Line circle intersection
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
- See Math Stack Exchange for development of the formulae needed.
- See Wolfram for the formulae needed.
C
<lang 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;
}</lang>
- 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++
<lang cpp>#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;
}</lang>
- 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)]
D
<lang 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));
}</lang>
- 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)]
Go
<lang 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))
}</lang>
- 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
<lang 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</lang>
- 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
<lang 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();
}
}</lang>
- 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)]
Julia
Uses the circles and points from the Go example. <lang julia>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
</lang>
- 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
<lang scala>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)}")
}</lang>
- 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
<lang 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()</lang>
- 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)]
Perl
<lang 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 ;
}</lang>
- 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
<lang Phix>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</lang>
- 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 <lang perl6>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";
}</lang>
- 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. <lang rexx>/*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</lang>
- 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
Swift
<lang 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))")</lang>
- 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)]
zkl
<lang 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) )
}</lang> <lang zkl>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)));
}</lang>
- 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))