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
- See Math Stack Exchange for development of the formulae needed.
- See Wolfram for the formulae needed.
AutoHotkey
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]
C
#include <math.h>
#include <stdbool.h>
#include <stdio.h>
const double eps = 1e-14;
typedef struct point_t {
double x, y;
} point;
point make_point(double x, double y) {
point p = { x, y };
return p;
}
void print_point(point p) {
double x = p.x;
double y = p.y;
if (x == 0) {
x = 0;
}
if (y == 0) {
y = 0;
}
printf("(%g, %g)", x, y);
}
double sq(double x) {
return x * x;
}
bool within(double x1, double y1, double x2, double y2, double x, double y) {
double d1 = sqrt(sq(x2 - x1) + sq(y2 - y1)); // 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++
#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#
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)
D
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)]
FreeBASIC
#define eps 1e-14
#define fx(a, b, c, d) -(a * d + c) / b
#define fy(a, b, c, d) -(b * d + c) / a
Type puntoT
Dim As Double x, y
End Type
Function construye_punto(x As Double, y As Double) As puntoT
Dim As puntoT p1 = (x, y)
Return p1
End Function
Sub imprime_punto(p As puntoT)
Dim As Double x = p.x
Dim As Double y = p.y
If x = 0 Then x = 0
If y = 0 Then y = 0
Print Using "(&, &)"; x; y;
End Sub
Function sq(x As Double) As Double
Return x * x
End Function
Function dentro(x1 As Double, y1 As Double, x2 As Double, y2 As Double, x As Double, y As Double) As Boolean
Dim d1 As Double = Sqr(sq(x2 - x1) + sq(y2 - y1)) ' distance between end-points
Dim d2 As Double = Sqr(sq(x - x1) + sq(y - y1)) ' distance from point to one end
Dim d3 As Double = Sqr(sq(x2 - x) + sq(y2 - y)) ' distance from point to other end
Dim delta As Double = d1 - d2 - d3
Return Abs(delta) < eps ' true if delta is less than a small tolerance
End Function
Function rxy(x1 As Double, y1 As Double, x2 As Double, y2 As Double, x As Double, y As Double, segmento As Boolean) As Integer
If Not segmento Or dentro(x1, y1, x2, y2, x, y) Then
imprime_punto(construye_punto(x, y))
Return 1
Else
Return 0
End If
End Function
' Prints the intersection puntos (if any) of a circle, center 'cp' with radius 'r',
' and either an infinite line containing the puntos 'p1' and 'p2'
' or a segmento drawn between those puntos.
Sub interseccion(p1 As puntoT, p2 As puntoT, cp As puntoT, r As Double, segmento As Boolean)
Dim As Double x0 = cp.x, y0 = cp.y
Dim As Double x1 = p1.x, y1 = p1.y
Dim As Double x2 = p2.x, y2 = p2.y
Dim As Double A1 = y2 - y1, B1 = x1 - x2, C1 = x2 * y1 - x1 * y2
Dim As Double a = sq(A1) + sq(B1)
Dim As Double b, c, d, x ,y
Dim As Boolean bnz = True
Dim As Integer cnt = 0
If Abs(B1) >= eps Then
' if B1 isn't zero or close to it
b = 2 * (A1 * C1 + A1 * B1 * y0 - B1 * B1 * x0)
c = sq(C1) + 2 * B1 * C1 * y0 - sq(B1) * (sq(r) - sq(x0) - sq(y0))
Else
b = 2 * (B1 * C1 + A1 * B1 * x0 - sq(A1) * y0)
c = sq(C1) + 2 * A1 * C1 * x0 - sq(A1) * (sq(r) - sq(x0) - sq(y0))
bnz = False
End If
d = sq(b) - 4 * a * c ' discriminant
Select Case d
Case Is < 0
' line & circle don't intersect
Print "[]";
Case 0
' line is tangent to circle, so just one intersect at most
If bnz Then
x = -b / (2 * a)
y = fx(A1, B1, C1, x)
cnt = rxy(x1, y1, x2, y2, x, y, segmento)
Else
y = -b / (2 * a)
x = fy(A1, B1, C1, y)
cnt = rxy(x1, y1, x2, y2, x, y, segmento)
End If
Case Else
' two interseccion at most
d = Sqr(d)
If bnz Then
x = (-b + d) / (2 * a)
y = fx(A1, B1, C1, x)
cnt = rxy(x1, y1, x2, y2, x, y, segmento)
x = (-b - d) / (2 * a)
y = fx(A1, B1, C1, x)
cnt += rxy(x1, y1, x2, y2, x, y, segmento)
Else
y = (-b + d) / (2 * a)
x = fy(A1, B1, C1, y)
cnt = rxy(x1, y1, x2, y2, x, y, segmento)
y = (-b - d) / (2 * a)
x = fy(A1, B1, C1, y)
cnt += rxy(x1, y1, x2, y2, x, y, segmento)
End If
End Select
If cnt <= 0 Then Print "[]";
End Sub
Dim As puntoT cp = construye_punto(3, -5)
Dim As Double r = 3.0
Print "The intersection puntos (if any) between:"
Print " A circle, center (3, -5) with radius 3, and:"
Print " a line containing the points (-10, 11) and (10, -9) is/are:"
Print Spc(6); : interseccion(construye_punto(-10, 11), construye_punto(10, -9), cp, r, False)
Print !"\n a segment starting at (-10, 11) and ending at (-11, 12) is/are"
Print Spc(6); : interseccion(construye_punto(-10, 11), construye_punto(-11, 12), cp, r, True)
Print !"\n a horizontal line containing the points (3, -2) and (7, -2) is/are:"
Print Spc(6); : interseccion(construye_punto(3, -2), construye_punto(7, -2), cp, r, False)
cp = construye_punto(0, 0)
r = 4.0
Print !"\n A circle, center (0, 0) with radius 4, and:"
Print " a vertical line containing the points (0, -3) and (0, 6) is/are:"
Print Spc(6); : interseccion(construye_punto(0, -3), construye_punto(0, 6), cp, r, False)
Print !"\n a vertical segmento starting at (0, -3) and ending at (0, 6) is/are:"
Print Spc(6); : interseccion(construye_punto(0, -3), construye_punto(0, 6), cp, r, True)
cp = construye_punto(4,2)
r = 5.0
Print !"\n A circle, center (4, 2) with radius 5, and:"
Print " a line containing the points (6, 3) and (10, 7) is/are:"
Print Spc(6); : interseccion(construye_punto(6, 3), construye_punto(10, 7), cp, r, False)
Print !"\n a segment starting at (7, 4) and ending at (11, 8) is/are:"
Print Spc(6); : interseccion(construye_punto(7, 4), construye_punto(11, 8), cp, r, True)
Sleep
- Output:
Same as C entry.
Go
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)]
Haskell
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)]
Java
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)]
jq
Adapted from Wren
Works with jq, the C implementation of jq
Works with gojq, the Go implementation of jq
In the following, points are represented by JSON objects of the form {x,y}, and a line or line segment is indicated by a JSON array of points.
# Tolerance
def eps: 1e-14;
# Output: a string
def intersects($p1; $p2; $cp; $r; $segment):
{res: [], bnz: true}
| $cp.x as $x0
| $cp.y as $y0
| $p1.x as $x1
| $p1.y as $y1
| $p2.x as $x2
| $p2.y as $y2
| ($y2 - $y1) as $A
| ($x1 - $x2) as $B
| ($x2 * $y1 - $x1 * $y2) as $C
| ($A * $A + $B * $B) as $a
| def within($x0; $y0):
((($x2 - $x1)*($x2 - $x1) + ($y2 - $y1)*($y2 - $y1)) |sqrt) as $d1 # distance between end-points
| ((($x0 - $x1)*($x0 - $x1) + ($y0 - $y1)*($y0 - $y1)) |sqrt) as $d2 # distance from point to one end
| ((($x2 - $x0)*($x2 - $x0) + ($y2 - $y0)*($y2 - $y0)) |sqrt) as $d3 # distance from point to other end
| ($d1 - $d2 - $d3) as $delta
| ($delta|length) < eps # true if delta is less than a small tolerance
;
def rxy:
if ($segment|not) or within(.x; .y)
then .res += [{x, y}]
end;
def fx: -($A * .x + $C) / $B;
def fy: -($B * .y + $C) / $A;
if ($B|length) >= eps
then
.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
end
| (.b * .b - 4 * $a * .c) as $d # discriminant
| if $d < 0
then "[]" # a string
else
# checks whether a point is within a segment
.x = 0
| .y = 0
| if $d == 0
then # line is tangent to circle, so just one intersect at most
if .bnz
then .x = -.b / (2 * $a)
| .y = fx
else
.y = -.b / (2 * $a)
| .x = fy
end
| rxy
else # two intersections at most
($d|sqrt) as $D
| if .bnz
then .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
end
end
# avoid negative zeros
| .res
| tostring
| gsub("-0,"; "0,") | gsub("-0}"; "0}")
end
;
def task:
({cp: {x:3, y:-5},
r: 3 }
| "The intersection points (if any) between:",
" A circle, center \(.cp) with radius \(.r), and:",
( .p1 = {x: -10, y: 11}
| .p2 = {x: 10, y: -9}
| " a line containing the points \(.p1) and \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; false))" ),
( .p1 = {x: -10, y: 11}
| .p2 = {x: -10, y:12}
| " a segment starting at \(.p1) and ending at \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; true))"),
( .p1 = {x:3, y: -2}
| .p2 = {x:7, y: -2}
| " a horizontal line containing the points \(.p1) and \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; false))" ) ),
({cp: {x:0, y:0},
r: 4 }
| " A circle, center \(.cp) with radius \(.r), and:",
( .p1 = {x:0, y:-3}
| .p2 = {x:0, y: 6}
| " a vertical line containing the points \(.p1) and \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; false))",
" a vertical segment containing the points \(.p1) and \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; true))") ),
({cp: {x:4, y: 2},
r: 5 }
| " A circle, center \(.cp) with radius \(.r), and:",
( .p1 = {x: 6, y: 3}
| .p2 = {x:10, y: 7}
| " a line containing the points \(.p1) and \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; false))"),
( .p1 = {x: 7, y: 4}
| .p2 = {x:11, y: 8}
| " a segment starting at \(.p1) and ending at \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; true))") ),
({cp: {x:10, y: 10},
r: 5 }
| " A circle, center \(.cp) with radius \(.r), and:",
( .p1 = {x: 5, y: 0}
| .p2 = {x: 5, y: 20}
| " a vertical line containing the points \(.p1) and \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; false))"),
( .p1 = {x:-5, y: 10}
| .p2 = {x: 5, y: 10}
| " a horizontal segment starting at \(.p1) and ending at \(.p2) is/are:",
" \(intersects(.p1; .p2; .cp; .r; true))") )
;
task- Output:
Invocation: jq -nr -f line-circle-intersection.jq
The intersection points (if any) between:
A circle, center {"x":3,"y":-5} with radius 3, and:
a line containing the points {"x":-10,"y":11} and {"x":10,"y":-9} is/are:
[{"x":6,"y":-5},{"x":3,"y":-2}]
a segment starting at {"x":-10,"y":11} and ending at {"x":-10,"y":12} is/are:
[]
a horizontal line containing the points {"x":3,"y":-2} and {"x":7,"y":-2} is/are:
[{"x":3,"y":-2}]
A circle, center {"x":0,"y":0} with radius 4, and:
a vertical line containing the points {"x":0,"y":-3} and {"x":0,"y":6} is/are:
[{"x":0,"y":4},{"x":0,"y":-4}]
a vertical segment containing the points {"x":0,"y":-3} and {"x":0,"y":6} is/are:
[{"x":0,"y":4}]
A circle, center {"x":4,"y":2} with radius 5, and:
a line containing the points {"x":6,"y":3} and {"x":10,"y":7} is/are:
[{"x":8,"y":5},{"x":1,"y":-2}]
a segment starting at {"x":7,"y":4} and ending at {"x":11,"y":8} is/are:
[{"x":8,"y":5}]
A circle, center {"x":10,"y":10} with radius 5, and:
a vertical line containing the points {"x":5,"y":0} and {"x":5,"y":20} is/are:
[{"x":5,"y":10}]
a horizontal segment starting at {"x":-5,"y":10} and ending at {"x":5,"y":10} is/are:
[{"x":5,"y":10}]
Julia
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)
Kotlin
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)]
Lua
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 Language
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}}]
Nim
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)]
Perl
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
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).
Raku
(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 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 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)]
REXX
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
Ruby
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]]
Rust
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
Swift
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 .NET
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)
Wren
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)]
zkl
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))