Convex hull

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Revision as of 21:27, 16 January 2017 by Walterpachl (talk | contribs) (→‎{{header|REXX}}: add Code to add Points that aren't on the hull but should be)
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



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 

 :: (Num a, Ord a)
 => [a] -> [a] -> [a] -> Ordering

compareFrom o l r = 

 compare ((x l - x o) * (y r - y o)) ((y l - y o) * (x r - x o))

distanceFrom 

 :: Floating a
 => [a] -> [a] -> a

distanceFrom from to = ((x to - x from) ** 2 + (y to - y from) ** 2) ** (1 / 2)

convexHull 

 :: (Floating a, Ord a)
 => a -> a

convexHull points = 

 let o = minimum points
     presorted = sortBy (compareFrom o) (filter (/= o) points)
     collinears = groupBy (((EQ ==) .) . compareFrom o) presorted
     outmost = maximumBy (comparing (distanceFrom o)) <$> collinears
 in dropConcavities [o] outmost

dropConcavities 

 :: (Num a, Ord a)
 => a -> a -> a

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 

 :: (Floating a, Ord a)
 => a

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]
   ]

main :: IO () main = mapM_ print example</lang>

Output:
[-3.0,-9.0]
[19.0,-8.0]
[17.0,5.0]
[12.0,17.0]
[5.0,19.0]
[-3.0,15.0]
[-9.0,-3.0]

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)

REXX

<lang rexx>/* REXX ---------------------------------------------------------------

  • Compute the Convex Hull for a set of points
  • Format of the input file:
  • (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)
  • --------------------------------------------------------------------*/
 Signal On Novalue
 Signal On Syntax

fid='chullx.in' fid='chullxx.in' fid='chulla.in' fid='chull.in' g.0oid=fn(fid)'.txt'; 'erase' g.0oid x.=0 yl.= Parse Value '1000 -1000' With g.0xmin g.0xmax Parse Value '1000 -1000' With g.0ymin g.0ymax /*---------------------------------------------------------------------

  • First read the input and store the points' coordinates
  • x.0 contains the number of points, x.i contains the x.coordinate
  • yl.x contains the y.coordinate(s) of points (x/y)
  • --------------------------------------------------------------------*/

Do while lines(fid)>0 

 l=linein(fid)
 Do While l<>
   Parse Var l '(' x ',' y ')' l
   Call store x,y
   End
 End

Do i=1 To x.0 /* loop over points */ 

 x=x.i
 yl.x=sortv(yl.x)                  /* sort y-coordinates            */
 End

Call sho

/*---------------------------------------------------------------------

  • Now we look for special border points:
  • lefthigh and leftlow: leftmost points with higheste and lowest y
  • ritehigh and ritelow: rightmost points with higheste and lowest y
  • yl.x contains the y.coordinate(s) of points (x/y)
  • --------------------------------------------------------------------*/

lefthigh=0 leftlow=0 Do i=1 To x.0 

 x=x.i
 If maxv(yl.x)=g.0ymax Then Do
   If lefthigh=0 Then lefthigh=x'/'g.0ymax
   ritehigh=x'/'g.0ymax
   End
 If minv(yl.x)=g.0ymin Then Do
   ritelow=x'/'g.0ymin
   If leftlow=0 Then leftlow=x'/'g.0ymin
   End
 End

/*---------------------------------------------------------------------

  • Now we look for special border points:
  • leftmost_n and leftmost_s: points with lowest x and highest/lowest y
  • ritemost_n and ritemost_s: points with largest x and highest/lowest y
  • n and s stand foNorth and South, respectively
  • --------------------------------------------------------------------*/

x=g.0xmi; leftmost_n=x'/'maxv(yl.x) x=g.0xmi; leftmost_s=x'/'minv(yl.x) x=g.0xma; ritemost_n=x'/'maxv(yl.x) x=g.0xma; ritemost_s=x'/'minv(yl.x)

/*---------------------------------------------------------------------

  • Now we compute the paths from lefthigh to ritehigh (n_high)
  • and ritelow to leftlow (s_low), respectively
  • --------------------------------------------------------------------*/

y=g.0ymax n_high= Do xi=1 To x.0 

 x=x.xi
 If maxv(yl.x)=y Then n_high=n_high x'/'y
 End

Call o 'n_high='n_high y=g.0ymin s_low= Do xi=x.0 To 1 By -1 

 x=x.xi
 If minv(yl.x)=y Then s_low=s_low x'/'y
 End

Call o 's_low='s_low

/*---------------------------------------------------------------------

  • Now we compute the paths from ritehigh to ritelow (n_end)
  • and leftlow to lefthigh (s_end), respectively
  • --------------------------------------------------------------------*/

x=g.0xma n_end= Do i=words(yl.x) To 1 By -1 

 n_end=n_end x'/'word(yl.x,i)
 End

Call o 'n_end='n_end x=g.0xmi s_end= Do i=1 To words(yl.x) 

 s_end=s_end x'/'word(yl.x,i)
 End

Call o 's_end='s_end

/*---------------------------------------------------------------------

  • Now we compute the upper part of the convex hull (nhull)
  • --------------------------------------------------------------------*/

res=mk_nhull(leftmost_n,lefthigh); nhull=leftmost_n res Do While res<>lefthigh 

 res=mk_nhull(res,lefthigh); nhull=nhull res
 End

nhull=nhull n_high res=mk_nhull(ritehigh,ritemost_n); nhull=nhull res Do While res<>ritemost_n 

 res=mk_nhull(res,ritemost_n); nhull=nhull res
 End

nhull=nhull n_end /* attach the right vertical border */

/*---------------------------------------------------------------------

  • Now we compute the lower part of the convex hull (shull)
  • --------------------------------------------------------------------*/

res=mk_shull(ritemost_s,ritelow); shull=ritemost_s res Do While res<>ritelow 

 res=mk_shull(res,ritelow);
 shull=shull res
 End

shull=shull s_low res=mk_shull(leftlow,leftmost_s); shull=shull leftlow res Do While res<>leftmost_s 

 res=mk_shull(res,leftmost_s);
 shull=shull res
 End

shull=shull s_end

chull=nhull shull /* concatenate upper and lower part */ 

                                 /* eliminate duplicates             */
                                 /* too lazy to take care before :-) */

Parse Var chull chullx chull have.=0 Do i=1 By 1 While chull> 

 Parse Var chull xy chull
 If have.xy=0 Then Do
   chullx=chullx xy
   have.xy=1
   End
 End
                                  /* show the result                */

Say 'Points of convex hull in clockwise order:' Call o chullx /*---------------------------------------------------------------------

  • Final polish: Insert points that are not yet in chullx but should be
  • First on the upper hull going from left to right
  • --------------------------------------------------------------------*/

i=1 Do While i<words(chullx) 

 xya=word(chullx,i)  ; Parse Var xya xa '/' ya
 If xa=g.0xmax Then Leave
 xyb=word(chullx,i+1); Parse Var xyb xb '/' yb
 Do j=1 To x.0
   If x.j>xa Then Do
     If x.j<xb Then Do
       xx=x.j
       parse Value kdx(xya,xyb) With k d x
       If (k*xx+d)=maxv(yl.xx) Then Do
         chullx=subword(chullx,1,i) xx'/'maxv(yl.xx) subword(chullx,i+1)
         i=i+1
         End
       End
     End
   Else
     i=i+1
   End
 End

Call o chullx

/*---------------------------------------------------------------------

  • Final polish: Insert points that are not yet in chullx but should be
  • Then on the lower hull going from right to left
  • --------------------------------------------------------------------*/

i=wordpos(ritemost_s,chullx) Do While i<words(chullx) 

 xya=word(chullx,i)  ; Parse Var xya xa '/' ya
 If xa=g.0xmin Then Leave
 xyb=word(chullx,i+1); Parse Var xyb xb '/' yb
 Do j=x.0 To 1 By -1
   If x.j<xa Then Do
     If x.j>xb Then Do
       xx=x.j
       parse Value kdx(xya,xyb) With k d x
       If (k*xx+d)=minv(yl.xx) Then Do
         chullx=subword(chullx,1,i) xx'/'minv(yl.xx) subword(chullx,i+1)
         i=i+1
         End
       End
     End
   Else
     i=i+1
   End
 End

Say chullx

Exit

store: Procedure Expose x. yl. g. /*---------------------------------------------------------------------

  • arrange the points in ascending order of x (in x.) and,
  • for each x in ascending order of y (in yl.x)
  • g.0xmin is the smallest x-value, etc.
  • g.0xmi is the x-coordinate
  • g.0ymin is the smallest y-value, etc.
  • g.0ymi is the x-coordinate of such a point
  • --------------------------------------------------------------------*/
 Parse Arg x,y
 Call o 'store' x y
 If x<g.0xmin Then Do; g.0xmin=x; g.0xmi=x; End
 If x>g.0xmax Then Do; g.0xmax=x; g.0xma=x; End
 If y<g.0ymin Then Do; g.0ymin=y; g.0ymi=x; End
 If y>g.0ymax Then Do; g.0ymax=y; g.0yma=x; End
 Do i=1 To x.0
   Select
     When x.i>x Then
       Leave
     When x.i=x Then Do
       yl.x=yl.x y
       Return
       End
     Otherwise
       Nop
     End
   End
 Do j=x.0 To i By -1
   ja=j+1
   x.ja=x.j
   End
 x.i=x
 yl.x=y
 x.0=x.0+1
 Return

sho: Procedure Expose x. yl. g. 

 Do i=1 To x.0
   x=x.i
   say 'x='format(x,3) 'yl='yl.x
   End
 Say 
 Return

maxv: Procedure Expose g. 

 Call trace 'O'
 Parse Arg l
 res=-1000
 Do While l<>
   Parse Var l v l
   If v>res Then res=v
   End
 Return res

minv: Procedure Expose g. 

 Call trace 'O'
 Parse Arg l
 res=1000
 Do While l<>
   Parse Var l v l
   If v<res Then res=v
   End
 Return res

sortv: Procedure Expose g. 

 Call trace 'O'
 Parse Arg l
 res=
 Do Until l=
   v=minv(l)
   res=res v
   l=remove(v,l)
   End
 Return space(res)

lastword: return word(arg(1),words(arg(1)))

kdx: Procedure Expose xy. g. /*---------------------------------------------------------------------

  • Compute slope and y-displacement of a straight line
  • that is defined by two points: y=k*x+d
  • Specialty; k='*' x=xa if xb=xa
  • --------------------------------------------------------------------*/
 Call trace 'O'
 Parse Arg xya,xyb
 Parse Var xya xa '/' ya
 Parse Var xyb xb '/' yb
 If xa=xb Then
   Parse Value '*' '-' xa With k d x
 Else Do
   k=(yb-ya)/(xb-xa)
   d=yb-k*xb
   x='*'
   End
 Return k d x

show: Procedure /*---------------------------------------------------------------------

  • Show the list of points in x. and yl.
  • --------------------------------------------------------------------*/
 Parse Arg xyl
 Do i=1 To words(xyl)
   Say format(i,2) word(xyl,i)
   End
 Return

remove: /*---------------------------------------------------------------------

  • Remove a specified element (e) from a given string (s)
  • --------------------------------------------------------------------*/
 Parse Arg e,s
 Parse Var s sa (e) sb
 Return space(sa sb)

o: Procedure Expose g. /*---------------------------------------------------------------------

  • Write a line to the debug file
  • --------------------------------------------------------------------*/
 Return lineout(g.0oid,arg(1))

is_ok: Procedure Expose x. yl. g. /*---------------------------------------------------------------------

  • Test if a given point (b) is above/on/or below a straight line
  • defined by two points (a and c)
  • --------------------------------------------------------------------*/
 Parse Arg a,b,c,op
 Parse Value kdx(a,c) With k d x
 Parse Var b x'/'y
 y=maxv(yl.x)
 Call o y x (k*x+d)
 If op='U' Then res=(y<=(k*x+d))|(abs(y-(k*x+d))<1.e-8)
           Else res=(y>=(k*x+d))|(abs(y-(k*x+d))<1.e-8)
 Return res

mk_nhull: Procedure Expose x. yl. g. /*---------------------------------------------------------------------

  • Compute the upper (north) hull between two points (xya and xyb)
  • Move x from xyb back to xya until all points within the current
  • range (x and xyb) are BELOW the straight line defined xya and x
  • Then make x the new starting point
  • --------------------------------------------------------------------*/
 Parse Arg xya,xyb
 Call o 'mk_nhull' xya xyb
 Parse Var xya xa '/' ya
 Parse Var xyb xb '/' yb
 iu=0
 iv=0
 Do xi=1 To x.0
   if x.xi>=xa & iu=0 Then iu=xi
   if x.xi<=xb Then iv=xi
   If x.xi>xb Then Leave
   End
 Call o iu iv
 Do xi=iu To iv
   x=x.xi; Call o xi format(x,3) '->' yl.x
   End
 xu=x.iu
 xyu=xu'/'maxv(yl.xu)
 Do h=iv To iu+1 By -1 Until good
   xh=x.h
   xyh=xh'/'maxv(yl.xh)
   Call o 'Testing' xyu xyh
   good=1
   Do hh=iv To iu+1 By -1 While good
     xhh=x.hh
     xyhh=xhh'/'maxv(yl.xhh)
     If is_ok(xyu,xyhh,xyh,'U') Then Do
       Call o xyhh 'is under' xyu xyh
       Nop
       End
     Else Do
       good=0
       Call o xyhh 'is above' xyu xyh '-' xyh 'ist nicht gut'
       End
     End
   End
 Call o xyh 'is the one'
 Return xyh

mk_shull: Procedure Expose x. yl. g. /*---------------------------------------------------------------------

  • Compute the lower (south) hull between two points (xya and xyb)
  • Move x from xyb back to xya until all points within the current
  • range (x and xyb) are ABOVE the straight line defined xya and x
  • Then make x the new starting point
  • --------------------------------------------------------------------*/
 Parse Arg xya,xyb
 Call o 'mk_shull' xya xyb
 Parse Var xya xa '/' ya
 Parse Var xyb xb '/' yb
 iu=0
 iv=0
 Do xi=x.0 To 1 By -1
   if x.xi<=xa & iu=0 Then iu=xi
   if x.xi>=xb Then iv=xi
   If x.xi<xb Then Leave
   End
 Call o iu iv '_' x.iu x.iv
 Call o 'iu iv' iu iv
 Do xi=iu To iv By -1
   x=x.xi; Call o xi format(x,3) '->' yl.x
   End
 xu=x.iu
 xyu=xu'/'minv(yl.xu)
 Do h=iv To iu+1 Until good
   xh=x.h
   xyh=xh'/'minv(yl.xh)
   Call o 'Testing' xyu xyh
   good=1
   Do hh=iv To iu+1 While good
     xhh=x.hh
     xyhh=xhh'/'minv(yl.xhh)
     Call o hh xyhh
     If is_ok(xyu,xyhh,xyh,'O') Then Do
       Call o xyhh 'is above' xyu xyh
       Nop
       End
     Else Do
       Call o xyhh 'is under' xyu xyh '-' xyh 'ist nicht gut'
       good=0
       End
     End
   End
 Call o xyh 'is the one'
 Return xyh

Novalue: 

 Say 'Novalue raised in line' sigl
 Say sourceline(sigl)
 Say 'Variable' condition('D')
 Signal lookaround

Syntax: 

 Say 'Syntax raised in line' sigl
 Say sourceline(sigl)
 Say 'rc='rc '('errortext(rc)')'

halt: lookaround: 

 If fore() Then Do
   Say 'You can look around now.'
   Trace ?R
   Nop
   End
 Exit 12</lang>
Output:
x=-9 yl=-3
x=-4 yl=-6 -2
x=-3 yl=-9 15
x= 0 yl=6 11
x= 3 yl=-4 16
x= 5 yl=19
x=12 yl=13 17
x=16 yl=-7 -3 3 6
x=17 yl=-4 5
x=19 yl=-8

Points of convex hull in clockwise order:
-9/-3 -3/15 5/19 12/17 17/5 19/-8 -3/-9 -9/-3

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))
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