Convex hull
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
D
<lang D>import std.algorithm.sorting; import std.stdio;
struct Point {
int x; int y;
int opCmp(Point rhs) { if (x < rhs.x) return -1; if (rhs.x < x) return 1; return 0; }
void toString(scope void delegate(const(char)[]) sink) const { import std.format; sink("("); formattedWrite(sink, "%d", x); sink(","); formattedWrite(sink, "%d", y); sink(")"); }
}
Point[] convexHull(Point[] p) {
if (p.length == 0) return []; p.sort; Point[] h;
// lower hull foreach (pt; p) { while (h.length >= 2 && !ccw(h[$-2], h[$-1], pt)) { h.length--; } h ~= pt; }
// upper hull auto t = h.length + 1; foreach_reverse (i; 0..(p.length - 1)) { auto pt = p[i]; while (h.length >= t && !ccw(h[$-2], h[$-1], pt)) { h.length--; } h ~= pt; }
h.length--; return h;
}
/* ccw returns true if the three points make a counter-clockwise turn */ auto ccw(Point a, Point b, Point c) {
return ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x));
}
void main() {
auto points = [ Point(16, 3), Point(12, 17), Point( 0, 6), Point(-4, -6), Point(16, 6), Point(16, -7), Point(16, -3), Point(17, -4), Point( 5, 19), Point(19, -8), Point( 3, 16), Point(12, 13), Point( 3, -4), Point(17, 5), Point(-3, 15), Point(-3, -9), Point( 0, 11), Point(-9, -3), Point(-4, -2), Point(12, 10) ]; auto hull = convexHull(points); writeln("Convex Hull: ", hull);
}</lang>
- Output:
Convex Hull: [(-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19), (-3,15)]
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.List (sortBy, groupBy, maximumBy) import Data.Ord (comparing)
(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
main :: IO () main =
mapM_ print $ 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>
- 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>
Java
<lang Java>import java.util.ArrayList; import java.util.Arrays; import java.util.List; import java.util.stream.Collectors;
import static java.util.Collections.emptyList;
public class ConvexHull {
private static class Point implements Comparable<Point> { private int x, y;
public Point(int x, int y) { this.x = x; this.y = y; }
@Override public int compareTo(Point o) { return Integer.compare(x, o.x); }
@Override public String toString() { return String.format("(%d, %d)", x, y); } }
private static List<Point> convexHull(List<Point> p) { if (p.isEmpty()) return emptyList(); p.sort(Point::compareTo); List<Point> h = new ArrayList<>();
// lower hull for (Point pt : p) { while (h.size() >= 2 && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) { h.remove(h.size() - 1); } h.add(pt); }
// upper hull int t = h.size() + 1; for (int i = p.size() - 1; i >= 0; i--) { Point pt = p.get(i); while (h.size() >= t && !ccw(h.get(h.size() - 2), h.get(h.size() - 1), pt)) { h.remove(h.size() - 1); } h.add(pt); }
h.remove(h.size() - 1); return h; }
// ccw returns true if the three points make a counter-clockwise turn private static boolean ccw(Point a, Point b, Point c) { return ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x)); }
public static void main(String[] args) { List<Point> points = Arrays.asList(new Point(16, 3), new Point(12, 17), new Point(0, 6), new Point(-4, -6), new Point(16, 6),
new Point(16, -7), new Point(16, -3), new Point(17, -4), new Point(5, 19), new Point(19, -8),
new Point(3, 16), new Point(12, 13), new Point(3, -4), new Point(17, 5), new Point(-3, 15),
new Point(-3, -9), new Point(0, 11), new Point(-9, -3), new Point(-4, -2), new Point(12, 10));
List<Point> hull = convexHull(points); System.out.printf("Convex Hull: %s\n", hull); }
}</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
Kotlin
<lang scala>// version 1.1.3
class Point(val x: Int, val y: Int) : Comparable<Point> {
override fun compareTo(other: Point) = this.x.compareTo(other.x)
override fun toString() = "($x, $y)"
}
fun convexHull(p: Array<Point>): List<Point> {
if (p.isEmpty()) return emptyList() p.sort() val h = mutableListOf<Point>()
// lower hull for (pt in p) { while (h.size >= 2 && !ccw(h[h.size - 2], h.last(), pt)) { h.removeAt(h.lastIndex) } h.add(pt) }
// upper hull val t = h.size + 1 for (i in p.size - 2 downTo 0) { val pt = p[i] while (h.size >= t && !ccw(h[h.size - 2], h.last(), pt)) { h.removeAt(h.lastIndex) } h.add(pt) }
h.removeAt(h.lastIndex) return h
}
/* ccw returns true if the three points make a counter-clockwise turn */ fun ccw(a: Point, b: Point, c: Point) =
((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))
fun main(args: Array<String>) {
val points = arrayOf( Point(16, 3), Point(12, 17), Point( 0, 6), Point(-4, -6), Point(16, 6), Point(16, -7), Point(16, -3), Point(17, -4), Point( 5, 19), Point(19, -8), Point( 3, 16), Point(12, 13), Point( 3, -4), Point(17, 5), Point(-3, 15), Point(-3, -9), Point( 0, 11), Point(-9, -3), Point(-4, -2), Point(12, 10) ) val hull = convexHull(points) println("Convex Hull: $hull")
}</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
Perl 6
Modified the angle sort method as the original could fail if there were multiple points on the same y coordinate as the starting point. Sorts on tangent rather than triangle area. Inexpensive since it still doesn't do any trigonometric math, just calculates the ratio of opposite over adjacent. The original returned the correct answer for the task example, but only by accident. If the points (14,-9), (1,-9) were added to the task example, it wouldn't give a correct answer. Now it does.
<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 tangent (Point $a, Point $b) {
my $opp = $b.x - $a.x; my $adj = $b.y - $a.y; $adj != 0 ?? $opp / $adj !! Inf
}
sub graham-scan (**@coords) {
# sort points by y, secondary sort on x my @sp = @coords.map( { Point.new( :x($_[0]), :y($_[1]) ) } ) .sort: {.y, .x};
# need at least 3 points to make a hull return @sp if +@sp < 3;
# first point on hull is minimum y point my @h = @sp.shift;
# re-sort the points by angle, secondary on x @sp = @sp.map( { $++ => [tangent(@h[0], $_), $_.x] } ) .sort( {-$_.value[0], $_.value[1] } ) .map: { @sp[$_.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;
@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), (14,-9), (1,-9) );
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)] Convex Hull (9 points): [(-3, -9) (1, -9) (14, -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))
Sidef
<lang ruby>class Point(Number x, Number y) {
method to_s { "(#{x}, #{y})" }
}
func ccw (Point a, Point b, Point c) {
(b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x)
}
func tangent (Point a, Point b) {
(b.x - a.x) / (b.y - a.y)
}
func graham_scan (*coords) {
## sort points by y, secondary sort on x var sp = coords.map { |a| Point(a...) }.sort { |a,b| (a.y <=> b.y) || (a.x <=> b.x) }
# need at least 3 points to make a hull if (sp.len < 3) { return sp }
# first point on hull is minimum y point var h = [sp.shift]
# re-sort the points by angle, secondary on x sp = sp.map_kv { |k,v| Pair(k, [tangent(h[0], v), v.x]) }.sort { |a,b| (b.value[0] <=> a.value[0]) || (a.value[1] <=> b.value[1]) }.map { |a| sp[a.key] }
# first point of re-sorted list is guaranteed to be on hull h << sp.shift
# check through the remaining list making sure that # there is always a positive angle sp.each { |point| loop { if (ccw(h.last(2)..., point) >= 0) { h << point break } else { h.pop } } }
return h
}
var 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.len} points): ", hull.join(" "))
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], [14,-9], [1,-9])
say("Convex Hull (#{hull.len} points): ", hull.join(" "))</lang>
- Output:
Convex Hull (7 points): (-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3) Convex Hull (9 points): (-3, -9) (1, -9) (14, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-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]
}
- 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))