Convex hull: Difference between revisions
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sub graham-scan (**@coords) { |
sub graham-scan (**@coords) { |
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# sorted point by y, secondary sort on x |
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my @sp = @coords.map( { Point.new( :x($_[0]), :y($_[1]) ) } ) |
my @sp = @coords.map( { Point.new( :x($_[0]), :y($_[1]) ) } ) |
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.sort: {.y, .x}; |
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# first point on hull is minimum y point |
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my @h = @sp.shift; |
my @h = @sp.shift; |
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my $X = @h[0].clone; |
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# re-sort the points by angle |
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$X.y -= 1; |
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@sp = @sp.map( { $++ => ccw( |
@sp = @sp.map( { $++ => ccw(@h[0], @sp.tail, $_) } ) |
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.sort( *.value ) |
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.map: { @sp[$_.key] }; |
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# first point of re-sorted list is guaranteed to be on hull |
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@h.push: @sp.shift; |
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# check through the remaining list making sure that |
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# there is always a positive angle |
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for @sp -> $point { |
for @sp -> $point { |
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if ccw( |@h.tail(2), $point ) > 0 { |
if ccw( |@h.tail(2), $point ) > 0 { |
Revision as of 17:46, 10 June 2017
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>
Perl 6
<lang perl6>class Point {
has Real $.x is rw; has Real $.y is rw; method gist { [~] '(', self.x,', ', self.y, ')' };
}
sub ccw (Point $a, Point $b, Point $c) {
($b.x - $a.x)*($c.y - $a.y) - ($b.y - $a.y)*($c.x - $a.x);
}
sub graham-scan (**@coords) {
# sorted point by y, secondary sort on x my @sp = @coords.map( { Point.new( :x($_[0]), :y($_[1]) ) } ) .sort: {.y, .x};
# first point on hull is minimum y point my @h = @sp.shift;
# re-sort the points by angle @sp = @sp.map( { $++ => ccw(@h[0], @sp.tail, $_) } ) .sort( *.value ) .map: { @sp[$_.key] };
# first point of re-sorted list is guaranteed to be on hull @h.push: @sp.shift;
# check through the remaining list making sure that # there is always a positive angle for @sp -> $point { if ccw( |@h.tail(2), $point ) > 0 { @h.push: $point; } else { @h.pop; redo; } } @h
}
my @hull = graham-scan(
(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) );
say "Convex Hull ({+@hull} points): ", @hull;</lang>
- Output:
Convex Hull (7 points): [(-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]
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
<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
version 1
<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
Parse Arg fid If fid= Then Do
fid='chullmin.in' /* miscellaneous test data */ fid='chullx.in' fid='chullt.in' fid='chulla.in' fid='chullxx.in' fid='sq.in' fid='tri.in' fid='line.in' fid='point.in' fid='chull.in' /* data from task description */ End
g.0debug= 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
Call lineout fid 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)
- --------------------------------------------------------------------*/
leftlow=0 lefthigh=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
Call o 'lefthigh='lefthigh Call o 'ritehigh='ritehigh Call o 'ritelow ='ritelow Call o 'leftlow ='leftlow /*---------------------------------------------------------------------
- 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 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
n_high= s_low= /*---------------------------------------------------------------------
- Now we compute the upper part of the convex hull (nhull)
- --------------------------------------------------------------------*/
Call o 'leftmost_n='leftmost_n Call o 'lefthigh ='lefthigh nhull=leftmost_n res=mk_nhull(leftmost_n,lefthigh); nhull=nhull res Call o 'A nhull='nhull Do While res<>lefthigh
res=mk_nhull(res,lefthigh); nhull=nhull res Call o 'B nhull='nhull End
res=mk_nhull(lefthigh,ritemost_n); nhull=nhull res Call o 'C nhull='nhull Do While res<>ritemost_n
res=mk_nhull(res,ritemost_n); nhull=nhull res Call o 'D nhull='nhull 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 Call o 'A shull='shull Do While res<>ritelow
res=mk_shull(res,ritelow) shull=shull res Call o 'B shull='shull End
res=mk_shull(ritelow,leftmost_s) shull=shull res Call o 'C shull='shull Do While res<>leftmost_s
res=mk_shull(res,leftmost_s); shull=shull res Call o 'D shull='shull 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 have.chullx=1 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:' Say chullx /**********************************************************************
- steps that were necessary in previous attempts
/*---------------------------------------------------------------------
- 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
Say 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
- /
Call lineout g.0oid
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 format(i,2) '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
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
- --------------------------------------------------------------------*/
If arg(2)=1 Then say arg(1) Return lineout(g.0oid,arg(1))
is_ok: Procedure Expose x. yl. g. sigl /*---------------------------------------------------------------------
- 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 Call o 'is_ok' a b c op Parse Value kdx(a,c) With k d x Parse Var b x'/'y If op='U' Then y=maxv(yl.x) Else y=minv(yl.x) Call o y x (k*x+d) If (abs(y-(k*x+d))<1.e-8) Then Return 0 If op='U' Then res=(y<=(k*x+d)) Else res=(y>=(k*x+d)) 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 If xya=xyb Then Return xya 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 xu=x.iu xyu=xu'/'maxv(yl.xu) Do h=iv To iu+1 By -1 Until good Call o 'iv='iv,g.0debug Call o ' h='h,g.0debug xh=x.h xyh=xh'/'maxv(yl.xh) Call o 'Testing' xyu xyh,g.0debug good=1 Do hh=h-1 To iu+1 By -1 While good xhh=x.hh xyhh=xhh'/'maxv(yl.xhh) Call o 'iu hh iv=' iu hh h,g.0debug If is_ok(xyu,xyhh,xyh,'U') Then Do Call o xyhh 'is under' xyu xyh,g.0debug 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
p: Return Say arg(1) Pull . Return
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 If xya=xyb Then Return xya 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 'mk_shull iv iu' iv iu xu=x.iu xyu=xu'/'minv(yl.xu) good=0 Do h=iv To iu-1 Until good xh=x.h xyh=xh'/'minv(yl.xh) Call o 'Testing' xyu xyh h iu good=1 Do hh=h+1 To iu-1 While good Call o 'iu hh h=' iu hh h xhh=x.hh xyhh=xhh'/'minv(yl.xhh) 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:
Say 'You can look around now.' Trace ?R Nop Exit 12</lang>
- Output:
1 x= -9 yl=-3 2 x= -4 yl=-6 -2 3 x= -3 yl=-9 15 4 x= 0 yl=6 11 5 x= 3 yl=-4 16 6 x= 5 yl=19 7 x= 12 yl=13 17 8 x= 16 yl=-7 -3 3 6 9 x= 17 yl=-4 5 10 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
version 2
After learning from https://www.youtube.com/watch?v=wRTGDig3jx8 <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)
- Alternate (better) method using slopes
- 1) Compute path from lowest/leftmost to leftmost/lowest
- 2) Compute leftmost vertical border
- 3) Compute path from rightmost/highest to highest/rightmost
- 4) Compute path from highest/rightmost to rightmost/highest
- 5) Compute rightmost vertical border
- 6) Compute path from rightmost/lowest to lowest_leftmost point
- --------------------------------------------------------------------*/
Parse Arg fid If fid= Then Do
fid='line.in' fid='point.in' fid='chullmin.in' /* miscellaneous test data */ fid='chullxx.in' fid='chullx.in' fid='chullt.in' fid='chulla.in' fid='sq.in' fid='tri.in' fid='z.in' fid='chull.in' /* data from task description */ End
g.0debug= 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
Call lineout fid g.0xlist= Do i=1 To x.0 /* loop over points */
x=x.i g.0xlist=g.0xlist x yl.x=sortv(yl.x) /* sort y-coordinates */ End
Call sho If x.0<3 Then Do
Say 'We need at least three points!' Exit End
Call o 'g.0xmin='g.0xmin Call o 'g.0xmi ='g.0xmi Call o 'g.0ymin='g.0ymin Call o 'g.0ymi ='g.0ymi
Do i=1 To x.0
x=x.i If minv(yl.x)=g.0ymin Then Leave End
lowest_leftmost=i
highest_rightmost=0 Do i=1 To x.0
x=x.i If maxv(yl.x)=g.0ymax Then highest_rightmost=i If maxv(yl.x)<g.0ymax Then If highest_rightmost>0 Then Leave End
Call o 'lowest_leftmost='lowest_leftmost Call o 'highest_rightmost ='highest_rightmost
x=x.lowest_leftmost Call o 'We start at' from x'/'minv(yl.x) path=x'/'minv(yl.x) /*---------------------------------------------------------------------
- 1) Compute path from lowest/leftmost to leftmost/lowest
- --------------------------------------------------------------------*/
Call min_path lowest_leftmost,1 /*---------------------------------------------------------------------
- 2) Compute leftmost vertical border
- --------------------------------------------------------------------*/
Do i=2 To words(yl.x)
path=path x'/'word(yl.x,i) End
cxy=x'/'maxv(yl.x) /*---------------------------------------------------------------------
- 3) Compute path from rightmost/highest to highest/rightmost
- --------------------------------------------------------------------*/
Call max_path ci,highest_rightmost /*---------------------------------------------------------------------
- 4) Compute path from highest/rightmost to rightmost/highest
- --------------------------------------------------------------------*/
Call max_path ci,x.0 /*---------------------------------------------------------------------
- 5) Compute rightmost vertical border
- --------------------------------------------------------------------*/
Do i=words(yl.x)-1 To 1 By -1
cxy=x'/'word(yl.x,i) path=path cxy End
/*---------------------------------------------------------------------
- 6) Compute path from rightmost/lowest to lowest_leftmost
- --------------------------------------------------------------------*/
Call min_path ci,lowest_leftmost
Parse Var path pathx path have.=0 Do i=1 By 1 While path>
Parse Var path xy path If have.xy=0 Then Do pathx=pathx xy have.xy=1 End End
Say 'Points of convex hull in clockwise order:' Say pathx Call lineout g.0oid Exit
min_path:
Parse Arg from,tgt ci=from cxy=x.ci Do Until ci=tgt kmax=-1000 Do i=ci-1 To 1 By sign(tgt-from) x=x.i k=k(cxy'/'minv(yl.cxy),x'/'minv(yl.x)) If k>kmax Then Do kmax=k ii=i End End ci=ii cxy=x.ii path=path cxy'/'minv(yl.cxy) End Return
max_path:
Parse Arg from,tgt Do Until ci=tgt kmax=-1000 Do i=ci+1 To tgt x=x.i k=k(cxy,x'/'maxv(yl.x)) If k>kmax Then Do kmax=k ii=i End End x=x.ii cxy=x'/'maxv(yl.x) path=path cxy ci=ii End Return
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 format(i,2) '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)))
k: Procedure /*---------------------------------------------------------------------
- Compute slope 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 k='*' Else k=(yb-ya)/(xb-xa) Return k
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
- --------------------------------------------------------------------*/
If arg(2)=1 Then say arg(1) Return lineout(g.0oid,arg(1))</lang>
- Output:
1 x= -9 yl=-3 2 x= -4 yl=-6 -2 3 x= -3 yl=-9 15 4 x= 0 yl=6 11 5 x= 3 yl=-4 16 6 x= 5 yl=19 7 x= 12 yl=13 17 8 x= 16 yl=-7 -3 3 6 9 x= 17 yl=-4 5 10 x= 19 yl=-8 Points of convex hull in clockwise order: -3/-9 -9/-3 -3/15 5/19 12/17 17/5 19/-8 -3/-9
Scala
Scala Implementation to find Convex hull of given points collection. Functional Paradigm followed <lang Scala> object convex_hull{
def get_hull(points:List[(Double,Double)], hull:List[(Double,Double)]):List[(Double,Double)] = points match{ case Nil => join_tail(hull,hull.size -1) case head :: tail => get_hull(tail,reduce(head::hull)) } def reduce(hull:List[(Double,Double)]):List[(Double,Double)] = hull match{ case p1::p2::p3::rest => { if(check_point(p1,p2,p3)) hull else reduce(p1::p3::rest) } case _ => hull } def check_point(pnt:(Double,Double), p2:(Double,Double),p1:(Double,Double)): Boolean = { val (x,y) = (pnt._1,pnt._2) val (x1,y1) = (p1._1,p1._2) val (x2,y2) = (p2._1,p2._2) ((x-x1)*(y2-y1) - (x2-x1)*(y-y1)) <= 0 } def m(p1:(Double,Double), p2:(Double,Double)):Double = { if(p2._1 == p1._1 && p1._2>p2._2) 90 else if(p2._1 == p1._1 && p1._2<p2._2) -90 else if(p1._1<p2._1) 180 - Math.toDegrees(Math.atan(-(p1._2 - p2._2)/(p1._1 - p2._1))) else Math.toDegrees(Math.atan((p1._2 - p2._2)/(p1._1 - p2._1))) } def join_tail(hull:List[(Double,Double)],len:Int):List[(Double,Double)] = { if(m(hull(len),hull(0)) > m(hull(len-1),hull(0))) join_tail(hull.slice(0,len),len-1) else hull } def main(args:Array[String]){ val points = List[(Double,Double)]((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)) val sorted_points = points.sortWith(m(_,(0.0,0.0)) < m(_,(0.0,0.0))) println(f"Points:\n" + points + f"\n\nConvex Hull :\n" +get_hull(sorted_points,List[(Double,Double)]())) }
} </lang>
- Output:
Points: List((16.0,3.0), (12.0,17.0), (0.0,6.0), (-4.0,-6.0), (16.0,6.0), (16.0,-7.0), (16.0,-3.0), (17.0,-4.0), (5.0,19.0), (19.0,-8.0), (3.0,16.0), (12.0,13.0), (3.0,-4.0), (17.0,5.0), (-3.0,15.0), (-3.0,-9.0), (0.0,11.0), (-9.0,-3.0), (-4.0,-2.0), (12.0,10.0)) Convex Hull : List((-3.0,-9.0), (-9.0,-3.0), (-3.0,15.0), (5.0,19.0), (12.0,17.0), (17.0,5.0), (19.0,-8.0))
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]
}
- Three points are a counter-clockwise turn if ccw > 0, clockwise if
- ccw < 0, and collinear if ccw = 0 because ccw is a determinant that
- 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))