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.

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>

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:
      [(6.000000, -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)]

zkl

<lang zkl></lang> <lang zkl></lang>

Output: