    result = abs(delta) < Eps                 # True if delta is less than a small tolerance.

  var x, y: float
  template fx: float = -(A * x + C) / B
  template fy: float = -(B * y + C) / A
  template rxy() =
    if not segment or within(x, y):
      result.add (x, y)

  if d == 0:
    # Line is tangent to circle, so just one intersect at most.
    if bnz:
      x = -b / (2 * a)
      y = fx()
      rxy()
    else:
      y = -b / (2 * a)
      x = fy()
      rxy()
  else:
    # Two intersects at most.
    let d = sqrt(d)
    if bnz:
      x = (-b + d) / (2 * a)
      y = fx()
      rxy()
      x = (-b - d) / (2 * a)
      y = fx()
      rxy()
    else:
      y = (-b + d) / (2 * a)
      x = fy()
      rxy()
      y = (-b - d) / (2 * a)
      x = fy()
      rxy()


when isMainModule:

  var cp: Point = (3.0, -5.0)
  var r = 3.0
  echo "The intersection points (if any) between:"
  echo "\n  A circle, center (3, -5) with radius 3, and:"
  echo "\n    a line containing the points (-10, 11) and (10, -9) is/are:"
  echo "     ", intersects((-10.0, 11.0), (10.0, -9.0), cp, r, false)
  echo "\n    a segment starting at (-10, 11) and ending at (-11, 12) is/are"
  echo "     ", intersects((-10.0, 11.0), (-11.0, 12.0), cp, r, true)
  echo "\n    a horizontal line containing the points (3, -2) and (7, -2) is/are:"
  echo "     ", intersects((3.0, -2.0), (7.0, -2.0), cp, r, false)
  cp = (0.0, 0.0)
  r = 4.0
  echo "\n  A circle, center (0, 0) with radius 4, and:"
  echo "\n    a vertical line containing the points (0, -3) and (0, 6) is/are:"
  echo "     ", intersects((0.0, -3.0), (0.0, 6.0), cp, r, false)
  echo "\n    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:"
  echo "     ", intersects((0.0, -3.0), (0.0, 6.0), cp, r, true)
  cp = (4.0, 2.0)
  r = 5.0
  echo "\n  A circle, center (4, 2) with radius 5, and:"
  echo "\n    a line containing the points (6, 3) and (10, 7) is/are:"
  echo "     ", intersects((6.0, 3.0), (10.0, 7.0), cp, r, false)
  echo "\n    a segment starting at (7, 4) and ending at (11, 8) is/are:"
  echo "     ", intersects((7.0, 4.0), (11.0, 8.0), cp, r, true)
Output:
The intersection points (if any) between:

  A circle, center (3, -5) with radius 3, and:

    a line containing the points (-10, 11) and (10, -9) is/are:
     @[(6.0, -5.0), (3.0, -2.0)]

    a segment starting at (-10, 11) and ending at (-11, 12) is/are
     @[]

    a horizontal line containing the points (3, -2) and (7, -2) is/are:
     @[(3.0, -2.0)]

  A circle, center (0, 0) with radius 4, and:

    a vertical line containing the points (0, -3) and (0, 6) is/are:
     @[(0.0, 4.0), (0.0, -4.0)]

    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:
     @[(0.0, 4.0)]

  A circle, center (4, 2) with radius 5, and:

    a line containing the points (6, 3) and (10, 7) is/are:
     @[(8.0, 5.0), (1.0, -2.0)]

    a segment starting at (7, 4) and ending at (11, 8) is/are:
     @[(8.0, 5.0)]

PerlEdit

use strict;
use warnings;
use feature 'say';
use List::Util 'sum';

sub find_intersection {
   my($P1, $P2, $center, $radius) = @_;
   my @d = ($$P2[0] -     $$P1[0], $$P2[1] -     $$P1[1]);
   my @f = ($$P1[0] - $$center[0], $$P1[1] - $$center[1]);
   my $a = sum map { $_**2 } @d;
   my $b = 2 * ($f[0]*$d[0] + $f[1]*$d[1]);
   my $c = sum(map { $_**2 } @f) - $radius**2;
   my $D =  $b**2 - 4*$a*$c;

   return unless $D >= 0;
   my ($t1, $t2) = ( (-$b - sqrt $D) / (2*$a), (-$b + sqrt $D) / (2*$a) );
   return unless $t1 >= 0 and $t1 <= 1 or $t2 >= 0 and $t2  <= 1;

   my ($dx, $dy) = ($$P2[0] - $$P1[0], $$P2[1] - $$P1[1]);
   return ([$dx*$t1 + $$P1[0], $dy*$t1 + $$P1[1]],
           [$dx*$t2 + $$P1[0], $dy*$t2 + $$P1[1]])
}

my @data = (
   [ [-10, 11], [ 10, -9], [3, -5], 3 ],
   [ [-10, 11], [-11, 12], [3, -5], 3 ],
   [ [  3, -2], [  7, -2], [3, -5], 3 ],
   [ [  3, -2], [  7, -2], [0,  0], 4 ],
   [ [  0, -3], [  0,  6], [0,  0], 4 ],
   [ [  6,  3], [ 10,  7], [4,  2], 5 ],
   [ [  7,  4], [ 11, 18], [4,  2], 5 ],
);

sub rnd { map { sprintf('%.2f', $_) =~ s/\.00//r } @_ }

for my $d (@data) {
   my @solution = find_intersection @$d[0] , @$d[1] , @$d[2], @$d[3];
   say 'For input: ' . join ', ', (map { '('. join(',', @$_) .')' } @$d[0,1,2]), @$d[3];
   say 'Solutions: ' . (@solution > 1 ? join ', ', map { '('. join(',', rnd @$_) .')' } @solution : 'None');
   say '';
}
Output:
For input: (-10,11), (10,-9), (3,-5), 3
Solutions: (3,-2), (6,-5)

For input: (-10,11), (-11,12), (3,-5), 3
Solutions: None

For input: (3,-2), (7,-2), (3,-5), 3
Solutions: (3,-2), (3,-2)

For input: (3,-2), (7,-2), (0,0), 4
Solutions: (-3.46,-2), (3.46,-2)

For input: (0,-3), (0,6), (0,0), 4
Solutions: (0,-4), (0,4)

For input: (6,3), (10,7), (4,2), 5
Solutions: (1,-2), (8,5)

For input: (7,4), (11,18), (4,2), 5
Solutions: (5.03,-2.89), (7.46,5.61)

PhixEdit

Translation of: Go
Translation of: zkl
with javascript_semantics
constant epsilon = 1e-14 -- say
atom cx, cy, r, x1, y1, x2, y2
 
function sq(atom x) return x*x end function
 
function within(atom x, y)
    --
    -- checks whether a point is within a segment
    -- ie:    <-------d1------->
    --        <--d2--><---d3--->    -- within, d2+d3 ~= d1
    --   x1,y1^      ^x,y      ^x2,y2
    -- vs:
    --  <-d2->
    --  <-----------d3--------->    -- not "", d2+d3 > d1
    --  ^x,y - and obviously ditto when x,y is (say) out here^
    --
    -- (obviously only works when x,y is on the same line as x1,y1 to x2,y2)
    --
    atom d1 := sqrt(sq(x2-x1) + sq(y2-y1)), -- distance between end-points
         d2 := sqrt(sq(x -x1) + sq(y -y1)), -- distance from point to one end
         d3 := sqrt(sq(x2-x ) + sq(y2-y )), -- distance from point to other end
         delta := (d2 + d3) - d1
    return abs(delta) < epsilon -- true if delta is less than a small tolerance
end function
 
function pf(atom x,y)
    return sprintf("(%g,%g)",{x,y})
end function
 
function intersects(bool bSegment)
--
-- Returns the intersection points (if any) of a circle, center (cx,cy) with radius r,
-- and line containing the points (x1,y1) and (x2,y2) being either infinite or limited
-- to the segment drawn between those points.
--
    sequence res = {}
    atom A = y2 - y1,       sqA = sq(A),
         B = x1 - x2,       sqB = sq(B),
         C = x2*y1 - x1*y2, sqC = sq(C),
         sqr = r*r-cx*cx-cy*cy,
         a := sqA + sqB,
         b, c
    bool bDivA = false
    if abs(B)<epsilon then  -- B is zero or close to it
        b = 2 * (B*C + A*B*cx - sqA*cy)
        c = sqC + 2*A*C*cx - sqA*sqr
        bDivA = true -- (and later divide by A instead!)
    else
        b = 2 * (A*C + A*B*cy - sqB*cx)
        c = sqC + 2*B*C*cy - sqB*sqr
    end if
    atom d := b*b - 4*a*c -- discriminant
    if d>=0 then -- (-ve means line & circle do not intersect)
        d = sqrt(d)
        atom ux,uy, vx,vy
        if bDivA then       
            {uy,vy} = sq_div(sq_sub({+d,-d},b),2*a)
            {ux,vx} = sq_div(sq_sub(sq_mul(-B,{uy,vy}),C),A)
        else
            {ux,vx} = sq_div(sq_sub({+d,-d},b),2*a)
            {uy,vy} = sq_div(sq_sub(sq_mul(-A,{ux,vx}),C),B)
        end if
        if not bSegment or within(ux,uy) then
            res = append(res,pf(ux,uy))
        end if
        if d!=0 and (not bSegment or within(vx,vy)) then
            res = append(res,pf(vx,vy))
        end if              
    end if
    return res
end function
 
--                cx cy r    x1 y1  x2 y2 bSegment
constant tests = {{3,-5,3,{{-10,11, 10,-9,false},
                           {-10,11,-11,12,true},
                           {  3,-2,  7,-2,false}}},
                  {0, 0,4,{{  0,-3,  0, 6,false},
                           {  0,-3,  0, 6,true}}},
                  {4, 2,5,{{  6, 3, 10, 7,false},
                           {  7, 4, 11, 8,true}}}}
 
for t=1 to length(tests) do
    sequence lines
    {cx, cy, r, lines} = tests[t]
    string circle = sprintf("Circle at %s radius %d",{pf(cx,cy),r})
    for l=1 to length(lines) do
        bool bSegment
        {x1, y1, x2, y2, bSegment} = lines[l]
        sequence res = intersects(bSegment)
        string ls = iff(bSegment?"segment":"   line"),
               at = iff(length(res)?"intersect at "&join(res," and ")
                                   :"do not intersect")
        printf(1,"%s and %s %s to %s %s.\n",{circle,ls,pf(x1,y1),pf(x2,y2),at})
        circle = repeat(' ',length(circle))
    end for
end for
Output:
Circle at (3,-5) radius 3 and    line (-10,11) to (10,-9) intersect at (6,-5) and (3,-2).
                          and segment (-10,11) to (-11,12) do not intersect.
                          and    line (3,-2) to (7,-2) intersect at (3,-2).
Circle at (0,0) radius 4 and    line (0,-3) to (0,6) intersect at (0,4) and (0,-4).
                         and segment (0,-3) to (0,6) intersect at (0,4).
Circle at (4,2) radius 5 and    line (6,3) to (10,7) intersect at (8,5) and (1,-2).
                         and segment (7,4) to (11,8) intersect at (8,5).

RakuEdit

(formerly Perl 6) Extend solution space to 3D. Reference: this SO question and answers

sub LineCircularOBJintersection(@P1, @P2, @Centre, \Radius) {
   my @d = @P2 »-« @P1 ;           # d
   my @f = @P1 »-« @Centre ;       # c

   my \a = [+] @d»²;               # d dot d
   my \b = 2 * ([+] @f »*« @d);    # 2 * f dot d
   my \c = ([+] @f»²) - Radius²;   # f dot f - r²
   my \Δ =  b²-(4*a*c);            # discriminant

   if (Δ < 0) {
      return [];
   } else {
      my (\t1,\t2) = (-b - Δ.sqrt)/(2*a), (-b + Δ.sqrt)/(2*a);
      if 0t1|t21 {
         return @P1 »+« ( @P2 »-« @P1 ) »*» t1, @P1 »+« ( @P2 »-« @P1 ) »*» t2
      } else {
         return []
      }
   }
}

my \DATA = [
   [ <-10 11>, < 10 -9>, <3 -5>, 3 ],
   [ <-10 11>, <-11 12>, <3 -5>, 3 ],
   [ <  3 -2>, <  7 -2>, <3 -5>, 3 ],
   [ <  3 -2>, <  7 -2>, <0  0>, 4 ],
   [ <  0 -3>, <  0  6>, <0  0>, 4 ],
   [ <  6  3>, < 10  7>, <4  2>, 5 ],
   [ <  7  4>, < 11 18>, <4  2>, 5 ],
   [ <5  2 −2.26 >, <0.77 2 4>, <1 4 0>, 4 ]
];

for DATA {
   my @solution = LineCircularOBJintersection $_[0] , $_[1] , $_[2], $_[3];
   say "For data set: ", $_;
   say "Solution(s) is/are: ", @solution.Bool ?? @solution !! "None";
}
Output:
For data set: [(-10 11) (10 -9) (3 -5) 3]
Solution(s) is/are: [(3 -2) (6 -5)]
For data set: [(-10 11) (-11 12) (3 -5) 3]
Solution(s) is/are: None
For data set: [(3 -2) (7 -2) (3 -5) 3]
Solution(s) is/are: [(3 -2) (3 -2)]
For data set: [(3 -2) (7 -2) (0 0) 4]
Solution(s) is/are: [(-3.4641016151377544 -2) (3.4641016151377544 -2)]
For data set: [(0 -3) (0 6) (0 0) 4]
Solution(s) is/are: [(0 -4) (0 4)]
For data set: [(6 3) (10 7) (4 2) 5]
Solution(s) is/are: [(1 -2) (8 5)]
For data set: [(7 4) (11 18) (4 2) 5]
Solution(s) is/are: [(5.030680985703315 -2.892616550038399) (7.459885052032535 5.60959768211387)]
For data set: [(5 2 −2.26) (0.77 2 4) (1 4 0) 4]
Solution(s) is/are: [(4.2615520237084015 2 -1.1671668246843006) (1.13386504516801 2 3.461514141193441)]

REXXEdit

The formulae used for this REXX version were taken from the MathWorld webpage:   circle line intersection.

/*REXX program calculates  where  (or if)  a  line  intersects  (or tengents)  a cirle. */
/*───────────────────────────────────── line= x1,y1  x2,y2;   circle is at 0,0, radius=r*/
parse arg x1 y1 x2 y2 cx cy r .                  /*obtain optional arguments from the CL*/
if x1=='' | x1==","  then x1=  0                 /*Not specified?  Then use the default.*/
if y1=='' | y1==","  then y1= -3                 /* "      "         "   "   "     "    */
if x2=='' | x2==","  then x2=  0                 /* "      "         "   "   "     "    */
if y2=='' | y2==","  then y2=  6                 /* "      "         "   "   "     "    */
if cx=='' | cx==","  then cx=  0                 /* "      "         "   "   "     "    */
if cy=='' | cy==","  then cy=  0                 /* "      "         "   "   "     "    */
if r =='' | r ==","  then r =  4                 /* "      "         "   "   "     "    */
x_1= x1;         x1= x1 + cx;        y_1= y1;        y1= y1 + cy
x_2= x2;         x2= x2 + cx;        y_2= y2;        y2= y2 + cy
dx= x2 - x1;     dy= y2 - y1
                                        dr2= dx**2 + dy**2
  D=  x1 * y2   -   x2 * y1;               r2= r**2;    D2= D**2
                                   $= sqrt(r2 * dr2  -  D2)
ix1= ( D * dy   +   sgn(dy) * dx * $) / dr2
ix2= ( D * dy   -   sgn(dy) * dx * $) / dr2
iy1= (-D * dx   +   abs(dy)      * $) / dr2
iy2= (-D * dx   -   abs(dy)      * $) / dr2
incidence= (r2 * dr2  -  D2)  /  1
say 'incidence='   incidence
                         @potla= 'points on the line are: '
if incidence<0  then do
                     say @potla ' ('||x_1","y_1')  and  ('||x_2","y_2')  are: '  ix1","iy1
                     say "The line doesn't intersect the circle with radius: "   r
                     end
if incidence=0  then do
                     say @potla ' ('||x_1","y_1')  and  ('||x_2","y_2')  are: '  ix1","iy1
                     say "The line is tangent to circle with radius: "           r
                     end
if incidence>0  then do
                     say @potla ' ('||x_1","y_1')  and  ('||x_2","y_2')  are: '  ix1","iy1
                     say "The line is secant to circle with radius: "            r
                     end
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
sgn:  procedure;  if arg(1)<0  then return -1;           return 1
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x;  if x=0  then return 0;  d=digits();  numeric digits;  h=d+6
      numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g "E" _ .; g=g *.5'e'_ %2
        do j=0  while h>9;      m.j= h;              h= h%2 + 1;      end  /*j*/
        do k=j+5  to 0  by -1;  numeric digits m.k;  g= (g+x/g) *.5;  end  /*k*/; return g
output   when using the default inputs:
incidence= 1296
points on the line are:   (0,-3)  and  (0,6)  are:  0,4
The line is secant to circle with radius:  4

RubyEdit

Translation of: C++
EPS = 1e-14

def sq(x)
    return x * x
end

def intersects(p1, p2, cp, r, segment)
    res = []
    (x0, y0) = cp
    (x1, y1) = p1
    (x2, y2) = p2
    aa = y2 - y1
    bb = x1 - x2
    cc = x2 * y1 - x1 * y2
    a = sq(aa) + sq(bb)
    if bb.abs >= EPS then
        b = 2 * (aa * cc + aa * bb * y0 - sq(bb) * x0)
        c = sq(cc) + 2 * bb * cc * y0 - sq(bb) * (sq(r) - sq(x0) - sq(y0))
        bnz = true
    else
        b = 2 * (bb * cc + aa * bb * x0 - sq(aa) * y0)
        c = sq(cc) + 2 * aa * cc * x0 - sq(aa) * (sq(r) - sq(x0) - sq(y0))
        bnz = false
    end
    d = sq(b) - 4 * a * c # disciminant
    if d < 0 then
        return res
    end

    # checks whether a point is within a segment
    within = ->(x, y) {
        d1 = Math.sqrt(sq(x2 - x1) + sq(y2 - y1))   # distance between end-points
        d2 = Math.sqrt(sq(x - x1) + sq(y - y1))     # distance from point to one end
        d3 = Math.sqrt(sq(x2 - x) + sq(y2 - y))     # distance from point to other end
        delta = d1 - d2 - d3
        return delta.abs < EPS                      # true if delta is less than a small tolerance
    }

    fx = ->(x) {
        return -(aa * x + cc) / bb
    }

    fy = ->(y) {
        return -(bb * y + cc) / aa
    }

    rxy = ->(x, y) {
        if not segment or within.call(x, y) then
            if x == 0.0 then
                x = 0.0
            end
            if y == 0.0 then
                y = 0.0
            end
            res << [x, y]
        end
    }

    if d == 0.0 then
        # line is tangent to circle, so just one intersect at most
        if bnz then
            x = -b / (2 * a)
            y = fx.call(x)
            rxy.call(x, y)
        else
            y = -b / (2 * a)
            x = fy.call(y)
            rxy.call(x, y)
        end
    else
        # two intersects at most
        d = Math.sqrt(d)
        if bnz then
            x = (-b + d) / (2 * a)
            y = fx.call(x)
            rxy.call(x, y)
            x = (-b - d) / (2 * a)
            y = fx.call(x)
            rxy.call(x, y)
        else
            y = (-b + d) / (2 * a)
            x = fy.call(y)
            rxy.call(x, y)
            y = (-b - d) / (2 * a)
            x = fy.call(y)
            rxy.call(x, y)
        end
    end

    return res
end

def main
    print "The intersection points (if any) between:\n"

    cp = [3.0, -5.0]
    r = 3.0
    print "  A circle, center %s with radius %f, and:\n" % [cp, r]

    p1 = [-10.0, 11.0]
    p2 = [10.0, -9.0]
    print "    a line containing the points %s and %s is/are:\n" % [p1, p2]
    print "      %s\n" % [intersects(p1, p2, cp, r, false)]

    p2 = [-10.0, 12.0]
    print "    a segment starting at %s and ending at %s is/are:\n" % [p1, p2]
    print "      %s\n" % [intersects(p1, p2, cp, r, true)]

    p1 = [3.0, -2.0]
    p2 = [7.0, -2.0]
    print "    a horizontal line containing the points %s and %s is/are:\n" % [p1, p2]
    print "      %s\n" % [intersects(p1, p2, cp, r, false)]

    cp = [0.0, 0.0]
    r = 4.0
    print "  A circle, center %s with radius %f, and:\n" % [cp, r]

    p1 = [0.0, -3.0]
    p2 = [0.0, 6.0]
    print "    a vertical line containing the points %s and %s is/are:\n" % [p1, p2]
    print "      %s\n" % [intersects(p1, p2, cp, r, false)]
    print "    a vertical line segment containing the points %s and %s is/are:\n" % [p1, p2]
    print "      %s\n" % [intersects(p1, p2, cp, r, true)]

    cp = [4.0, 2.0]
    r = 5.0
    print "  A circle, center %s with radius %f, and:\n" % [cp, r]

    p1 = [6.0, 3.0]
    p2 = [10.0, 7.0]
    print "    a line containing the points %s and %s is/are:\n" % [p1, p2]
    print "      %s\n" % [intersects(p1, p2, cp, r, false)]

    p1 = [7.0, 4.0]
    p2 = [11.0, 8.0]
    print "    a segment starting at %s and ending at %s is/are:\n" % [p1, p2]
    print "      %s\n" % [intersects(p1, p2, cp, r, true)]
end

main()
Output:
The intersection points (if any) between:
  A circle, center [3.0, -5.0] with radius 3.000000, and:
    a line containing the points [-10.0, 11.0] and [10.0, -9.0] is/are:
      [[6.0, -5.0], [3.0, -2.0]]
    a segment starting at [-10.0, 11.0] and ending at [-10.0, 12.0] is/are:
      []
    a horizontal line containing the points [3.0, -2.0] and [7.0, -2.0] is/are:
      [[3.0, -2.0]]
  A circle, center [0.0, 0.0] with radius 4.000000, and:
    a vertical line containing the points [0.0, -3.0] and [0.0, 6.0] is/are:
      [[0.0, 4.0], [0.0, -4.0]]
    a vertical line segment containing the points [0.0, -3.0] and [0.0, 6.0] is/are:
      [[0.0, 4.0]]
  A circle, center [4.0, 2.0] with radius 5.000000, and:
    a line containing the points [6.0, 3.0] and [10.0, 7.0] is/are:
      [[8.0, 5.0], [1.0, -2.0]]
    a segment starting at [7.0, 4.0] and ending at [11.0, 8.0] is/are:
      [[8.0, 5.0]]

RustEdit

Translation of: C++
use assert_approx_eq::assert_approx_eq;

const EPS: f64 = 1e-14;

pub struct Point {
    x: f64,
    y: f64,
}

pub struct Line {
    p1: Point,
    p2: Point,
}

impl Line {
    pub fn circle_intersections(&self, mx: f64, my: f64, r: f64, segment: bool) -> Vec<Point> {
        let mut intersections: Vec<Point> = Vec::new();

        let x0 = mx;
        let y0 = my;
        let x1 = self.p1.x;
        let y1 = self.p1.y;
        let x2 = self.p2.x;
        let y2 = self.p2.y;

        let ca = y2 - y1;
        let cb = x1 - x2;
        let cc = x2 * y1 - x1 * y2;

        let a = ca.powi(2) + cb.powi(2);
        let mut b = 0.0;
        let mut c = 0.0;
        let mut bnz = true;

        if cb.abs() >= EPS {
            b = 2.0 * (ca * cc + ca * cb * y0 - cb.powi(2) * x0);
            c = cc.powi(2) + 2.0 * cb * cc * y0
                - cb.powi(2) * (r.powi(2) - x0.powi(2) - y0.powi(2));
        } else {
            b = 2.0 * (cb * cc + ca * cb * x0 - ca.powi(2) * y0);
            c = cc.powi(2) + 2.0 * ca * cc * x0
                - ca.powi(2) * (r.powi(2) - x0.powi(2) - y0.powi(2));
            bnz = false;
        }
        let mut d = b.powi(2) - 4.0 * a * c;
        if d < 0.0 {
            return intersections;
        }

        fn within(x: f64, y: f64, x1: f64, y1: f64, x2: f64, y2: f64) -> bool {
            let d1 = ((x2 - x1).powi(2) + (y2 - y1).powi(2)).sqrt(); // distance between end-points
            let d2 = ((x - x1).powi(2) + (y - y1).powi(2)).sqrt(); // distance from point to one end
            let d3 = ((x2 - x).powi(2) + (y2 - y).powi(2)).sqrt(); // distance from point to other end
            let delta = d1 - d2 - d3;
            return delta.abs() < EPS;
        }

        fn fx(x: f64, ca: f64, cb: f64, cc: f64) -> f64 {
            -(ca * x + cc) / cb
        }

        fn fy(y: f64, ca: f64, cb: f64, cc: f64) -> f64 {
            -(cb * y + cc) / ca
        }

        fn rxy(
            x: f64,
            y: f64,
            x1: f64,
            y1: f64,
            x2: f64,
            y2: f64,
            segment: bool,
            intersections: &mut Vec<Point>,
        ) {
            if !segment || within(x, y, x1, y1, x2, y2) {
                let point = Point { x: x, y: y };
                intersections.push(point);
            }
        }

        if d == 0.0 {
            if bnz {
                let x = -b / (2.0 * a);
                let y = fx(x, ca, cb, cc);
                rxy(x, y, x1, y1, x2, y2, segment, &mut intersections);
            } else {
                let y = -b / (2.0 * a);
                let x = fy(y, ca, cb, cc);
                rxy(x, y, x1, y1, x2, y2, segment, &mut intersections);
            }
        } else {
            d = d.sqrt();
            if bnz {
                let x = (-b + d) / (2.0 * a);
                let y = fx(x, ca, cb, cc);
                rxy(x, y, x1, y1, x2, y2, segment, &mut intersections);
                let x = (-b - d) / (2.0 * a);
                let y = fx(x, ca, cb, cc);
                rxy(x, y, x1, y1, x2, y2, segment, &mut intersections);
            } else {
                let y = (-b + d) / (2.0 * a);
                let x = fy(y, ca, cb, cc);
                rxy(x, y, x1, y1, x2, y2, segment, &mut intersections);
                let y = (-b - d) / (2.0 * a);
                let x = fy(y, ca, cb, cc);
                rxy(x, y, x1, y1, x2, y2, segment, &mut intersections);
            }
        }

        intersections.sort_unstable_by(|a, b| a.x.partial_cmp(&b.x).unwrap());
        intersections
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_circle_line_intersections() {
        let mut p1 = Point { x: -10.0, y: 11.0 };
        let mut p2 = Point { x: 10.0, y: -9.0 };
        let mut line = Line { p1: p1, p2: p2 };

        let result1 = line.circle_intersections(3.0, -5.0, 3.0, false);
        assert_eq!(result1.len(), 2);
        assert_approx_eq!(result1[0].x, 3.0);
        assert_approx_eq!(result1[0].y, -2.0);
        assert_approx_eq!(result1[1].x, 6.0);
        assert_approx_eq!(result1[1].y, -5.0);

        p1 = Point { x: -10.0, y: 11.0 };
        p2 = Point { x: -11.0, y: -12.0 };
        line = Line { p1: p1, p2: p2 };
        let result2 = line.circle_intersections(3.0, -5.0, 3.0, true);
        assert_eq!(result2.len(), 0);

        p1 = Point { x: 3.0, y: -2.0 };
        p2 = Point { x: 7.0, y: -2.0 };
        line = Line { p1, p2 };
        let result3 = line.circle_intersections(3.0, -5.0, 3.0, true);
        assert_eq!(result3.len(), 1);
        assert_approx_eq!(result3[0].x, 3.0);
        assert_approx_eq!(result3[0].y, -2.0);

        p1 = Point { x: 0.0, y: -3.0 };
        p2 = Point { x: 0.0, y: 6.0 };
        line = Line { p1, p2 };
        let result4 = line.circle_intersections(0.0, 0.0, 4.0, false);
        assert_eq!(result4.len(), 2);
        assert_approx_eq!(result4[0].x, 0.0);
        assert_approx_eq!(result4[1].x, 0.0);

        let result5 = line.circle_intersections(0.0, 0.0, 4.0, true);
        assert_eq!(result5.len(), 1);

        p1 = Point { x: 6.0, y: 3.0 };
        p2 = Point { x: 10.0, y: 7.0 };
        line = Line { p1, p2 };
        let result6 = line.circle_intersections(4.0, 2.0, 5.0, false);
        assert_eq!(result6.len(), 2);
        assert_approx_eq!(result6[0].x, 1.0);
        assert_approx_eq!(result6[0].y, -2.0);
        assert_approx_eq!(result6[1].x, 8.0);
        assert_approx_eq!(result6[1].y, 5.0);

        p1 = Point { x: 7.0, y: 4.0 };
        p2 = Point { x: 11.0, y: 8.0 };
        line = Line { p1, p2 };
        let result7 = line.circle_intersections(4.0, 2.0, 5.0, true);
        assert_eq!(result7.len(), 1);
        assert_approx_eq!(result7[0].x, 8.0);
        assert_approx_eq!(result7[0].y, 5.0);
    }
}
Output:
running 1 test
test tests::test_circle_line_intersections ... ok
test result: ok. 1


SwiftEdit

Translation of: Java
import Foundation
import CoreGraphics

func lineCircleIntersection(start: NSPoint, end: NSPoint, center: NSPoint,
                            radius: CGFloat, segment: Bool) -> [NSPoint] {
    var result: [NSPoint] = []
    let angle = atan2(end.y - start.y, end.x - start.x)
    var at = AffineTransform(rotationByRadians: angle)
    at.invert()
    at.translate(x: -center.x, y: -center.y)
    let p1 = at.transform(start)
    let p2 = at.transform(end)
    let minX = min(p1.x, p2.x), maxX = max(p1.x, p2.x)
    let y = p1.y
    at.invert()
    func addPoint(x: CGFloat, y: CGFloat) {
        if !segment || (x <= maxX && x >= minX) {
            result.append(at.transform(NSMakePoint(x, y)))
        }
    }
    if y == radius || y == -radius {
        addPoint(x: 0, y: y)
    } else if y < radius && y > -radius {
        let x = (radius * radius - y * y).squareRoot()
        addPoint(x: -x, y: y)
        addPoint(x: x, y: y)
    }
    return result
}

func toString(points: [NSPoint]) -> String {
    var result = "["
    result += points.map{String(format: "(%.4f, %.4f)", $0.x, $0.y)}.joined(separator: ", ")
    result += "]"
    return result
}

var center = NSMakePoint(3, -5)
var radius: CGFloat = 3

print("The intersection points (if any) between:")
print("\n  A circle, center (3, -5) with radius 3, and:")
print("\n    a line containing the points (-10, 11) and (10, -9) is/are:")
var points = lineCircleIntersection(start: NSMakePoint(-10, 11), end: NSMakePoint(10, -9),
                                    center: center, radius: radius,
                                    segment: false)
print("     \(toString(points: points))")
print("\n    a segment starting at (-10, 11) and ending at (-11, 12) is/are")
points = lineCircleIntersection(start: NSMakePoint(-10, 11), end: NSMakePoint(-11, 12),
                                center: center, radius: radius,
                                segment: true)
print("     \(toString(points: points))")
print("\n    a horizontal line containing the points (3, -2) and (7, -2) is/are:")
points = lineCircleIntersection(start: NSMakePoint(3, -2), end: NSMakePoint(7, -2),
                                center: center, radius: radius,
                                segment: false)
print("     \(toString(points: points))")

center.x = 0
center.y = 0
radius = 4

print("\n  A circle, center (0, 0) with radius 4, and:")
print("\n    a vertical line containing the points (0, -3) and (0, 6) is/are:")
points = lineCircleIntersection(start: NSMakePoint(0, -3), end: NSMakePoint(0, 6),
                                center: center, radius: radius,
                                segment: false)
print("     \(toString(points: points))")
print("\n    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:")
points = lineCircleIntersection(start: NSMakePoint(0, -3), end: NSMakePoint(0, 6),
                                center: center, radius: radius,
                                segment: true)
print("     \(toString(points: points))")

center.x = 4
center.y = 2
radius = 5

print("\n  A circle, center (4, 2) with radius 5, and:")
print("\n    a line containing the points (6, 3) and (10, 7) is/are:")
points = lineCircleIntersection(start: NSMakePoint(6, 3), end: NSMakePoint(10, 7),
                                center: center, radius: radius,
                                segment: false)
print("     \(toString(points: points))")
print("\n    a segment starting at (7, 4) and ending at (11, 8) is/are:")
points = lineCircleIntersection(start: NSMakePoint(7, 4), end: NSMakePoint(11, 8),
                                center: center, radius: radius,
                                segment: true)
print("     \(toString(points: points))")
Output:
The intersection points (if any) between:

  A circle, center (3, -5) with radius 3, and:

    a line containing the points (-10, 11) and (10, -9) is/are:
     [(3.0000, -2.0000), (6.0000, -5.0000)]

    a segment starting at (-10, 11) and ending at (-11, 12) is/are
     []

    a horizontal line containing the points (3, -2) and (7, -2) is/are:
     [(3.0000, -2.0000)]

  A circle, center (0, 0) with radius 4, and:

    a vertical line containing the points (0, -3) and (0, 6) is/are:
     [(-0.0000, -4.0000), (0.0000, 4.0000)]

    a vertical segment starting at (0, -3) and ending at (0, 6) is/are:
     [(0.0000, 4.0000)]

  A circle, center (4, 2) with radius 5, and:

    a line containing the points (6, 3) and (10, 7) is/are:
     [(1.0000, -2.0000), (8.0000, 5.0000)]

    a segment starting at (7, 4) and ending at (11, 8) is/are:
     [(8.0000, 5.0000)]

Visual Basic .NETEdit

Translation of: C#
Module Module1

    Structure Point
        Implements IComparable(Of Point)

        Public Sub New(mx As Double, my As Double)
            X = mx
            Y = my
        End Sub

        Public ReadOnly Property X As Double
        Public ReadOnly Property Y As Double

        Public Function CompareTo(other As Point) As Integer Implements IComparable(Of Point).CompareTo
            Dim c = X.CompareTo(other.X)
            If c <> 0 Then
                Return c
            End If
            Return Y.CompareTo(other.Y)
        End Function

        Public Overrides Function ToString() As String
            Return String.Format("({0}, {1})", X, Y)
        End Function
    End Structure

    Structure Line
        Public Sub New(mp1 As Point, mp2 As Point, Optional segment As Boolean = False)
            If P2.CompareTo(P1) < 0 Then
                P1 = mp2
                P2 = mp1
            Else
                P1 = mp1
                P2 = mp2
            End If
            IsSegment = segment
            If P1.X = P2.X Then
                Slope = Double.PositiveInfinity
                YIntercept = Double.NaN
            Else
                Slope = (P2.Y - P1.Y) / (P2.X - P1.X)
                YIntercept = P2.Y - Slope * P2.X
            End If
        End Sub

        Public ReadOnly Property P1 As Point
        Public ReadOnly Property P2 As Point
        Public ReadOnly Property Slope As Double
        Public ReadOnly Property YIntercept As Double
        Public ReadOnly Property IsSegment As Boolean

        Public Function IsVertical() As Boolean
            Return P1.X = P2.X
        End Function

        Public Overrides Function ToString() As String
            Return String.Format("[{0}, {1}]", P1, P2)
        End Function
    End Structure

    Structure Circle
        Public Sub New(c As Point, r As Double)
            Center = c
            Radius = r
        End Sub

        Public ReadOnly Property Center As Point
        Public ReadOnly Property Radius As Double

        Public Function X() As Double
            Return Center.X
        End Function

        Public Function Y() As Double
            Return Center.Y
        End Function

        Public Overrides Function ToString() As String
            Return String.Format("{{ C:{0}, R:{1} }}", Center, Radius)
        End Function
    End Structure

    Function Intersection(oc As Circle, ol As Line) As IEnumerable(Of Point)
        Dim LineIntersection = Iterator Function(ic As Circle, il As Line) As IEnumerable(Of Point)
                                   Dim m = il.Slope
                                   Dim c = il.YIntercept
                                   Dim p = ic.X
                                   Dim q = ic.Y
                                   Dim r = ic.Radius

                                   If il.IsVertical Then
                                       Dim x = il.P1.X
                                       Dim B = -2 * q
                                       Dim CC = p * p + q * q - r * r + x * x - 2 * p * x
                                       Dim D = B * B - 4 * CC
                                       If D = 0 Then
                                           Yield New Point(x, -q)
                                       ElseIf D > 0 Then
                                           D = Math.Sqrt(D)
                                           Yield New Point(x, (-B - D) / 2)
                                           Yield New Point(x, (-B + D) / 2)
                                       End If
                                   Else
                                       Dim A = m * m + 1
                                       Dim B = 2 * (m * c - m * q - p)
                                       Dim CC = p * p + q * q - r * r + c * c - 2 * c * q
                                       Dim D = B * B - 4 * A * CC
                                       If D = 0 Then
                                           Dim x = -B / (2 * A)
                                           Dim y = m * x + c
                                           Yield New Point(x, y)
                                       ElseIf D > 0 Then
                                           D = Math.Sqrt(D)
                                           Dim x = (-B - D) / (2 * A)
                                           Dim y = m * x + c
                                           Yield New Point(x, y)
                                           x = (-B + D) / (2 * A)
                                           y = m * x + c
                                           Yield New Point(x, y)
                                       End If
                                   End If
                               End Function

        Dim int = LineIntersection(oc, ol)
        If ol.IsSegment Then
            Return int.Where(Function(p) p.CompareTo(ol.P1) >= 0 AndAlso p.CompareTo(ol.P2) <= 0)
        Else
            Return int
        End If
    End Function

    Sub Print(c As Circle, lines() As Line)
        Console.WriteLine("Circle: {0}", c)
        For Each line In lines
            Console.Write(vbTab)
            If line.IsSegment Then
                Console.Write("Segment: ")
            Else
                Console.Write("Line: ")
            End If
            Console.WriteLine(line)

            Dim points = Intersection(c, line).ToList

            Console.Write(vbTab + vbTab)
            If points.Count = 0 Then
                Console.WriteLine("do not intersect")
            Else
                Console.WriteLine("intersect at {0}", String.Join(" and ", points))
            End If
        Next
        Console.WriteLine()
    End Sub

    Sub Main()
        Dim c = New Circle(New Point(3, -5), 3)
        Dim lines() As Line = {
            New Line(New Point(-10, 11), New Point(10, -9)),
            New Line(New Point(-10, 11), New Point(-11, 12), True),
            New Line(New Point(3, -2), New Point(7, -2))
            }
        Print(c, lines)

        c = New Circle(New Point(0, 0), 4)
        lines = {
            New Line(New Point(0, -3), New Point(0, 6)),
            New Line(New Point(0, -3), New Point(0, 6), True)
            }
        Print(c, lines)

        c = New Circle(New Point(4, 2), 5)
        lines = {
            New Line(New Point(6, 3), New Point(10, 7)),
            New Line(New Point(7, 4), New Point(11, 8), True)
            }
        Print(c, lines)
    End Sub

End Module
Output:
Circle: { C:(3, -5), R:3 }
        Line: [(-10, 11), (10, -9)]
                intersect at (3, -2) and (6, -5)
        Segment: [(-10, 11), (-11, 12)]
                do not intersect
        Line: [(3, -2), (7, -2)]
                intersect at (3, -2)

Circle: { C:(0, 0), R:4 }
        Line: [(0, -3), (0, 6)]
                intersect at (0, -4) and (0, 4)
        Segment: [(0, -3), (0, 6)]
                intersect at (0, 4)

Circle: { C:(4, 2), R:5 }
        Line: [(6, 3), (10, 7)]
                intersect at (1, -2) and (8, 5)
        Segment: [(7, 4), (11, 8)]
                intersect at (8, 5)

WrenEdit

Translation of: Kotlin
Library: Wren-dynamic
import "/dynamic" for Tuple

var Point = Tuple.create("Point", ["x", "y"])

var eps = 1e-14

var intersects = Fn.new { |p1, p2, cp, r, segment|
    var res = []
    var x0 = cp.x
    var y0 = cp.y
    var x1 = p1.x
    var y1 = p1.y
    var x2 = p2.x
    var y2 = p2.y
    var A = y2 - y1
    var B = x1 - x2
    var C = x2 * y1 - x1 * y2
    var a = A * A + B * B
    var b
    var c
    var bnz = true
    if (B.abs >= eps) {
        b = 2 * (A * C + A * B * y0 - B * B * x0)
        c = C * C + 2 * B * C * y0 - B * B * (r * r - x0 * x0 - y0 * y0)
    } else {
        b = 2 * (B * C + A * B * x0 - A * A * y0)
        c = C * C + 2 * A * C * x0 - A * A * (r * r - x0 * x0 - y0 * y0)
        bnz = false
    }
    var d = b * b - 4 * a * c // discriminant
    if (d < 0) {
        return "[]"
    }

    // checks whether a point is within a segment
    var within = Fn.new { |x0, y0|
        var d1 = ((x2 - x1)*(x2 - x1) + (y2 - y1)*(y2 - y1)).sqrt  // distance between end-points
        var d2 = ((x0 - x1)*(x0 - x1) + (y0 - y1)*(y0 - y1)).sqrt  // distance from point to one end
        var d3 = ((x2 - x0)*(x2 - x0) + (y2 - y0)*(y2 - y0)).sqrt  // distance from point to other end
        var delta = d1 - d2 - d3
        return delta.abs < eps // true if delta is less than a small tolerance
    }

    var x = 0
    var fx = Fn.new { -(A * x + C) / B }

    var y = 0
    var fy = Fn.new { -(B * y + C) / A }

    var rxy = Fn.new {
        if (!segment || within.call(x, y)) {
            res.add(Point.new(x, y))
        }
    }
 
    if (d == 0) {
        // line is tangent to circle, so just one intersect at most
        if (bnz) {
            x = -b / (2 * a)
            y = fx.call()
        } else {
            y = -b / (2 * a)
            x = fy.call()
        }
        rxy.call()
    } else {
        // two intersects at most
        d = d.sqrt
        if (bnz) {
            x = (-b + d) / (2 * a)
            y = fx.call()
            rxy.call()
            x = (-b - d) / (2 * a)
            y = fx.call()
            rxy.call()
        } else {
            y = (-b + d) / (2 * a)
            x = fy.call()
            rxy.call()
            y = (-b - d) / (2 * a)
            x = fy.call()
            rxy.call()
        }
    }

    // get rid of any negative zeros and return as a string
    return res.toString.replace("-0,", "0,").replace("-0]", "0]")
}

System.print("The intersection points (if any) between:")

var cp = Point.new(3, -5)
var r = 3
System.print("  A circle, center %(cp) with radius %(r), and:")

var p1 = Point.new(-10, 11)
var p2 = Point.new( 10, -9)
System.print("    a line containing the points %(p1) and %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, false))")

p2 = Point.new(-10, 12)
System.print("    a segment starting at %(p1) and ending at %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, true))")

p1 = Point.new(3, -2)
p2 = Point.new(7, -2)
System.print("    a horizontal line containing the points %(p1) and %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, false))")

cp = Point.new(0, 0)
r = 4
System.print("  A circle, center %(cp) with radius %(r), and:")

p1 = Point.new(0, -3)
p2 = Point.new(0,  6)
System.print("    a vertical line containing the points %(p1) and %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, false))")
System.print("    a vertical segment containing the points %(p1) and %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, true))")

cp = Point.new(4, 2)
r = 5
System.print("  A circle, center %(cp) with radius %(r), and:")

p1 = Point.new( 6, 3)
p2 = Point.new(10, 7)
System.print("    a line containing the points %(p1) and %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, false))")

p1 = Point.new( 7, 4)
p2 = Point.new(11, 8)
System.print("    a segment starting at %(p1) and ending at %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, true))")

cp = Point.new(10, 10)
r = 5
System.print("  A circle, center %(cp) with radius %(r), and:")

p1 = Point.new( 5,  0)
p2 = Point.new( 5, 20)
System.print("    a vertical line containing the points %(p1) and %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, false))")

p1 = Point.new(-5, 10)
p2 = Point.new( 5, 10)
System.print("    a horizontal segment starting at %(p1) and ending at %(p2) is/are:")
System.print("     %(intersects.call(p1, p2, cp, r, true))")
Output:
The intersection points (if any) between:
  A circle, center (3, -5) with radius 3, and:
    a line containing the points (-10, 11) and (10, -9) is/are:
     [(6, -5), (3, -2)]
    a segment starting at (-10, 11) and ending at (-10, 12) is/are:
     []
    a horizontal line containing the points (3, -2) and (7, -2) is/are:
     [(3, -2)]
  A circle, center (0, 0) with radius 4, and:
    a vertical line containing the points (0, -3) and (0, 6) is/are:
     [(0, 4), (0, -4)]
    a vertical segment containing the points (0, -3) and (0, 6) is/are:
     [(0, 4)]
  A circle, center (4, 2) with radius 5, and:
    a line containing the points (6, 3) and (10, 7) is/are:
     [(8, 5), (1, -2)]
    a segment starting at (7, 4) and ending at (11, 8) is/are:
     [(8, 5)]
  A circle, center (10, 10) with radius 5, and:
    a vertical line containing the points (5, 0) and (5, 20) is/are:
     [(5, 10)]
    a horizontal segment starting at (-5, 10) and ending at (5, 10) is/are:
     [(5, 10)]

zklEdit

Translation of: Go
const EPS=1e-14;	// a close-ness to zero
    // p1,p2 are (x,y), circle is ( (x,y),r )
fcn intersectLineCircle(p1,p2, circle, segment=False)	// assume line
{
   cx,cy := circle[0].apply("toFloat");
   r     := circle[1].toFloat();
   x1,y1 := p1.apply("toFloat");   x2,y2 := p2.apply("toFloat");
   A,B,C,a := (y2 - y1),  (x1 - x2),  (x2*y1 - x1*y2),  (A*A + B*B);
   b,c,bnz := 0,0,True;
   if(B.closeTo(0,EPS)){ // B is zero or close to it
      b   = 2.0 * (B*C + A*B*cx - A*A*cy);
      c   = C*C + 2.0*A*C*cx - A*A*(r*r - cx*cx - cy*cy);
      bnz = False
   }else{
      b   = 2.0*( A*C + A*B*cy - B*B*cx );
      c   = C*C + 2.0*B*C*cy - B*B*( r*r - cx*cx - cy*cy );
   }
   d := b*b - 4.0*a*c;		// discriminant
   if(d<0.0){			// no real solution?  zero --> one solution
      if (d>-0.005) d=0.0;	// close enough to zero
      else return(T);		// no intersection
   }
   d=d.sqrt();

   reg ux,uy, vx,vy;
   if(bnz){
      ux,vx = (-b + d) / (2*a),  (-b - d) / (2*a);
      uy,vy = -(A*ux + C) / B,   -(A*vx + C) / B;
   }else{
      uy,vy = (-b + d) / (2*a), (-b - d) / (2*a);
      ux,vx = -(B*uy + C) / A,  -(B*vy + C) / A;
   }

   if(segment){
      within:='wrap(x,y){  // is (x,y) within segment p1 p2?
	 d1:=( (x2 - x1).pow(2) + (y2 - y1).pow(2) ).sqrt();
	 d2:=( (x  - x1).pow(2) + (y  - y1).pow(2) ).sqrt();
	 d3:=( (x2 - x) .pow(2) + (y2 - y) .pow(2) ).sqrt();
	 (d1 - d2 - d3).closeTo(0,EPS);
      };

      i1,i2 := within(ux,uy), within(vx,vy);
      if(d==0) return(if(i1) T(ux,uy) else T);
      return(T( i1 and T(ux,uy), i2 and T(vx,vy) ).filter())
   }

   if(d==0)  return( T( T(ux,uy) ) );
   return( T(ux,uy), T(vx,vy) )
}
circle:=T( T(3,-5),3 ); p1,p2 := T(-10,11), T( 10,-9);
println("Circle @ ",circle); lcpp(p1,p2,circle);
p2:=T(-11,12);               lcpp(p1,p2,circle,True);
p1,p2 := T(3,-2), T(7,-2);   lcpp(p1,p2,circle);

circle:=T( T(0,0),4 ); p1,p2 := T(0,-3), T(0,6);
println("\nCircle @ ",circle);
lcpp(p1,p2,circle);
lcpp(p1,p2,circle,True);

circle:=T( T(4,2),5 ); p1,p2 := T(6,3), T(10,7);
println("\nCircle @ ",circle);
lcpp(p1,p2,circle);
p1,p2 := T(7,4), T(11,8); lcpp(p1,p2,circle,True);

fcn lcpp(p1,p2,circle,segment=False){
   println("  %s %s -- %s intersects at %s"
      .fmt(segment and "Segment" or "Line   ",
           p1,p2,intersectLineCircle(p1,p2, circle,segment)));
}
Output:
Circle @ L(L(3,-5),3)
  Line    L(-10,11) -- L(10,-9) intersects at L(L(6,-5),L(3,-2))
  Segment L(-10,11) -- L(-11,12) intersects at L()
  Line    L(3,-2) -- L(7,-2) intersects at L(L(3,-2))

Circle @ L(L(0,0),4)
  Line    L(0,-3) -- L(0,6) intersects at L(L(0,4),L(0,-4))
  Segment L(0,-3) -- L(0,6) intersects at L(L(0,4))

Circle @ L(L(4,2),5)
  Line    L(6,3) -- L(10,7) intersects at L(L(8,5),L(1,-2))
  Segment L(7,4) -- L(11,8) intersects at L(L(8,5))