Line circle intersection: Difference between revisions

{{draft task}}

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 [https://math.stackexchange.com/questions/228841/how-do-i-calculate-the-intersections-of-a-straight-line-and-a-circle Math Stack Exchange] for development of the formulae needed.

=={{header|Go}}==

<lang go>package main

import (

"fmt"

"math"

)

const eps = 1e-14 // say

type point struct{ x, y float64 }

func (p point) String() string {

return fmt.Sprintf("(%f, %f)", 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>

{{out}}

<pre>

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:

[(6.000000, -5.000000) (3.000000, -2.000000)]

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:

[(3.000000, -2.000000)]

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

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

[(-0.000000, 4.000000) (0.000000, -4.000000)]

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

[(-0.000000, 4.000000)]

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

a line containing the points (6, 3) and (10, 7) is/are:

[(8.000000, 5.000000) (1.000000, -2.000000)]

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

[(8.000000, 5.000000)]

</pre>