Convex hull

Convex hull 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.

Find the points which form a convex hull from a set of arbitrary two dimensional points.

For example, given the points (16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2) and (12,10) the convex hull would be (-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19) and (-3,15).

See also

• Convex Hull (youtube)
• http://www.geeksforgeeks.org/convex-hull-set-2-graham-scan/

Go

<lang go>package main

import ( "fmt" "image" "sort" )

// ConvexHull returns the set of points that define the // convex hull of p in CCW order starting from the left most. func (p points) ConvexHull() points { // From https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain // with only minor deviations. sort.Sort(p) var h points

// Lower hull for _, pt := range p { for len(h) >= 2 && !ccw(h[len(h)-2], h[len(h)-1], pt) { h = h[:len(h)-1] } h = append(h, pt) }

// Upper hull for i, t := len(p)-2, len(h)+1; i >= 0; i-- { pt := p[i] for len(h) >= t && !ccw(h[len(h)-2], h[len(h)-1], pt) { h = h[:len(h)-1] } h = append(h, pt) }

return h[:len(h)-1] }

// ccw returns true if the three points make a counter-clockwise turn func ccw(a, b, c image.Point) bool { return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X)) }

type points []image.Point

func (p points) Len() int { return len(p) } func (p points) Swap(i, j int) { p[i], p[j] = p[j], p[i] } func (p points) Less(i, j int) bool { if p[i].X == p[j].X { return p[i].Y < p[i].Y } return p[i].X < p[j].X }

func main() { pts := points{ {16, 3}, {12, 17}, {0, 6}, {-4, -6}, {16, 6}, {16, -7}, {16, -3}, {17, -4}, {5, 19}, {19, -8}, {3, 16}, {12, 13}, {3, -4}, {17, 5}, {-3, 15}, {-3, -9}, {0, 11}, {-9, -3}, {-4, -2}, {12, 10}, } hull := pts.ConvexHull() fmt.Println("Convex Hull:", hull) }</lang>

Output:

Convex Hull: [(-9,-3) (-3,-9) (19,-8) (17,5) (12,17) (5,19) (-3,15)]

Haskell

<lang Haskell> import Data.Ord;import Data.List;(x,y)=((!!0),(!!1)); compareFrom o l r=compare((x l-x o)*(y r-y o))((y l-y o)*(x r-x o)); distanceFrom from to = ((x to-x from)**2+(y to-y from)**2)**(1/2); convexHull points =

   let{o = minimum points;
       presorted=sortBy(compareFrom o)(filter(/=o)points);
       collinears=groupBy(((EQ==).).compareFrom o)presorted;
       outmost=map(maximumBy(comparing(distanceFrom o)))collinears;
   }in dropConcavities[o]outmost;

dropConcavities (left:lefter) (right:righter:rightest) =

   case compareFrom left right righter of{
   LT -> dropConcavities (right:left:lefter) (righter:rightest);
   EQ -> dropConcavities (left:lefter) (righter:rightest);
   GT -> dropConcavities lefter (left:righter:rightest);
   };dropConcavities output lastInput = lastInput++output;

example=convexHull[[16,3],[12,17],[0,6],[-4,-6],[16,6],[16,-7],[16,-3],

[17,-4],[5,19],[19,-8],[3,16],[12,13],[3,-4],[17,5],[-3,15],[-3,-9],
[0,11],[-9,-3],[-4,-2],[12,10]]

</lang>

J

Restated from the implementation at http://kukuruku.co/hub/funcprog/introduction-to-j-programming-language-2004 which in turn is a translation of http://dr-klm.livejournal.com/42312.html

<lang J>counterclockwise =: ({. , }. /: 12 o. }. - {.) @ /:~ crossproduct =: 11"_ o. [: (* +)/ }. - {. removeinner =: #~ 1, 0 > 3 crossproduct\ ], 1: hull =: [: removeinner^:_ counterclockwise</lang>

Example use:

<lang J> hull 16j3 12j17 0j6 _4j_6 16j6 16j_7 16j_3 17j_4 5j19 19j_8 3j16 12j13 3j_4 17j5 _3j15 _3j_9 0j11 _9j_3 _4j_2 12j10 _9j_3 _3j_9 19j_8 17j5 12j17 5j19 _3j15</lang>

Python

An approach that uses the shapely library:

<lang python>from __future__ import print_function from shapely.geometry import MultiPoint

if __name__=="__main__": pts = MultiPoint([(16,3), (12,17), (0,6), (-4,-6), (16,6), (16,-7), (16,-3), (17,-4), (5,19), (19,-8), (3,16), (12,13), (3,-4), (17,5), (-3,15), (-3,-9), (0,11), (-9,-3), (-4,-2), (12,10)]) print (pts.convex_hull)</lang>

Output:

POLYGON ((-3 -9, -9 -3, -3 15, 5 19, 12 17, 17 5, 19 -8, -3 -9))

Racket

Also an implementation of https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain (therefore kinda

Translation of: Go

<lang racket>#lang typed/racket (define-type Point (Pair Real Real)) (define-type Points (Listof Point))

(: ⊗ (Point Point Point -> Real)) (define/match (⊗ o a b)

 [((cons o.x o.y) (cons a.x a.y) (cons b.x b.y))
  (- (* (- a.x o.x) (- b.y o.y)) (* (- a.y o.y) (- b.x o.x)))])

(: Point<? (Point Point -> Boolean)) (define (Point<? a b)

 (cond [(< (car a) (car b)) #t] [(> (car a) (car b)) #f] [else (< (cdr a) (cdr b))]))

Input

a list P of points in the plane.

(define (convex-hull [P:unsorted : Points])

 ;; Sort the points of P by x-coordinate (in case of a tie, sort by y-coordinate).
 (define P (sort P:unsorted Point<?))
 ;; for i = 1, 2, ..., n:
 ;;     while L contains at least two points and the sequence of last two points
 ;;             of L and the point P[i] does not make a counter-clockwise turn:
 ;;        remove the last point from L
 ;;     append P[i] to L
 ;; TB: U is identical with (reverse P) 
 (define (upper/lower-hull [P : Points])
   (reverse
    (for/fold ((rev : Points null))
      ((P.i (in-list P)))
      (let u/l : Points ((rev rev))
        (match rev
          [(list p-2 p-1 ps ...) #:when (not (positive? (⊗ p-2 P.i p-1))) (u/l (list* p-1 ps))]
          [(list ps ...) (cons P.i ps)])))))

 ;; Initialize U and L as empty lists.
 ;; The lists will hold the vertices of upper and lower hulls respectively.
 (let ((U (upper/lower-hull (reverse P)))
       (L (upper/lower-hull P)))
   ;; Remove the last point of each list (it's the same as the first point of the other list).
   ;; Concatenate L and U to obtain the convex hull of P.
   (append (drop-right L 1) (drop-right U 1)))) ; Points in the result will be listed in CCW order.)

(module+ test

 (require typed/rackunit)
 (check-equal?
  (convex-hull
   (list '(16 . 3) '(12 . 17) '(0 . 6) '(-4 . -6) '(16 . 6) '(16 . -7) '(16 . -3) '(17 . -4)
         '(5 . 19) '(19 . -8) '(3 . 16) '(12 . 13) '(3 . -4) '(17 . 5) '(-3 . 15) '(-3 . -9)
         '(0 . 11) '(-9 . -3) '(-4 . -2) '(12 . 10)))
  (list '(-9 . -3) '(-3 . -9) '(19 . -8) '(17 . 5) '(12 . 17) '(5 . 19) '(-3 . 15))))</lang>

Output:

silence implies tests pass (and output is as expected)

zkl

<lang zkl>// Use Graham Scan to sort points into a convex hull // https://en.wikipedia.org/wiki/Graham_scan, O(n log n) // http://www.geeksforgeeks.org/convex-hull-set-2-graham-scan/ // http://geomalgorithms.com/a10-_hull-1.html fcn grahamScan(points){

  N:=points.len();
  # find the point with the lowest y-coordinate, x is tie breaker
  p0:=points.reduce(fcn([(a,b)]ab,[(x,y)]xy){

if(b<y)ab else if(b==y and a<x)ab else xy });

  #sort points by polar angle with p0, ie ccw from p0
  points.sort('wrap(p1,p2){ ccw(p0,p1,p2)>0 });

  # We want points[0] to be a sentinel point that will stop the loop.
  points.insert(0,points[-1]);
  M:=1; # M will denote the number of points on the convex hull.
  foreach i in ([2..N]){
     # Find next valid point on convex hull.
     while(ccw(points[M-1], points[M], points[i])<=0){

if(M>1) M-=1; else if(i==N) break; # All points are collinear else i+=1;

     }
     points.swap(M+=1,i); # Update M and swap points[i] to the correct place.
  }
  points[0,M]

}

1. Three points are a counter-clockwise turn if ccw > 0, clockwise if
2. ccw < 0, and collinear if ccw = 0 because ccw is a determinant that
3. gives twice the signed area of the triangle formed by p1, p2 and p3.

fcn ccw(a,b,c){ // a,b,c are points: (x,y)

  ((b[0] - a[0])*(c[1] - a[1])) - ((b[1] - a[1])*(c[0] - a[0]))

}</lang> <lang zkl>pts:=List( T(16,3), T(12,17), T(0,6), T(-4,-6), T(16,6), T(16, -7), T(16,-3),T(17,-4), T(5,19), T(19,-8), T(3,16), T(12,13), T(3,-4), T(17,5), T(-3,15), T(-3,-9), T(0,11), T(-9,-3), T(-4,-2), T(12,10), ) .apply(fcn(xy){ xy.apply("toFloat") }).copy(); hull:=grahamScan(pts); println("Convex Hull (%d points): %s".fmt(hull.len(),hull.toString(*)));</lang>

Output:

Convex Hull (7 points): L(L(-3,-9),L(19,-8),L(17,5),L(12,17),L(5,19),L(-3,15),L(-9,-3))