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Line circle intersection

From Rosetta Code
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.

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

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).

References


C[edit]

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)

Go[edit]

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))
}
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)]

Haskell[edit]

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)]


Julia[edit]

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 ? 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)

Kotlin[edit]

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)]

Perl[edit]

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)

Phix[edit]

Translation of: Go
Translation of: zkl
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
{cx, cy, r, sequence lines} = tests[t]
string circle = sprintf("Circle at %s radius %d",{pf(cx,cy),r})
for l=1 to length(lines) do
{x1, y1, x2, y2, bool 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).

Raku[edit]

(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=-(4*a*c); # discriminant
 
if (Δ < 0) {
return [];
} else {
my (\t1,\t2) = (-b - Δ.sqrt)/(2*a), (-b + Δ.sqrt)/(2*a);
if 0 ≤ t1|t2 ≤ 1 {
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 22.26 >, <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)]

REXX[edit]

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

zkl[edit]

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))