Convex hull
You are encouraged to solve this task according to the task description, using any language you may know.
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
11l
<lang 11l>F orientation(p, q, r)
V val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y)
I val == 0
R 0
R I val > 0 {1} E 2
F calculateConvexHull(points)
[(Int, Int)] result
I points.len < 3
R points
V indexMinX = 0
L(p) points
V i = L.index
I p.x < points[indexMinX].x
indexMinX = i
V p = indexMinX V q = 0
L
result.append(points[p])
q = (p + 1) % points.len
L(i) 0 .< points.len
I orientation(points[p], points[i], points[q]) == 2
q = i
p = q
I p == indexMinX
L.break
R result
V points = [(16, 3),
(12, 17),
(0, 6),
(-4, -6),
(16, 6),
(16, -7),
(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)]
V hull = calculateConvexHull(points) L(i) hull
print(i)</lang>
- Output:
(-9, -3) (-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15)
Action!
<lang Action!>DEFINE POINTSIZE="4" TYPE PointI=[INT x,y]
CARD FUNC GetAddr(INT ARRAY points BYTE index) RETURN (points+POINTSIZE*index)
BYTE FUNC GetMinXIndex(INT ARRAY points BYTE pLen)
PointI POINTER p BYTE i,index INT minX
p=GetAddr(points,0)
minX=p.x
index=0
FOR i=1 TO pLen-1
DO
p=GetAddr(points,i)
IF p.x<minX THEN
minX=p.x
index=i
FI
OD
RETURN (index)
BYTE FUNC Orientation(PointI POINTER p,q,r)
INT v v=(q.y-p.y)*(r.x-q.x) v==-(q.x-p.x)*(r.y-q.y)
IF v=0 THEN RETURN (0) ELSEIF v>0 THEN RETURN (1) FI
RETURN (2)
PROC ConvexHull(INT ARRAY points BYTE pLen
INT ARRAY res BYTE POINTER resLen) PointI POINTER pSrc,pDst,p1,p2,p3 BYTE minIndex,i,p,q
IF pLen<3 THEN resLen^=pLen MoveBlock(res,points,pLen*POINTSIZE) RETURN FI
resLen^=0 minIndex=GetMinXIndex(points,pLen) p=minIndex q=0 DO pSrc=GetAddr(points,p) pDst=GetAddr(res,resLen^) pDst.x=pSrc.x pDst.y=pSrc.y resLen^==+1
q=(p+1) MOD pLen
FOR i=0 TO pLen-1
DO
p1=GetAddr(points,p)
p2=GetAddr(points,i)
p3=GetAddr(points,q)
IF Orientation(p1,p2,p3)=2 THEN
q=i
FI
OD
p=q
UNTIL p=minIndex
OD
RETURN
PROC PrintPoints(INT ARRAY points BYTE len)
PointI POINTER p BYTE i
FOR i=0 TO len-1
DO
p=GetAddr(points,i)
PrintF("(%I,%I) ",p.x,p.y)
OD
RETURN
PROC Main()
INT ARRAY points=[ 16 3 12 17 0 6 65532 65530 16 6 16 65529 17 65532 5 19 19 65528 3 16 12 13 3 65532 17 5 65533 15 65533 65527 0 11 65527 65533 65532 65534 12 10] INT ARRAY result(38) BYTE pLen=[19],rlen
ConvexHull(points,pLen,result,@rlen)
PrintE("Points:")
PrintPoints(points,pLen)
PutE() PutE()
PrintE("Convex hull:")
PrintPoints(result,rLen)
RETURN</lang>
- Output:
Screenshot from Atari 8-bit computer
Points: (16,3) (12,17) (0,6) (-4,-6) (16,6) (16,-7) (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) Convex hull: (-9,-3) (-3,-9) (19,-8) (17,5) (12,17) (5,19) (-3,15)
Ada
<lang Ada>with Ada.Text_IO; with Ada.Containers.Vectors;
procedure Convex_Hull is
type Point is record
X, Y : Integer;
end record;
package Point_Vectors is
new Ada.Containers.Vectors (Index_Type => Positive,
Element_Type => Point);
use Point_Vectors;
function Find_Convex_Hull (Vec : in Vector) return Vector is
function Counter_Clock_Wise (A, B, C : in Point) return Boolean is
((B.X - A.X) * (C.Y - A.Y) > (B.Y - A.Y) * (C.X - A.X));
function Less_Than (Left, Right : Point) return Boolean is
(Left.X < Right.X);
package Sorter is
new Point_Vectors.Generic_Sorting (Less_Than);
Sorted : Vector := Vec;
Result : Vector := Empty_vector;
use type Ada.Containers.Count_Type;
begin
if Vec = Empty_Vector then
return Empty_Vector;
end if;
Sorter.Sort (Sorted);
-- Lower hull
for Index in Sorted.First_Index .. Sorted.Last_Index loop
while
Result.Length >= 2 and then
not Counter_Clock_Wise (Result (Result.Last_Index - 1),
Result (Result.Last_Index),
Sorted (Index))
loop
Result.Delete_Last;
end loop;
Result.Append (Sorted (Index));
end loop;
-- Upper hull
declare
T : constant Ada.Containers.Count_Type := Result.Length + 1;
begin
for Index in reverse Sorted.First_Index .. Sorted.Last_Index loop
while
Result.Length >= T and then
not Counter_Clock_Wise (Result (Result.Last_Index - 1),
Result (Result.Last_Index),
Sorted (Index))
loop
Result.Delete_Last;
end loop;
Result.Append (Sorted (Index));
end loop;
end;
Result.Delete_Last;
return Result;
end Find_Convex_Hull;
procedure Show (Vec : in Vector) is
use Ada.Text_IO;
begin
Put ("[ ");
for Point of Vec loop
Put ("(" & Point.X'Image & "," & Point.Y'Image & ")");
end loop;
Put (" ]");
end Show;
Vec : constant Vector :=
(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);
begin
Show (Find_Convex_Hull (Vec)); Ada.Text_IO.New_Line;
end Convex_Hull;</lang>
- Output:
[ (-9,-3)(-3,-9)( 19,-8)( 17, 5)( 12, 17)( 5, 19)(-3, 15) ]
Arturo
<lang rebol>define :point [x y][]
orientation: function [P Q R][
val: sub (Q\y - P\y)*(R\x - Q\x) (Q\x - P\x)*(R\y - Q\y) if val=0 -> return 0 if val>0 -> return 1 return 2
]
calculateConvexHull: function [points][
result: new []
if 3 > size points [
loop points 'p -> 'result ++ p
]
indexMinX: 0
loop.with:'i points 'p [
if p\x < get points\[indexMinX] 'x ->
indexMinX: i
]
p: indexMinX q: 0
while [true][
'result ++ points\[p]
q: (p + 1) % size points
loop 0..dec size points 'i [
if 2 = orientation points\[p] points\[i] points\[q] ->
q: i
]
p: q
if p=indexMinX -> break ]
return result
]
points: @[
to :point [16 3], to :point [12 17], to :point [0 6], to :point @[neg 4 neg 6], to :point [16 6], to :point @[16 neg 7], to :point @[17 neg 4], to :point [5 19], to :point @[19 neg 8], to :point [3 16], to :point [12 13], to :point @[3 neg 4], to :point [17 5], to :point @[neg 3 15], to :point @[neg 3 neg 9], to :point [0 11], to :point @[neg 9 neg 3], to :point @[neg 4 neg 2], to :point [12 10]
]
hull: calculateConvexHull points loop hull 'i ->
print i</lang>
- Output:
[x:-9 y:-3] [x:-3 y:-9] [x:19 y:-8] [x:17 y:5] [x:12 y:17] [x:5 y:19] [x:-3 y:15]
C
<lang C>#include <assert.h>
- include <stdbool.h>
- include <stdio.h>
- include <stdlib.h>
typedef struct tPoint {
int x, y;
} Point;
bool ccw(const Point *a, const Point *b, const Point *c) {
return (b->x - a->x) * (c->y - a->y)
> (b->y - a->y) * (c->x - a->x);
}
int comparePoints(const void *lhs, const void *rhs) {
const Point* lp = lhs;
const Point* rp = rhs;
if (lp->x < rp->x)
return -1;
if (rp->x < lp->x)
return 1;
if (lp->y < rp->y)
return -1;
if (rp->y < lp->y)
return 1;
return 0;
}
void fatal(const char* message) {
fprintf(stderr, "%s\n", message); exit(1);
}
void* xmalloc(size_t n) {
void* ptr = malloc(n);
if (ptr == NULL)
fatal("Out of memory");
return ptr;
}
void* xrealloc(void* p, size_t n) {
void* ptr = realloc(p, n);
if (ptr == NULL)
fatal("Out of memory");
return ptr;
}
void printPoints(const Point* points, int len) {
printf("[");
if (len > 0) {
const Point* ptr = points;
const Point* end = points + len;
printf("(%d, %d)", ptr->x, ptr->y);
++ptr;
for (; ptr < end; ++ptr)
printf(", (%d, %d)", ptr->x, ptr->y);
}
printf("]");
}
Point* convexHull(Point p[], int len, int* hsize) {
if (len == 0) {
*hsize = 0;
return NULL;
}
int i, size = 0, capacity = 4; Point* hull = xmalloc(capacity * sizeof(Point));
qsort(p, len, sizeof(Point), comparePoints);
/* lower hull */
for (i = 0; i < len; ++i) {
while (size >= 2 && !ccw(&hull[size - 2], &hull[size - 1], &p[i]))
--size;
if (size == capacity) {
capacity *= 2;
hull = xrealloc(hull, capacity * sizeof(Point));
}
assert(size >= 0 && size < capacity);
hull[size++] = p[i];
}
/* upper hull */
int t = size + 1;
for (i = len - 1; i >= 0; i--) {
while (size >= t && !ccw(&hull[size - 2], &hull[size - 1], &p[i]))
--size;
if (size == capacity) {
capacity *= 2;
hull = xrealloc(hull, capacity * sizeof(Point));
}
assert(size >= 0 && size < capacity);
hull[size++] = p[i];
}
--size;
assert(size >= 0);
hull = xrealloc(hull, size * sizeof(Point));
*hsize = size;
return hull;
}
int main() {
Point 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}
};
int hsize;
Point* hull = convexHull(points, sizeof(points)/sizeof(Point), &hsize);
printf("Convex Hull: ");
printPoints(hull, hsize);
printf("\n");
free(hull);
return 0;
}</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
C#
<lang csharp>using System; using System.Collections.Generic;
namespace ConvexHull {
class Point : IComparable<Point> {
private int x, y;
public Point(int x, int y) {
this.x = x;
this.y = y;
}
public int X { get => x; set => x = value; }
public int Y { get => y; set => y = value; }
public int CompareTo(Point other) {
return x.CompareTo(other.x);
}
public override string ToString() {
return string.Format("({0}, {1})", x, y);
}
}
class Program {
private static List<Point> ConvexHull(List<Point> p) {
if (p.Count == 0) return new List<Point>();
p.Sort();
List<Point> h = new List<Point>();
// lower hull
foreach (var pt in p) {
while (h.Count >= 2 && !Ccw(h[h.Count - 2], h[h.Count - 1], pt)) {
h.RemoveAt(h.Count - 1);
}
h.Add(pt);
}
// upper hull
int t = h.Count + 1;
for (int i = p.Count - 1; i >= 0; i--) {
Point pt = p[i];
while (h.Count >= t && !Ccw(h[h.Count - 2], h[h.Count - 1], pt)) {
h.RemoveAt(h.Count - 1);
}
h.Add(pt);
}
h.RemoveAt(h.Count - 1);
return h;
}
private static bool Ccw(Point a, Point b, Point c) {
return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X));
}
static void Main(string[] args) {
List<Point> points = new List<Point>() {
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);
Console.Write("Convex Hull: [");
for (int i = 0; i < hull.Count; i++) {
if (i > 0) {
Console.Write(", ");
}
Point pt = hull[i];
Console.Write(pt);
}
Console.WriteLine("]");
}
}
}</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
C++
<lang cpp>#include <algorithm>
- include <iostream>
- include <ostream>
- include <vector>
- include <tuple>
typedef std::tuple<int, int> point;
std::ostream& print(std::ostream& os, const point& p) {
return os << "(" << std::get<0>(p) << ", " << std::get<1>(p) << ")";
}
std::ostream& print(std::ostream& os, const std::vector<point>& v) {
auto it = v.cbegin(); auto end = v.cend();
os << "[";
if (it != end) {
print(os, *it);
it = std::next(it);
}
while (it != end) {
os << ", ";
print(os, *it);
it = std::next(it);
}
return os << "]";
}
// returns true if the three points make a counter-clockwise turn bool ccw(const point& a, const point& b, const point& c) {
return ((std::get<0>(b) - std::get<0>(a)) * (std::get<1>(c) - std::get<1>(a)))
> ((std::get<1>(b) - std::get<1>(a)) * (std::get<0>(c) - std::get<0>(a)));
}
std::vector<point> convexHull(std::vector<point> p) {
if (p.size() == 0) return std::vector<point>();
std::sort(p.begin(), p.end(), [](point& a, point& b){
if (std::get<0>(a) < std::get<0>(b)) return true;
return false;
});
std::vector<point> h;
// lower hull
for (const auto& pt : p) {
while (h.size() >= 2 && !ccw(h.at(h.size() - 2), h.at(h.size() - 1), pt)) {
h.pop_back();
}
h.push_back(pt);
}
// upper hull
auto t = h.size() + 1;
for (auto it = p.crbegin(); it != p.crend(); it = std::next(it)) {
auto pt = *it;
while (h.size() >= t && !ccw(h.at(h.size() - 2), h.at(h.size() - 1), pt)) {
h.pop_back();
}
h.push_back(pt);
}
h.pop_back(); return h;
}
int main() {
using namespace std;
vector<point> points = {
make_pair(16, 3), make_pair(12, 17), make_pair(0, 6), make_pair(-4, -6), make_pair(16, 6),
make_pair(16, -7), make_pair(16, -3), make_pair(17, -4), make_pair(5, 19), make_pair(19, -8),
make_pair(3, 16), make_pair(12, 13), make_pair(3, -4), make_pair(17, 5), make_pair(-3, 15),
make_pair(-3, -9), make_pair(0, 11), make_pair(-9, -3), make_pair(-4, -2), make_pair(12, 10)
};
auto hull = convexHull(points); auto it = hull.cbegin(); auto end = hull.cend();
cout << "Convex Hull: "; print(cout, hull); cout << endl;
return 0;
}</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
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)]
Delphi
<lang Delphi> program ConvexHulls;
{$APPTYPE CONSOLE}
{$R *.res}
uses
System.Types, System.SysUtils, System.Generics.Defaults, System.Generics.Collections;
function Ccw(const a, b, c: TPoint): Boolean; begin Result := ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X)); end;
function ConvexHull(const p: TList<TPoint>): TList<TPoint>; var pt: TPoint; i, t: Integer; begin Result := TList<TPoint>.Create;
if (p.Count = 0) then Exit;
p.Sort(TComparer<TPoint>.Construct(
function(const Left, Right: TPoint): Integer
begin
Result := Left.X - Right.X;
end
));
// lower hull
for i := 0 to p.Count-1 do
begin
pt := p[i];
while ((Result.Count >= 2) and (not Ccw(Result[Result.Count - 2], Result[Result.Count - 1], pt))) do
begin
Result.Delete(Result.Count - 1);
end;
Result.Add(pt);
end;
// upper hull
t := Result.Count + 1;
for i := p.Count-1 downto 0 do
begin
pt := p[i];
while ((Result.Count >= t) and (not Ccw(Result[Result.Count - 2], Result[Result.Count - 1], pt))) do
begin
Result.Delete(Result.Count - 1);
end;
Result.Add(pt);
end;
Result.Delete(Result.Count - 1); end;
var
points: TList<TPoint>; hull: TList<TPoint>; i: Integer;
begin
hull := nil; points := TList<TPoint>.Create; try
points.AddRange([
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)
]);
hull := ConvexHull(points);
// Output the result
Write('Convex Hull: [');
for i := 0 to hull.Count-1 do
begin
if (i > 0) then Write(', ');
Write(Format('(%d, %d)', [hull[i].X, hull[i].Y]));
end;
WriteLn(']');
finally hull.Free; points.Free; end;
end. </lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
F#
<lang F#>open System
type Point =
struct
val X : int
val Y : int
new (x : int, y : int ) = {X = x; Y = y}
end
let (poly : Point list) = [ 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)]
let affiche (lst : Point list) =
let mutable (str : string) = List.fold (fun acc (p : Point) -> acc + sprintf "(%d, %d) " p.X p.Y) "Convex Hull: [" lst printfn "%s" (str.[0.. str.Length - 2] + "]")
let ccw (p1 : Point) (p2 : Point) (p3 : Point) =
(p2.X - p1.X) * (p3.Y - p1.Y) > (p2.Y - p1.Y) * (p3.X - p1.X)
let convexHull (poly : Point list) =
let mutable (outHull : Point list) = List.Empty let mutable (k : int) = 0
for p in poly do
while k >= 2 && not (ccw outHull.[k-2] outHull.[k-1] p) do
k <- k - 1
if k >= outHull.Length
then outHull <- outHull @ [p]
else outHull <- outHull.[0..k - 1] @ [p]
k <- k + 1
let (t : int) = k + 1
for p in List.rev poly do
while k >= t && not (ccw outHull.[k-2] outHull.[k-1] p) do
k <- k - 1
if k >= outHull.Length
then outHull <- outHull @ [p]
else outHull <- outHull.[0..k - 1] @ [p]
k <- k + 1
outHull.[0 .. k - 2]
affiche (convexHull (List.sortBy (fun (x : Point) -> x.X, x.Y) poly))
</lang>
- Output:
Convex Hull: [(-9,-3), (-3,-9), (19,-8), (17,5), (12,17), (5,19), (-3,15)]
Fortran
<lang fortran>module convex_hulls
! ! Convex hulls by Andrew's monotone chain algorithm. ! ! For a description of the algorithm, see ! https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169 ! ! For brevity in the task, I shall use the built-in "complex" type ! to represent objects in the plane. One could have fun rewriting ! this implementation in terms of geometric algebra. !
implicit none private
public :: find_convex_hull
contains
elemental function x (u) complex, intent(in) :: u real :: x
x = real (u) end function x
elemental function y (u) complex, intent(in) :: u real :: y
y = aimag (u) end function y
elemental function cross (u, v) result (p) complex, intent(in) :: u, v real :: p
! The cross product as a signed scalar. p = (x (u) * y (v)) - (y (u) * x (v)) end function cross
subroutine sort_points (num_points, points) integer, intent(in) :: num_points complex, intent(inout) :: points(0:*)
! Sort first in ascending order by x-coordinates, then in ! ascending order by y-coordinates. Any decent sort algorithm will ! suffice; for the sake of interest, here is the Shell sort of ! https://en.wikipedia.org/w/index.php?title=Shellsort&oldid=1084744510
integer, parameter :: gaps(1:8) = (/ 701, 301, 132, 57, 23, 10, 4, 1 /)
integer :: i, j, k, gap, offset complex :: temp logical :: done
do k = 1, 8
gap = gaps(k)
do offset = 0, gap - 1
do i = offset, num_points - 1, gap
temp = points(i)
j = i
done = .false.
do while (.not. done)
if (j < gap) then
done = .true.
else if (x (points(j - gap)) < x (temp)) then
done = .true.
else if (x (points(j - gap)) == x (temp) .and. &
& (y (points(j - gap)) <= y (temp))) then
done = .true.
else
points(j) = points(j - gap)
j = j - gap
end if
end do
points(j) = temp
end do
end do
end do
end subroutine sort_points
subroutine delete_neighbor_duplicates (n, pt) integer, intent(inout) :: n complex, intent(inout) :: pt(0:*)
call delete_trailing_duplicates call delete_nontrailing_duplicates
contains
subroutine delete_trailing_duplicates
integer :: i
logical :: done
i = n - 1
done = .false.
do while (.not. done)
if (i == 0) then
n = 1
done = .true.
else if (pt(i - 1) /= pt(i)) then
n = i + 1
done = .true.
else
i = i - 1
end if
end do
end subroutine delete_trailing_duplicates
subroutine delete_nontrailing_duplicates
integer :: i, j, num_deleted
logical :: done
i = 0
do while (i < n - 1)
j = i + 1
done = .false.
do while (.not. done)
if (j == n) then
done = .true.
else if (pt(j) /= pt(i)) then
done = .true.
else
j = j + 1
end if
end do
if (j /= i + 1) then
num_deleted = j - i - 1
do while (j /= n)
pt(j - num_deleted) = pt(j)
j = j + 1
end do
n = n - num_deleted
end if
i = i + 1
end do
end subroutine delete_nontrailing_duplicates
end subroutine delete_neighbor_duplicates
subroutine construct_lower_hull (n, pt, hull_size, hull) integer, intent(in) :: n ! Number of points. complex, intent(in) :: pt(0:*) integer, intent(inout) :: hull_size complex, intent(inout) :: hull(0:*)
integer :: i, j logical :: done
j = 1
hull(0:1) = pt(0:1)
do i = 2, n - 1
done = .false.
do while (.not. done)
if (j == 0) then
j = j + 1
hull(j) = pt(i)
done = .true.
else if (0.0 < cross (hull(j) - hull(j - 1), &
& pt(i) - hull(j - 1))) then
j = j + 1
hull(j) = pt(i)
done = .true.
else
j = j - 1
end if
end do
end do
hull_size = j + 1
end subroutine construct_lower_hull
subroutine construct_upper_hull (n, pt, hull_size, hull) integer, intent(in) :: n ! Number of points. complex, intent(in) :: pt(0:*) integer, intent(inout) :: hull_size complex, intent(inout) :: hull(0:*)
integer :: i, j logical :: done
j = 1
hull(0:1) = pt(n - 1 : n - 2 : -1)
do i = n - 3, 0, -1
done = .false.
do while (.not. done)
if (j == 0) then
j = j + 1
hull(j) = pt(i)
done = .true.
else if (0.0 < cross (hull(j) - hull(j - 1), &
& pt(i) - hull(j - 1))) then
j = j + 1
hull(j) = pt(i)
done = .true.
else
j = j - 1
end if
end do
end do
hull_size = j + 1
end subroutine construct_upper_hull
subroutine contruct_hull (n, pt, hull_size, hull) integer, intent(in) :: n ! Number of points. complex, intent(in) :: pt(0:*) integer, intent(inout) :: hull_size complex, intent(inout) :: hull(0:*)
integer :: lower_hull_size, upper_hull_size complex :: lower_hull(0 : n - 1), upper_hull(0 : n - 1) integer :: ihull0
ihull0 = lbound (hull, 1)
! A side note: the calls to construct_lower_hull and ! construct_upper_hull could be done in parallel. call construct_lower_hull (n, pt, lower_hull_size, lower_hull) call construct_upper_hull (n, pt, upper_hull_size, upper_hull)
hull_size = lower_hull_size + upper_hull_size - 2
hull(:ihull0 + lower_hull_size - 2) = &
& lower_hull(:lower_hull_size - 2)
hull(ihull0 + lower_hull_size - 1 : ihull0 + hull_size - 1) = &
& upper_hull(0 : upper_hull_size - 2)
end subroutine contruct_hull
subroutine find_convex_hull (n, points, hull_size, hull) integer, intent(in) :: n ! Number of points. complex, intent(in) :: points(*) ! Input points. integer, intent(inout) :: hull_size ! The size of the hull. complex, intent(inout) :: hull(*) ! Points of the hull.
! ! Yes, you can call this with something like ! ! call find_convex_hull (n, points, n, points) ! ! and in the program below I shall demonstrate that. !
complex :: pt(0 : n - 1) integer :: ipoints0, ihull0, numpt
ipoints0 = lbound (points, 1) ihull0 = lbound (hull, 1)
pt = points(:ipoints0 + n - 1) numpt = n
call sort_points (numpt, pt) call delete_neighbor_duplicates (numpt, pt)
if (numpt == 0) then
hull_size = 0
else if (numpt <= 2) then
hull_size = numpt
hull(:ihull0 + numpt - 1) = pt(:numpt - 1)
else
call contruct_hull (numpt, pt, hull_size, hull)
end if
end subroutine find_convex_hull
end module convex_hulls
program convex_hull_task
use, non_intrinsic :: convex_hulls implicit none
complex, parameter :: example_points(20) = &
& (/ (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) /)
integer :: n, i complex :: points(0:100) character(len = 100) :: fmt
n = 20 points(1:n) = example_points call find_convex_hull (n, points(1:n), n, points(1:n))
write (fmt, '("(", I20, ("(", F3.0, 1X, F3.0, ") "), ")")') n
write (*, fmt) (points(i), i = 1, n)
end program convex_hull_task</lang>
- Output:
If you compile with -Wextra you may get warnings about the use of == on floating-point numbers. I hate those warnings and turn them off if I am using -Wextra. (Unfortunately, there is much incorrect information about == and floating point that is widely taught as inviolable doctrine.)
$ gfortran -fcheck=all -std=f2018 convex_hull_task.f90 && ./a.out (-9. -3.) (-3. -9.) (19. -8.) (17. 5.) (12. 17.) ( 5. 19.) (-3. 15.)
FreeBASIC
<lang freebasic>
- include "crt.bi"
Screen 20 Window (-20,-20)-(30,30)
Type Point
As Single x,y As Long done
End Type
- macro rotate(pivot,p,a,scale)
Type<Point>(scale*(Cos(a*.0174533)*(p.x-pivot.x)-Sin(a*.0174533)*(p.y-pivot.y))+pivot.x, _ scale*(Sin(a*.0174533)*(p.x-pivot.x)+Cos(a*.0174533)*(p.y-pivot.y))+pivot.y)
- endmacro
Dim As Point p(1 To ...)={(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)}
Function south(p() As Point,Byref idx As Long) As Point
Dim As Point s=Type(0,100)
For n As Long=Lbound(p) To Ubound(p)
Circle(p(n).x,p(n).y),.2,7,,,,f
If s.y>p(n).y Then s=p(n):idx=n
Next n
Return s
End Function
Function segment_distance(lx1 As Single, _
ly1 As Single, _
lx2 As Single, _
ly2 As Single, _
px As Single,_
py As Single, _
Byref ox As Single=0,_
Byref oy As Single=0) As Single
Dim As Single M1,M2,C1,C2,B
B=(Lx2-Lx1):If B=0 Then B=1e-20
M2=(Ly2-Ly1)/B:If M2=0 Then M2=1e-20
M1=-1/M2
C1=py-M1*px
C2=(Ly1*Lx2-Lx1*Ly2)/B
Var L1=((px-lx1)*(px-lx1)+(py-ly1)*(py-ly1)),L2=((px-lx2)*(px-lx2)+(py-ly2)*(py-ly2))
Var a=((lx1-lx2)*(lx1-lx2) + (ly1-ly2)*(ly1-ly2))
Var a1=a+L1
Var a2=a+L2
Var f1=a1>L2,f2=a2>L1
If f1 Xor f2 Then
Var d1=((px-Lx1)*(px-Lx1)+(py-Ly1)*(py-Ly1))
Var d2=((px-Lx2)*(px-Lx2)+(py-Ly2)*(py-Ly2))
If d1<d2 Then Ox=Lx1:Oy=Ly1 : Return Sqr(d1) Else Ox=Lx2:Oy=Ly2:Return Sqr(d2)
End If
Var M=M1-M2:If M=0 Then M=1e-20
Ox=(C2-C1)/(M1-M2)
Oy=(M1*C2-M2*C1)/M
Return Sqr((px-Ox)*(px-Ox)+(py-Oy)*(py-Oy))
End Function
Dim As Long idx
Var s= south(p(),idx)
p(idx).done=1
Redim As Point ans(1 To 1)
ans(1)=s
Dim As Point e=s
e.x=1000
Dim As Long count=1
Dim As Single z
Circle(s.x,s.y),.4,5
Do
z+=.05
Var pt=rotate(s,e,z,1)
For n As Long=Lbound(p) To Ubound(p)
If segment_distance(s.x,s.y,pt.x,pt.y,p(n).x,p(n).y)<.05 Then
s=p(n)
If p(n).done=0 Then
count+=1
Redim Preserve ans(1 To count)
ans(count)=p(n)
p(n).done=1
End If
End If
Circle(s.x,s.y),.4,5
Next n
Loop Until z>360
For n As Long=Lbound(ans) To Ubound(ans)
printf (!"(%2.0f , %2.0f )\n", ans(n).x, ans(n).y)
Next Sleep </lang>
- Output:
Graphical, plus console: (-3 , -9 ) (19 , -8 ) (17 , 5 ) (12 , 17 ) ( 5 , 19 ) (-3 , 15 ) (-9 , -3 )
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)]
Groovy
<lang groovy>class ConvexHull {
private static class Point implements Comparable<Point> {
private int x, y
Point(int x, int y) {
this.x = x
this.y = y
}
@Override
int compareTo(Point o) {
return Integer.compare(x, o.x)
}
@Override
String toString() {
return String.format("(%d, %d)", x, y)
}
}
private static List<Point> convexHull(List<Point> p) {
if (p.isEmpty()) return Collections.emptyList()
p.sort(new Comparator<Point>() {
@Override
int compare(Point o1, Point o2) {
return o1 <=> o2
}
})
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))
}
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)
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]
Haxe
<lang haxe>typedef Point = {x:Float, y:Float};
class Main {
// Calculate orientation for 3 points
// 0 -> Straight line
// 1 -> Clockwise
// 2 -> Counterclockwise
static function orientation(pt1:Point, pt2:Point, pt3:Point): Int
{
var val = ((pt2.x - pt1.x) * (pt3.y - pt1.y)) -
((pt2.y - pt1.y) * (pt3.x - pt1.x));
if (val == 0)
return 0;
else if (val > 0)
return 1;
else return 2;
}
static function convexHull(pts:Array<Point>):Array<Point> {
var result = new Array<Point>();
// There must be at least 3 points
if (pts.length < 3)
for (i in pts) result.push(i);
// Find the leftmost point
var indexMinX = 0;
for (i in 0...(pts.length - 1))
if (pts[i].x < pts[indexMinX].x)
indexMinX = i;
var p = indexMinX;
var q = 0;
while (true) {
// The leftmost point must be part of the hull.
result.push(pts[p]);
q = (p + 1) % pts.length;
for (i in 0...(pts.length - 1))
if (orientation(pts[p], pts[i], pts[q]) == 2) q = i;
p = q;
// Break from loop once we reach the first point again.
if (p == indexMinX)
break;
}
return result;
}
static function main() {
var pts = new Array<Point>();
pts.push({x: 16, y: 3});
pts.push({x: 12, y: 17});
pts.push({x: 0, y: 6});
pts.push({x: -4, y: -6});
pts.push({x: 16, y: 6});
pts.push({x: 16, y: -7});
pts.push({x: 16, y: -3});
pts.push({x: 17, y: -4});
pts.push({x: 5, y: 19});
pts.push({x: 19, y: -8});
pts.push({x: 3, y: 16});
pts.push({x: 12, y: 13});
pts.push({x: 3, y: -4});
pts.push({x: 17, y: 5});
pts.push({x: -3, y: 15});
pts.push({x: -3, y: -9});
pts.push({x: 0, y: 11});
pts.push({x: -9, y: -3});
pts.push({x: -4, y: -2});
pts.push({x: 12, y: 10});
var hull = convexHull(pts);
Sys.print('Convex Hull: [');
var length = hull.length;
var i = 0;
while (length > 0) {
if (i > 0)
Sys.print(', ');
Sys.print('(${hull[i].x}, ${hull[i].y})');
length--;
i++;
}
Sys.println(']');
}
}</lang>
- Output:
Convex Hull: [(-9, -3), (-3, 15), (5, 19), (12, 17), (17, 5), (19, -8), (-3, -9)]
J
Restated from the implementation at [1] which in turn is Kukuruku Hub's [2] translation of Dr. K. L. Metlov's original Russian article [3].
<lang J>counterclockwise =: ({. , }. /: 12 o. }. - {.) @ /:~ crossproduct =: 11 o. [: (* +)/ }. - {. removeinner =: #~ (1 , (0 > 3 crossproduct\ ]) , 1:) hull =: [: removeinner^:_ counterclockwise</lang>
Example use:
<lang J> hull 16j3 12j17 0j6 _4j_6 16j6 16j_7 16j_3 17j_4 5j19 19j_8 3j16 12j13 3j_4 17j5 _3j15 _3j_9 0j11 _9j_3 _4j_2 12j10 _9j_3 _3j_9 19j_8 17j5 12j17 5j19 _3j15
] A =: ~. 20 2 ?.@$ 9
0 2 1 3 2 1 4 1 8 7 3 5 4 8 7 5 6 1 2 6 1 7 0 1 6 8 4 0 8 6 7 6 5 8 0 4 5 3
hull&.:(+.inv"1) A
0 1 4 0 6 1 8 6 8 7 6 8 4 8 1 7 0 4 </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)]
JavaScript
<lang Javascript> function convexHull(points) {
points.sort(comparison);
var L = [];
for (var i = 0; i < points.length; i++) {
while (L.length >= 2 && cross(L[L.length - 2], L[L.length - 1], points[i]) <= 0) {
L.pop();
}
L.push(points[i]);
}
var U = [];
for (var i = points.length - 1; i >= 0; i--) {
while (U.length >= 2 && cross(U[U.length - 2], U[U.length - 1], points[i]) <= 0) {
U.pop();
}
U.push(points[i]);
}
L.pop();
U.pop();
return L.concat(U);
}
function comparison(a, b) {
return a.x == b.x ? a.y - b.y : a.x - b.x;
}
function cross(a, b, o) {
return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
} </lang>
For usage: convexhull.js <lang Javascript> var points = []; var hull = [];
function setup() {
createCanvas(1132, 700); frameRate(10);
strokeWeight(4); stroke(220);
}
function draw() {
background(40);
// draw points
for (i = 0; i < points.length; i++) {
point(points[i].x, points[i].y);
};
console.log(hull);
// draw hull
noFill();
beginShape();
for (i = 0; i < hull.length; i++) {
vertex(hull[i].x, hull[i].y);
};
endShape(CLOSE);
}
function mouseClicked() {
points.push(createVector(mouseX, mouseY));
hull = convexHull(points);
noFill();
//console.log(hull);
beginShape();
for (var i = 0; i < hull.length; i++) {
vertex(hull[i].x, hull[i].y);
}
endShape(CLOSE);
return false;
}
// https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain function convexHull(points) {
points.sort(comparison);
var L = [];
for (var i = 0; i < points.length; i++) {
while (L.length >= 2 && cross(L[L.length - 2], L[L.length - 1], points[i]) <= 0) {
L.pop();
}
L.push(points[i]);
}
var U = [];
for (var i = points.length - 1; i >= 0; i--) {
while (U.length >= 2 && cross(U[U.length - 2], U[U.length - 1], points[i]) <= 0) {
U.pop();
}
U.push(points[i]);
}
L.pop();
U.pop();
return L.concat(U);
}
function comparison(a, b) {
return a.x == b.x ? a.y - b.y : a.x - b.x;
}
function cross(a, b, o) {
return (a.x - o.x) * (b.y - o.y) - (a.y - o.y) * (b.x - o.x);
} </lang>
convexhull.html <lang html> <html>
<head>
<script src="https://cdnjs.cloudflare.com/ajax/libs/p5.js/0.7.2/p5.js"></script> <script src="convexhull.js"></script>
</head>
<body>
Convex Hull |
Left mouse: Add points |
|---|
</body>
</html> </lang>
jq
Works with gojq, the Go implementation of jq <lang jq># ccw returns true if the three points make a counter-clockwise turn def ccw(a; b; c):
a as [$ax, $ay] | b as [$bx, $by] | c as [$cx, $cy] | (($bx - $ax) * ($cy - $ay)) > (($by - $ay) * ($cx - $ax)) ;
def convexHull:
if . == [] then []
else sort as $pts
# lower hull:
| reduce $pts[] as $pt ([];
until (length < 2 or ccw(.[-2]; .[-1]; $pt); .[:-1] )
| . + [$pt] )
# upper hull
| (length + 1) as $t
| reduce range($pts|length-2; -1; -1) as $i (.;
$pts[$i] as $pt
| until (length < $t or ccw(.[-2]; .[-1]; $pt); .[:-1] )
| . + [$pt])
| .[:-1]
end ;</lang>
The task <lang jq>def pts: [
[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]
];
"Convex Hull: \(pts|convexHull)"</lang>
- Output:
Convex Hull: [[-9,-3],[-3,-9],[19,-8],[17,5],[12,17],[5,19],[-3,15]]
Julia
<lang Julia># v1.0.4
using Polyhedra, CDDLib
A = vrep([[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]]) P = polyhedron(A, CDDLib.Library()) Pch = convexhull(P, P) removevredundancy!(Pch) println("$Pch")</lang>
- Output:
convexhull([5.0, 19.0], [19.0, -8.0], [17.0, 5.0], [-3.0, 15.0], [-9.0, -3.0], [12.0, 17.0], [-3.0, -9.0])
Kotlin
<lang scala>// version 1.1.3
class Point(val x: Int, val y: Int) : Comparable<Point> {
override fun compareTo(other: Point) = this.x.compareTo(other.x)
override fun toString() = "($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)]
WARRNING: Wrong results for case:
val points = arrayOf( Point(1387, 2842), Point(1247, 2842), // A Point(1247, 2382), // B (it should be in result) Point(1387, 2462), Point(1247, 2462), Point(1527, 2762), Point(1387, 2382), )
Also if we swap points A and B then result will be different (!). Probably the problem is that when we use p.sort() we sort only by x - but for case when A.x == B.x we should also sort by y (like in e.g. JavaScript version). So below code in class Point method compareTo probably should be used
override fun compareTo(other: Point): Int {
val x = this.x.compareTo(other.x)
if (x==0) {
return this.y.compareTo(other.y)
} else {
return x
}
}
Lua
<lang lua>function print_point(p)
io.write("("..p.x..", "..p.y..")")
return nil
end
function print_points(pl)
io.write("[")
for i,p in pairs(pl) do
if i>1 then
io.write(", ")
end
print_point(p)
end
io.write("]")
return nil
end
function ccw(a,b,c)
return (b.x - a.x) * (c.y - a.y) > (b.y - a.y) * (c.x - a.x)
end
function pop_back(ta)
table.remove(ta,#ta) return ta
end
function convexHull(pl)
if #pl == 0 then
return {}
end
table.sort(pl, function(left,right)
return left.x < right.x
end)
local h = {}
-- lower hull
for i,pt in pairs(pl) do
while #h >= 2 and not ccw(h[#h-1], h[#h], pt) do
table.remove(h,#h)
end
table.insert(h,pt)
end
-- upper hull
local t = #h + 1
for i=#pl, 1, -1 do
local pt = pl[i]
while #h >= t and not ccw(h[#h-1], h[#h], pt) do
table.remove(h,#h)
end
table.insert(h,pt)
end
table.remove(h,#h) return h
end
-- main local points = {
{x=16,y= 3},{x=12,y=17},{x= 0,y= 6},{x=-4,y=-6},{x=16,y= 6},
{x=16,y=-7},{x=16,y=-3},{x=17,y=-4},{x= 5,y=19},{x=19,y=-8},
{x= 3,y=16},{x=12,y=13},{x= 3,y=-4},{x=17,y= 5},{x=-3,y=15},
{x=-3,y=-9},{x= 0,y=11},{x=-9,y=-3},{x=-4,y=-2},{x=12,y=10}
} local hull = convexHull(points)
io.write("Convex Hull: ") print_points(hull) print()</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
Maple
Function are available in the geometry, ComputationalGeometry and simplex packages.
<lang maple>pts:=[[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]]:
with(geometry): map(coordinates,convexhull([seq(point(P||i,pts[i]),i=1..nops(pts))]));
- [[-9, -3], [-3, -9], [19, -8], [17, 5], [12, 17], [5, 19], [-3, 15]]
with(ComputationalGeometry): pts[ConvexHull(pts)];
- [[12, 17], [5, 19], [-3, 15], [-9, -3], [-3, -9], [19, -8], [17, 5]]
simplex:-convexhull(pts);
- [[-9, -3], [-3, -9], [19, -8], [17, 5], [12, 17], [5, 19], [-3, 15]]</lang>
Mathematica/Wolfram Language
<lang Mathematica>hullPoints[data_] := MeshCoordinates[ConvexHullMesh[data]]; hullPoints[{{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:
{{12., 17.}, {5., 19.}, {19., -8.}, {17., 5.}, {-3., 15.}, {-3., -9.}, {-9., -3.}}
Modula-2
<lang modula2>MODULE ConvexHull; FROM FormatString IMPORT FormatString; FROM Storage IMPORT ALLOCATE, DEALLOCATE; FROM SYSTEM IMPORT TSIZE; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE WriteInt(n : INTEGER); VAR buf : ARRAY[0..15] OF CHAR; BEGIN
FormatString("%i", buf, n);
WriteString(buf);
END WriteInt;
TYPE
Point = RECORD
x, y : INTEGER;
END;
PROCEDURE WritePoint(pt : Point); BEGIN
WriteString("(");
WriteInt(pt.x);
WriteString(", ");
WriteInt(pt.y);
WriteString(")");
END WritePoint;
TYPE
NextNode = POINTER TO PNode;
PNode = RECORD
value : Point;
next : NextNode;
END;
PROCEDURE WriteNode(it : NextNode); BEGIN
IF it = NIL THEN
RETURN
END;
WriteString("[");
WritePoint(it^.value); it := it^.next;
WHILE it # NIL DO
WriteString(", ");
WritePoint(it^.value);
it := it^.next
END;
WriteString("]")
END WriteNode;
PROCEDURE AppendNode(pn : NextNode; p : Point) : NextNode; VAR it,nx : NextNode; BEGIN
IF pn = NIL THEN
ALLOCATE(it,TSIZE(PNode));
it^.value := p;
it^.next := NIL;
RETURN it
END;
it := pn;
WHILE it^.next # NIL DO
it := it^.next
END;
ALLOCATE(nx,TSIZE(PNode)); nx^.value := p; nx^.next := NIL;
it^.next := nx; RETURN pn
END AppendNode;
PROCEDURE DeleteNode(VAR pn : NextNode); BEGIN
IF pn = NIL THEN RETURN END; DeleteNode(pn^.next);
DEALLOCATE(pn,TSIZE(PNode)); pn := NIL
END DeleteNode;
PROCEDURE SortNode(VAR pn : NextNode); VAR
it : NextNode; tmp : Point; done : BOOLEAN;
BEGIN
REPEAT
done := TRUE;
it := pn;
WHILE (it # NIL) AND (it^.next # NIL) DO
IF it^.next^.value.x < it^.value.x THEN
tmp := it^.value;
it^.value := it^.next^.value;
it^.next^.value := tmp;
done := FALSE
END;
it := it^.next;
END
UNTIL done;
END SortNode;
PROCEDURE NodeLength(it : NextNode) : INTEGER; VAR length : INTEGER; BEGIN
length := 0;
WHILE it # NIL DO
INC(length);
it := it^.next;
END;
RETURN length
END NodeLength;
PROCEDURE ReverseNode(fp : NextNode) : NextNode; VAR rp,tmp : NextNode; BEGIN
IF fp = NIL THEN RETURN NIL END;
ALLOCATE(tmp,TSIZE(PNode)); tmp^.value := fp^.value; tmp^.next := NIL; rp := tmp; fp := fp^.next;
WHILE fp # NIL DO
ALLOCATE(tmp,TSIZE(PNode));
tmp^.value := fp^.value;
tmp^.next := rp;
rp := tmp;
fp := fp^.next;
END;
RETURN rp
END ReverseNode;
(* ccw returns true if the three points make a counter-clockwise turn *) PROCEDURE CCW(a,b,c : Point) : BOOLEAN; BEGIN
RETURN ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))
END CCW;
PROCEDURE ConvexHull(p : NextNode) : NextNode; VAR
hull,it,h1,h2 : NextNode; t : INTEGER;
BEGIN
IF p = NIL THEN RETURN NIL END; SortNode(p); hull := NIL;
(* lower hull *)
it := p;
WHILE it # NIL DO
IF hull # NIL THEN
WHILE hull^.next # NIL DO
(* At least two points in the list *)
h2 := hull;
h1 := hull^.next;
WHILE h1^.next # NIL DO
h2 := h1;
h1 := h2^.next;
END;
IF CCW(h2^.value, h1^.value, it^.value) THEN
BREAK
ELSE
h2^.next := NIL;
DeleteNode(h1);
h1 := NIL
END
END
END;
hull := AppendNode(hull, it^.value);
it := it^.next;
END;
(* upper hull *)
t := NodeLength(hull) + 1;
p := ReverseNode(p);
it := p;
WHILE it # NIL DO
WHILE NodeLength(hull) >= t DO
h2 := hull;
h1 := hull^.next;
WHILE h1^.next # NIL DO
h2 := h1;
h1 := h2^.next;
END;
IF CCW(h2^.value, h1^.value, it^.value) THEN
BREAK
ELSE
h2^.next := NIL;
DeleteNode(h1);
h1 := NIL
END
END;
hull := AppendNode(hull, it^.value);
it := it^.next;
END;
DeleteNode(p);
h2 := hull;
h1 := h2^.next;
WHILE h1^.next # NIL DO
h2 := h1;
h1 := h1^.next;
END;
h2^.next := NIL;
DeleteNode(h1);
RETURN hull
END ConvexHull;
(* Main *) VAR nodes,hull : NextNode; BEGIN
nodes := AppendNode(NIL, Point{16, 3});
AppendNode(nodes, Point{12,17});
AppendNode(nodes, Point{ 0, 6});
AppendNode(nodes, Point{-4,-6});
AppendNode(nodes, Point{16, 6});
AppendNode(nodes, Point{16,-7});
AppendNode(nodes, Point{16,-3});
AppendNode(nodes, Point{17,-4});
AppendNode(nodes, Point{ 5,19});
AppendNode(nodes, Point{19,-8});
AppendNode(nodes, Point{ 3,16});
AppendNode(nodes, Point{12,13});
AppendNode(nodes, Point{ 3,-4});
AppendNode(nodes, Point{17, 5});
AppendNode(nodes, Point{-3,15});
AppendNode(nodes, Point{-3,-9});
AppendNode(nodes, Point{ 0,11});
AppendNode(nodes, Point{-9,-3});
AppendNode(nodes, Point{-4,-2});
AppendNode(nodes, Point{12,10});
hull := ConvexHull(nodes); WriteNode(hull); DeleteNode(hull);
DeleteNode(nodes); ReadChar
END ConvexHull.</lang>
- Output:
[(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
Nim
<lang nim>type
Point = object x: float y: float
- Calculate orientation for 3 points
- 0 -> Straight line
- 1 -> Clockwise
- 2 -> Counterclockwise
proc orientation(p, q, r: Point): int =
let val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y) if val == 0: 0 elif val > 0: 1 else: 2
proc calculateConvexHull(points: openArray[Point]): seq[Point] =
result = newSeq[Point]()
# There must be at least 3 points
if len(points) < 3:
for i in points: result.add(i)
# Find the leftmost point
var indexMinX = 0
for i, _ in points:
if points[i].x < points[indexMinX].x:
indexMinX = i
var p = indexMinX
var q = 0
while true:
# The leftmost point must be part of the hull.
result.add(points[p])
q = (p + 1) mod len(points)
for i in 0..<len(points):
if orientation(points[p], points[i], points[q]) == 2:
q = i
p = q
# Break from loop once we reach the first point again
if p == indexMinX:
break
var points = @[Point(x: 16, y: 3),
Point(x: 12, y: 17),
Point(x: 0, y: 6),
Point(x: -4, y: -6),
Point(x: 16, y: 6),
Point(x: 16, y: -7),
Point(x: 17, y: -4),
Point(x: 5, y: 19),
Point(x: 19, y: -8),
Point(x: 3, y: 16),
Point(x: 12, y: 13),
Point(x: 3, y: -4),
Point(x: 17, y: 5),
Point(x: -3, y: 15),
Point(x: -3, y: -9),
Point(x: 0, y: 11),
Point(x: -9, y: -3),
Point(x: -4, y: -2),
Point(x: 12, y: 10)]
let hull = calculateConvexHull(points) for i in hull:
echo i</lang>
- Output:
(x: -9.0, y: -3.0) (x: -3.0, y: -9.0) (x: 19.0, y: -8.0) (x: 17.0, y: 5.0) (x: 12.0, y: 17.0) (x: 5.0, y: 19.0) (x: -3.0, y: 15.0)
OCaml
<lang ocaml>(*
* Convex hulls by Andrew's monotone chain algorithm. * * For a description of the algorithm, see * https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169 *)
(*------------------------------------------------------------------*) (* Just enough plane geometry for our purpose. *)
module Plane_point =
struct type t = float * float
let make (xy : t) = xy let to_tuple ((x, y) : t) = (x, y)
let x ((x, _) : t) = x let y ((_, y) : t) = y
let equal (u : t) (v : t) = (x u = x v && y u = y v)
(* Impose a total order on points, making it one that will work
for Andrew's monotone chain algorithm. *)
let order (u : t) (v : t) =
(x u < x v) || (x u = x v && y u < y v)
(* The Array module's sort routines expect a "cmp" function. *)
let cmp u v =
if order u v then
(-1)
else if order v u then
(1)
else
(0)
(* Subtraction is really a vector or multivector operation. *) let sub (u : t) (v : t) = make (x u -. x v, y u -. y v)
(* Cross product is really a multivector operation. *) let cross (u : t) (v : t) = (x u *. y v) -. (y u *. x v)
let to_string ((x, y) : t) =
"(" ^ string_of_float x ^ " " ^ string_of_float y ^ ")"
end
(*------------------------------------------------------------------*) (* We want something akin to array_delete_neighbor_dups of Scheme's
SRFI-132. *)
let array_delete_neighbor_dups equal arr =
(* Returns a Seq.t rather than an array. *)
let rec loop i lst =
(* Cons a list of non-duplicates, going backwards through the
array so the list will be in forwards order. *)
if i = 0 then
arr.(i) :: lst
else if equal arr.(i - 1) arr.(i) then
loop (i - 1) lst
else
loop (i - 1) (arr.(i) :: lst)
in
let n = Array.length arr in
List.to_seq (if n = 0 then [] else loop (n - 1) [])
(*------------------------------------------------------------------*) (* The convex hull algorithm. *)
let cross_test pt_i hull j =
let hull_j = hull.(j)
and hull_j1 = hull.(j - 1) in
0.0 < Plane_point.(cross (sub hull_j hull_j1)
(sub pt_i hull_j1))
let construct_lower_hull n pt =
let hull = Array.make n (Plane_point.make (0.0, 0.0)) in
let () = hull.(0) <- pt.(0)
and () = hull.(1) <- pt.(1) in
let rec outer_loop i j =
if i = n then
j + 1
else
let pt_i = pt.(i) in
let rec inner_loop j =
if j = 0 || cross_test pt_i hull j then
begin
hull.(j + 1) <- pt_i;
j + 1
end
else
inner_loop (j - 1)
in
outer_loop (i + 1) (inner_loop j)
in
let hull_size = outer_loop 2 1 in
(hull_size, hull)
let construct_upper_hull n pt =
let hull = Array.make n (Plane_point.make (0.0, 0.0)) in
let () = hull.(0) <- pt.(n - 1)
and () = hull.(1) <- pt.(n - 2) in
let rec outer_loop i j =
if i = (-1) then
j + 1
else
let pt_i = pt.(i) in
let rec inner_loop j =
if j = 0 || cross_test pt_i hull j then
begin
hull.(j + 1) <- pt_i;
j + 1
end
else
inner_loop (j - 1)
in
outer_loop (i - 1) (inner_loop j)
in
let hull_size = outer_loop (n - 3) 1 in
(hull_size, hull)
let construct_hull n pt =
(* Side note: Construction of the lower and upper hulls can be done
in parallel. *)
let (lower_hull_size, lower_hull) = construct_lower_hull n pt
and (upper_hull_size, upper_hull) = construct_upper_hull n pt in
let hull_size = lower_hull_size + upper_hull_size - 2 in let hull = Array.make hull_size (Plane_point.make (0.0, 0.0)) in
begin
Array.blit lower_hull 0 hull 0 (lower_hull_size - 1);
Array.blit upper_hull 0 hull (lower_hull_size - 1)
(upper_hull_size - 1);
hull
end
let plane_convex_hull points =
(* Takes an arbitrary sequence of points, which may be in any order
and may contain duplicates. Returns an ordered array of points
that make up the convex hull. If the sequence of points is empty,
the returned array is empty. *)
let pt = Array.of_seq points in
let () = Array.sort Plane_point.cmp pt in
let pt = Array.of_seq
(array_delete_neighbor_dups Plane_point.equal pt) in
let n = Array.length pt in
if n <= 2 then
pt
else
construct_hull n pt
(*------------------------------------------------------------------*)
let example_points =
[Plane_point.make (16.0, 3.0); Plane_point.make (12.0, 17.0); Plane_point.make (0.0, 6.0); Plane_point.make (-4.0, -6.0); Plane_point.make (16.0, 6.0); Plane_point.make (16.0, -7.0); Plane_point.make (16.0, -3.0); Plane_point.make (17.0, -4.0); Plane_point.make (5.0, 19.0); Plane_point.make (19.0, -8.0); Plane_point.make (3.0, 16.0); Plane_point.make (12.0, 13.0); Plane_point.make (3.0, -4.0); Plane_point.make (17.0, 5.0); Plane_point.make (-3.0, 15.0); Plane_point.make (-3.0, -9.0); Plane_point.make (0.0, 11.0); Plane_point.make (-9.0, -3.0); Plane_point.make (-4.0, -2.0); Plane_point.make (12.0, 10.0)]
let hull = plane_convex_hull (List.to_seq example_points)
Array.iter
(fun p -> (print_string (Plane_point.to_string p);
print_string " "))
hull;
print_newline ()
(*------------------------------------------------------------------*)</lang>
- Output:
$ ocaml convex_hull_task.ml (-9. -3.) (-3. -9.) (19. -8.) (17. 5.) (12. 17.) (5. 19.) (-3. 15.)
Perl
<lang perl>use strict; use warnings; use feature 'say';
{ package Point; use Class::Struct; struct( x => '$', y => '$',); sub print { '(' . $_->x . ', ' . $_->y . ')' } }
sub ccw {
my($a, $b, $c) = @_; ($b->x - $a->x)*($c->y - $a->y) - ($b->y - $a->y)*($c->x - $a->x);
}
sub tangent {
my($a, $b) = @_; my $opp = $b->x - $a->x; my $adj = $b->y - $a->y; $adj != 0 ? $opp / $adj : 1E99;
}
sub graham_scan {
our @coords; local *coords = shift;
my @sp = sort { $a->y <=> $b->y or $a->x <=> $b->x } map { Point->new( x => $_->[0], y => $_->[1] ) } @coords;
# need at least 3 points to make a hull return @sp if @sp < 3;
# first point on hull is minimum y point my @h = shift @sp;
# re-sort the points by angle, secondary on x (classic Schwartzian)
@sp =
map { $sp[$_->[0]] }
sort { $b->[1] <=> $a->[1] or $a->[2] <=> $b->[2] }
map { [$_, tangent($h[0], $sp[$_]), $sp[$_]->x] }
0..$#sp;
# first point of re-sorted list is guaranteed to be on hull push @h, shift @sp;
# check through the remaining list making sure that there is always a positive angle
for my $point (@sp) {
if (ccw( @h[-2,-1], $point ) >= 0) {
push @h, $point;
} else {
pop @h;
redo;
}
}
@h
}
my @hull_1 = 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]] );
my @hull_2 = 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]] );
my $list = join ' ', map { Point::print($_) } @hull_1; say "Convex Hull (@{[scalar @hull_1]} points): [$list]";
$list = join ' ', map { Point::print($_) } @hull_2;
say "Convex Hull (@{[scalar @hull_2]} points): [$list]"; </lang>
- Output:
Convex Hull (7 points): [(-3, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)] Convex Hull (9 points): [(-3, -9) (1, -9) (14, -9) (19, -8) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]
Phix
You can run this online here.
-- -- demo/rosetta/Convex_hull.exw -- ============================ -- with javascript_semantics constant tests = {{{16, 3}, {12, 17}, { 0, 6}, {-4, -6}, {16, 6}, {16, -7}, {16, -3}, {17, -4}, { 5, 19}, {19, -8}, { 3, 16}, {12, 13}, { 3, -4}, {17, 5}, {-3, 15}, {-3, -9}, { 0, 11}, {-9, -3}, {-4, -2}, {12, 10}}, {{16, 3}, {12, 17}, { 0, 6}, {-4, -6}, {16, 6}, {16, -7}, {16, -3}, {17, -4}, { 5, 19}, {19, -8}, { 3, 16}, {12, 13}, { 3, -4}, {17, 5}, {-3, 15}, {-3, -9}, { 0, 11}, {-9, -3}, {-4, -2}, {12, 10}, {1,-9}, {1,-9}}, {{0,0}, {5,5}, {5,0}, {0,5}}, {{0,0}, {0,1}, {0,2}, {1,0}, {1,1}, {1,2}, {2,0}, {2,1}, {2,2}}, {{-8,-8}, {-8,4}, {-8,16}, { 4,-8}, { 4,4}, { 4,16}, {16,-8}, {16,4}, {16,16}}} integer t = 1 -- idx to tests, for the GTITLE. enum x, y function ccw(sequence a, b, c) return (b[x] - a[x]) * (c[y] - a[y]) > (b[y] - a[y]) * (c[x] - a[x]) end function function convex_hull(sequence points) sequence h = {} points = sort(deep_copy(points)) /* lower hull */ for i=1 to length(points) do while length(h)>=2 and not ccw(h[$-1], h[$], points[i]) do h = h[1..$-1] end while h = append(h, points[i]) end for /* upper hull */ integer t = length(h)+1 for i=length(points) to 1 by -1 do while length(h)>=t and not ccw(h[$-1],h[$],points[i]) do h = h[1..$-1] end while h = append(h, points[i]) end for h = h[1..$-1] return h end function for i=1 to length(tests) do printf(1,"For test set: ") pp(tests[i],{pp_Indent,13,pp_Maxlen,111}) printf(1,"Convex Hull is: ") pp(convex_hull(tests[i])) end for -- plots the test points in red crosses and the convex hull in green circles include pGUI.e include IupGraph.e Ihandle dlg, graph function get_data(Ihandle /*graph*/) IupSetStrAttribute(graph, "GTITLE", "Marks Mode (%d/%d)", {t, length(tests)}) sequence tt = tests[t], {x,y} = columnize(tt), {cx,cy} = columnize(convex_hull(tt)) -- and close the loop: cx &= cx[1] cy &= cy[1] return {{x,y,CD_RED}, {cx,cy,CD_GREEN,"HOLLOW_CIRCLE","MARKLINE"}} end function function key_cb(Ihandle /*ih*/, atom c) if c=K_ESC then return IUP_CLOSE end if -- (standard practice for me) if c=K_F5 then return IUP_DEFAULT end if -- (let browser reload work) if c=K_LEFT then t = iff(t=1?length(tests):t-1) end if if c=K_RIGHT then t = iff(t=length(tests)?1:t+1) end if IupUpdate(graph) return IUP_CONTINUE end function IupOpen() graph = IupGraph(get_data,"RASTERSIZE=640x480,MODE=MARK") dlg = IupDialog(graph,`TITLE="Convex Hull",MINSIZE=270x430`) IupSetAttributes(graph,"XTICK=2,XMIN=-12,XMAX=20,XMARGIN=25,YTICK=2,YMIN=-12,YMAX=20") IupSetInt(graph,"GRIDCOLOR",CD_LIGHT_GREY) IupShow(dlg) IupSetCallback(dlg, "K_ANY", Icallback("key_cb")) IupSetAttribute(graph,`RASTERSIZE`,NULL) if platform()!=JS then IupMainLoop() IupClose() end if
- Output:
For test set: {{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}}
Convex Hull is: {{-9,-3}, {-3,-9}, {19,-8}, {17,5}, {12,17}, {5,19}, {-3,15}}
For test set: {{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}}
Convex Hull is: {{-9,-3}, {-3,-9}, {14,-9}, {19,-8}, {17,5}, {12,17}, {5,19}, {-3,15}}
For test set: {{0,0}, {5,5}, {5,0}, {0,5}}
Convex Hull is: {{0,0}, {5,0}, {5,5}, {0,5}}
For test set: {{0,0}, {0,1}, {0,2}, {1,0}, {1,1}, {1,2}, {2,0}, {2,1}, {2,2}}
Convex Hull is: {{0,0}, {2,0}, {2,2}, {0,2}}
For test set: {{-8,-8}, {-8,4}, {-8,16}, {4,-8}, {4,4}, {4,16}, {16,-8}, {16,4}, {16,16}}
Convex Hull is: {{-8,-8}, {16,-8}, {16,16}, {-8,16}}
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)
Raku
(formerly 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.
<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;
@h.push: $point and next if +@h < 2;
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), (20,-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 (7 points): [(-3, -9) (20, -9) (17, 5) (12, 17) (5, 19) (-3, 15) (-9, -3)]
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
Ruby
<lang ruby>class Point
include Comparable attr :x, :y
def initialize(x, y)
@x = x
@y = y
end
def <=>(other)
x <=> other.x
end
def to_s
"(%d, %d)" % [@x, @y]
end
def to_str
to_s()
end
end
def ccw(a, b, c)
((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x))
end
def convexHull(p)
if p.length == 0 then
return []
end
p = p.sort h = []
# Lower hull
p.each { |pt|
while h.length >= 2 and not ccw(h[-2], h[-1], pt)
h.pop()
end
h << pt
}
# upper hull
t = h.length + 1
p.reverse.each { |pt|
while h.length >= t and not ccw(h[-2], h[-1], pt)
h.pop()
end
h << pt
}
h.pop() h
end
def main
points = [
Point.new(16, 3), Point.new(12, 17), Point.new( 0, 6), Point.new(-4, -6), Point.new(16, 6),
Point.new(16, -7), Point.new(16, -3), Point.new(17, -4), Point.new( 5, 19), Point.new(19, -8),
Point.new( 3, 16), Point.new(12, 13), Point.new( 3, -4), Point.new(17, 5), Point.new(-3, 15),
Point.new(-3, -9), Point.new( 0, 11), Point.new(-9, -3), Point.new(-4, -2), Point.new(12, 10)
]
hull = convexHull(points)
print "Convex Hull: [", hull.join(", "), "]\n"
end
main()</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
Rust
Calculates convex hull from list of points (f32, f32). This can be executed entirely in the Rust Playground. <lang Rust>
- [derive(Debug, Clone)]
struct Point {
x: f32, y: f32
}
fn calculate_convex_hull(points: &Vec<Point>) -> Vec<Point> {
//There must be at least 3 points
if points.len() < 3 { return points.clone(); }
let mut hull = vec![];
//Find the left most point in the polygon
let (left_most_idx, _) = points.iter()
.enumerate()
.min_by(|lhs, rhs| lhs.1.x.partial_cmp(&rhs.1.x).unwrap())
.expect("No left most point");
let mut p = left_most_idx; let mut q = 0_usize;
loop {
//The left most point must be part of the hull
hull.push(points[p].clone());
q = (p + 1) % points.len();
for i in 0..points.len() {
if orientation(&points[p], &points[i], &points[q]) == 2 {
q = i;
}
}
p = q;
//Break from loop once we reach the first point again
if p == left_most_idx { break; }
}
return hull;
}
//Calculate orientation for 3 points //0 -> Straight line //1 -> Clockwise //2 -> Counterclockwise fn orientation(p: &Point, q: &Point, r: &Point) -> usize {
let val = (q.y - p.y) * (r.x - q.x) -
(q.x - p.x) * (r.y - q.y);
if val == 0. { return 0 };
if val > 0. { return 1; } else { return 2; }
}
fn main(){
let points = vec![pt(16,3), pt(12,17), pt(0,6), pt(-4,-6), pt(16,6), pt(16,-7), pt(16,-3), pt(17,-4), pt(5,19), pt(19,-8), pt(3,16), pt(12,13), pt(3,-4), pt(17,5), pt(-3,15), pt(-3,-9), pt(0,11), pt(-9,-3), pt(-4,-2), pt(12,10)];
let hull = calculate_convex_hull(&points);
hull.iter()
.for_each(|pt| println!("{:?}", pt));
}
fn pt(x: i32, y: i32) -> Point {
return Point {x:x as f32, y:y as f32};
} </lang>
- Output:
Point { x: -9.0, y: -3.0 }
Point { x: -3.0, y: -9.0 }
Point { x: 19.0, y: -8.0 }
Point { x: 17.0, y: 5.0 }
Point { x: 12.0, y: 17.0 }
Point { x: 5.0, y: 19.0 }
Point { x: -3.0, y: 15.0 }
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))
Scheme
The program is in R7RS Scheme. For CHICKEN, you need to install the r7rs and srfi-132 eggs.
<lang scheme>(define-library (convex-hulls)
(export vector-convex-hull) (export list-convex-hull)
(import (scheme base)) (import (srfi 132)) ; Sorting.
(begin
;; ;; The implementation is based on Andrew's monotone chain ;; algorithm, and is adapted from a Fortran implementation I wrote ;; myself in 2011. ;; ;; For a description of the algorithm, see ;; https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=4016938 ;;
(define x@ car) (define y@ cadr)
(define (cross u v)
;; Cross product (as a signed scalar).
(- (* (x@ u) (y@ v)) (* (y@ u) (x@ v))))
(define (point- u v)
(list (- (x@ u) (x@ v)) (- (y@ u) (y@ v))))
(define (sort-points-vector points-vector)
;; Ascending sort on x-coordinates, followed by ascending sort
;; on y-coordinates.
(vector-sort (lambda (u v)
(or (< (x@ u) (x@ v))
(and (= (x@ u) (x@ v))
(< (y@ u) (y@ v)))))
points-vector))
(define (construct-lower-hull sorted-points-vector)
(let* ((pt sorted-points-vector)
(n (vector-length pt))
(hull (make-vector n))
(j 1))
(vector-set! hull 0 (vector-ref pt 0))
(vector-set! hull 1 (vector-ref pt 1))
(do ((i 2 (+ i 1)))
((= i n))
(let inner-loop ()
(if (or (zero? j)
(positive?
(cross (point- (vector-ref hull j)
(vector-ref hull (- j 1)))
(point- (vector-ref pt i)
(vector-ref hull (- j 1))))))
(begin
(set! j (+ j 1))
(vector-set! hull j (vector-ref pt i)))
(begin
(set! j (- j 1))
(inner-loop)))))
(values (+ j 1) hull))) ; Hull size, hull points.
(define (construct-upper-hull sorted-points-vector)
(let* ((pt sorted-points-vector)
(n (vector-length pt))
(hull (make-vector n))
(j 1))
(vector-set! hull 0 (vector-ref pt (- n 1)))
(vector-set! hull 1 (vector-ref pt (- n 2)))
(do ((i (- n 3) (- i 1)))
((= i -1))
(let inner-loop ()
(if (or (zero? j)
(positive?
(cross (point- (vector-ref hull j)
(vector-ref hull (- j 1)))
(point- (vector-ref pt i)
(vector-ref hull (- j 1))))))
(begin
(set! j (+ j 1))
(vector-set! hull j (vector-ref pt i)))
(begin
(set! j (- j 1))
(inner-loop)))))
(values (+ j 1) hull))) ; Hull size, hull points.
(define (construct-hull sorted-points-vector)
;; Notice that the lower and upper hulls could be constructed in
;; parallel.
(let-values (((lower-hull-size lower-hull)
(construct-lower-hull sorted-points-vector))
((upper-hull-size upper-hull)
(construct-upper-hull sorted-points-vector)))
(let* ((hull-size (+ lower-hull-size upper-hull-size -2))
(hull (make-vector hull-size)))
(vector-copy! hull 0 lower-hull 0 (- lower-hull-size 1))
(vector-copy! hull (- lower-hull-size 1) upper-hull
0 (- upper-hull-size 1))
hull)))
(define (vector-convex-hull points)
(let* ((points-vector (if (vector? points)
points
(list->vector points)))
(sorted-points-vector
(vector-delete-neighbor-dups
equal?
(sort-points-vector points-vector))))
(if (<= (vector-length sorted-points-vector) 2)
sorted-points-vector
(construct-hull sorted-points-vector))))
(define (list-convex-hull points)
(vector->list (vector-convex-hull points)))
)) ;; end of library convex-hulls.
- A demonstration.
(import (scheme base)) (import (scheme write)) (import (convex-hulls))
(define example-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)))
(write (list-convex-hull example-points)) (newline)</lang>
- Output:
Using Gauche Scheme:
$ gosh convex-hull-task.scm ((-9 -3) (-3 -9) (19 -8) (17 5) (12 17) (5 19) (-3 15))
Compiling to native code with CHICKEN Scheme:
$ csc -O3 -R r7rs convex-hull-task.scm && ./convex-hull-task ((-9 -3) (-3 -9) (19 -8) (17 5) (12 17) (5 19) (-3 15))
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)
Standard ML
<lang sml>(*
* Convex hulls by Andrew's monotone chain algorithm. * * For a description of the algorithm, see * https://en.wikibooks.org/w/index.php?title=Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain&stableid=40169 *)
(*------------------------------------------------------------------*) (*
* Just enough plane geometry for our purpose. *)
signature PLANE_POINT = sig
type planePoint val planePoint : real * real -> planePoint val toTuple : planePoint -> real * real val x : planePoint -> real val y : planePoint -> real val == : planePoint * planePoint -> bool
(* Impose a total order on points, making it one that will work for
Andrew's monotone chain algorithm. *)
val order : planePoint * planePoint -> bool
(* Subtraction is really a vector or multivector operation. *) val subtract : planePoint * planePoint -> planePoint
(* Cross product is really a multivector operation. *) val cross : planePoint * planePoint -> real
val toString : planePoint -> string
end
structure PlanePoint : PLANE_POINT = struct type planePoint = real * real
fun planePoint xy = xy fun toTuple (x, y) = (x, y) fun x (x, _) = x fun y (_, y) = y
fun == ((a : real, b : real), (c : real, d : real)) =
Real.== (a, c) andalso Real.== (b, d)
fun order ((a : real, b : real), (c : real, d : real)) =
Real.< (a, c) orelse (Real.== (a, c) andalso Real.< (b, d))
fun subtract ((a : real, b : real), (c : real, d : real)) =
(a - c, b - d)
fun cross ((a : real, b : real), (c : real, d : real)) =
(a * d) - (b * c)
fun toString (x, y) =
"(" ^ Real.toString x ^ " " ^ Real.toString y ^ ")"
end
(*------------------------------------------------------------------*) (*
* Rather than rely on compiler extensions for sorting, let us write * our own. * * For no special reason, let us use the Shell sort of * https://en.wikipedia.org/w/index.php?title=Shellsort&oldid=1084744510 *)
val ciura_gaps = Array.fromList [701, 301, 132, 57, 23, 10, 4, 1]
fun sort_in_place less arr =
let
open Array
fun span_gap gap =
let
fun iloop i =
if length arr <= i then
()
else
let
val temp = sub (arr, i)
fun jloop j =
if j < gap orelse
less (sub (arr, j - gap), temp) then
update (arr, j, temp)
else
(update (arr, j, sub (arr, j - gap));
jloop (j - gap))
in
jloop i;
iloop (i + gap)
end
in
iloop 0
end
in
app span_gap ciura_gaps
end
(*------------------------------------------------------------------*) (*
* To go with our sort routine, we want something akin to * array_delete_neighbor_dups of Scheme's SRFI-132. *)
fun array_delete_neighbor_dups equal arr =
let
open Array
fun loop i lst =
(* Cons a list of non-duplicates, going backwards through
the array so the list will be in forwards order. *)
if i = 0 then
sub (arr, i) :: lst
else if equal (sub (arr, i - 1), sub (arr, i)) then
loop (i - 1) lst
else
loop (i - 1) (sub (arr, i) :: lst)
val n = length arr
in
fromList (if n = 0 then [] else loop (n - 1) [])
end
(*------------------------------------------------------------------*) (*
* The convex hull algorithm. *)
fun cross_test (pt_i, hull, j) =
let
open PlanePoint
val hull_j = Array.sub (hull, j)
val hull_j1 = Array.sub (hull, j - 1)
in
0.0 < cross (subtract (hull_j, hull_j1),
subtract (pt_i, hull_j1))
end
fun construct_lower_hull (n, pt) =
let
open PlanePoint
val hull = Array.array (n, planePoint (0.0, 0.0))
val () = Array.update (hull, 0, Array.sub (pt, 0))
val () = Array.update (hull, 1, Array.sub (pt, 1))
fun outer_loop i j =
if i = n then
j + 1
else
let
val pt_i = Array.sub (pt, i)
fun inner_loop j =
if j = 0 orelse cross_test (pt_i, hull, j) then
(Array.update (hull, j + 1, pt_i);
j + 1)
else
inner_loop (j - 1)
in
outer_loop (i + 1) (inner_loop j)
end
val hull_size = outer_loop 2 1
in
(hull_size, hull)
end
fun construct_upper_hull (n, pt) =
let
open PlanePoint
val hull = Array.array (n, planePoint (0.0, 0.0))
val () = Array.update (hull, 0, Array.sub (pt, n - 1))
val () = Array.update (hull, 1, Array.sub (pt, n - 2))
fun outer_loop i j =
if i = ~1 then
j + 1
else
let
val pt_i = Array.sub (pt, i)
fun inner_loop j =
if j = 0 orelse cross_test (pt_i, hull, j) then
(Array.update (hull, j + 1, pt_i);
j + 1)
else
inner_loop (j - 1)
in
outer_loop (i - 1) (inner_loop j)
end
val hull_size = outer_loop (n - 3) 1
in
(hull_size, hull)
end
fun construct_hull (n, pt) =
let
(* Side note: Construction of the lower and upper hulls can be
done in parallel. *)
val (lower_hull_size, lower_hull) = construct_lower_hull (n, pt)
and (upper_hull_size, upper_hull) = construct_upper_hull (n, pt)
val hull_size = lower_hull_size + upper_hull_size - 2
val hull =
Array.array (hull_size, PlanePoint.planePoint (0.0, 0.0))
fun copy_lower i =
if i = lower_hull_size - 1 then
()
else
(Array.update (hull, i, Array.sub (lower_hull, i));
copy_lower (i + 1))
fun copy_upper i =
if i = upper_hull_size - 1 then
()
else
(Array.update (hull, i + lower_hull_size - 1,
Array.sub (upper_hull, i));
copy_upper (i + 1))
in
copy_lower 0;
copy_upper 0;
hull
end
fun plane_convex_hull points_lst =
(* Takes an arbitrary list of points, which may be in any order
and may contain duplicates. Returns an ordered array of points
that make up the convex hull. If the list of points is empty,
the returned array is empty. *)
let
val pt = Array.fromList points_lst
val () = sort_in_place PlanePoint.order pt
val pt = array_delete_neighbor_dups (PlanePoint.==) pt
val n = Array.length pt
in
if n <= 2 then
pt
else
construct_hull (n, pt)
end
(*------------------------------------------------------------------*)
fun main () =
let
open PlanePoint
val example_points =
[planePoint (16.0, 3.0),
planePoint (12.0, 17.0),
planePoint (0.0, 6.0),
planePoint (~4.0, ~6.0),
planePoint (16.0, 6.0),
planePoint (16.0, ~7.0),
planePoint (16.0, ~3.0),
planePoint (17.0, ~4.0),
planePoint (5.0, 19.0),
planePoint (19.0, ~8.0),
planePoint (3.0, 16.0),
planePoint (12.0, 13.0),
planePoint (3.0, ~4.0),
planePoint (17.0, 5.0),
planePoint (~3.0, 15.0),
planePoint (~3.0, ~9.0),
planePoint (0.0, 11.0),
planePoint (~9.0, ~3.0),
planePoint (~4.0, ~2.0),
planePoint (12.0, 10.0)]
val hull = plane_convex_hull example_points
in
Array.app (fn p => (print (toString p);
print " "))
hull;
print "\n"
end;
main ();
(*------------------------------------------------------------------*) (* local variables: *) (* mode: sml *) (* sml-indent-level: 2 *) (* sml-indent-args: 2 *) (* end: *)</lang>
- Output:
Output with MLton:
$ mlton convex_hull_task.sml && ./convex_hull_task (~9 ~3) (~3 ~9) (19 ~8) (17 5) (12 17) (5 19) (~3 15)
Output with Poly/ML (yes, the result is printed twice):
$ polyc convex_hull_task.sml && ./a.out (~9.0 ~3.0) (~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) (~3.0 ~9.0) (19.0 ~8.0) (17.0 5.0) (12.0 17.0) (5.0 19.0) (~3.0 15.0)
When you use Poly/ML, if you want the result printed only once, comment out the call to main() near the bottom of the source file. The call is redundant, because a program compiled with Poly/ML automatically calls main().
Swift
<lang swift>public struct Point: Equatable, Hashable {
public var x: Double public var y: Double
public init(fromTuple t: (Double, Double)) {
self.x = t.0
self.y = t.1
}
}
public func calculateConvexHull(fromPoints points: [Point]) -> [Point] {
guard points.count >= 3 else {
return points
}
var hull = [Point]()
let (leftPointIdx, _) = points.enumerated().min(by: { $0.element.x < $1.element.x })!
var p = leftPointIdx var q = 0
repeat {
hull.append(points[p])
q = (p + 1) % points.count
for i in 0..<points.count where calculateOrientation(points[p], points[i], points[q]) == .counterClockwise {
q = i
}
p = q } while p != leftPointIdx
return hull
}
private func calculateOrientation(_ p: Point, _ q: Point, _ r: Point) -> Orientation {
let val = (q.y - p.y) * (r.x - q.x) - (q.x - p.x) * (r.y - q.y)
if val == 0 {
return .straight
} else if val > 0 {
return .clockwise
} else {
return .counterClockwise
}
}
private enum Orientation {
case straight, clockwise, counterClockwise
}
let 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)
].map(Point.init(fromTuple:))
print("Input: \(points)") print("Output: \(calculateConvexHull(fromPoints: points))")</lang>
- Output:
Input: [Point(x: 16.0, y: 3.0), Point(x: 12.0, y: 17.0), Point(x: 0.0, y: 6.0), Point(x: -4.0, y: -6.0), Point(x: 16.0, y: 6.0), Point(x: 16.0, y: -7.0), Point(x: 16.0, y: -3.0), Point(x: 17.0, y: -4.0), Point(x: 5.0, y: 19.0), Point(x: 19.0, y: -8.0), Point(x: 3.0, y: 16.0), Point(x: 12.0, y: 13.0), Point(x: 3.0, y: -4.0), Point(x: 17.0, y: 5.0), Point(x: -3.0, y: 15.0), Point(x: -3.0, y: -9.0), Point(x: 0.0, y: 11.0), Point(x: -9.0, y: -3.0), Point(x: -4.0, y: -2.0), Point(x: 12.0, y: 10.0)] Output: [Point(x: -9.0, y: -3.0), Point(x: -3.0, y: -9.0), Point(x: 19.0, y: -8.0), Point(x: 17.0, y: 5.0), Point(x: 12.0, y: 17.0), Point(x: 5.0, y: 19.0), Point(x: -3.0, y: 15.0)]
Tcl
Translation of: https://en.wikibooks.org/wiki/Algorithm_Implementation/Geometry/Convex_hull/Monotone_chain#Python
<lang tcl>catch {namespace delete test_convex_hull} ;# Start with a clean namespace
namespace eval test_convex_hull {
package require Tcl 8.5 ;# maybe earlier?
interp alias {} @ {} lindex;# An essential readability helper for list indexing
proc cross {o a b} {
### 2D cross product of OA and OB vectors ###
return [expr {([@ $a 0] - [@ $o 0]) * ([@ $b 1] - [@ $o 1]) - ([@ $a 1] - [@ $o 1]) * ([@ $b 0] - [@ $o 0]) }]
}
proc calc_hull_edge {points} {
### Build up hull edge ###
set edge [list]
foreach p $points {
while {[llength $edge ] >= 2 && [cross [@ $edge end-1] [@ $edge end] $p] <= 0} {
set edge [lreplace $edge end end] ;# pop
}
lappend edge $p
}
return $edge
}
proc convex_hull {points} {
### Convex hull of a set of 2D points ###
# Unique points
set points [lsort -unique $points]
# Sorted points
set points [lsort -real -index 0 [lsort -real -index 1 $points]]
# No calculation necessary
if {[llength $points] <= 1} {
return $points
}
set lower [calc_hull_edge $points]
set upper [calc_hull_edge [lreverse $points]]
return [concat [lrange $lower 0 end-1] [lrange $upper 0 end-1]] }
# Testing
set tpoints {{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}}
puts "Test points:" puts [lrange $tpoints 0 end] ;# prettier puts "Convex Hull:" puts [convex_hull $tpoints]
} </lang>
- Output:
Test 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}
Convex Hull:
{-9 -3} {-3 -9} {19 -8} {17 5} {12 17} {5 19} {-3 15}
Visual Basic .NET
<lang vbnet>Imports ConvexHull
Module Module1
Class Point : Implements IComparable(Of Point)
Public Property X As Integer
Public Property Y As Integer
Public Sub New(x As Integer, y As Integer)
Me.X = x
Me.Y = y
End Sub
Public Function CompareTo(other As Point) As Integer Implements IComparable(Of Point).CompareTo
Return X.CompareTo(other.X)
End Function
Public Overrides Function ToString() As String
Return String.Format("({0}, {1})", X, Y)
End Function
End Class
Function ConvexHull(p As List(Of Point)) As List(Of Point)
If p.Count = 0 Then
Return New List(Of Point)
End If
p.Sort()
Dim h As New List(Of Point)
' Lower hull
For Each pt In p
While h.Count >= 2 AndAlso Not Ccw(h(h.Count - 2), h(h.Count - 1), pt)
h.RemoveAt(h.Count - 1)
End While
h.Add(pt)
Next
' Upper hull
Dim t = h.Count + 1
For i = p.Count - 1 To 0 Step -1
Dim pt = p(i)
While h.Count >= t AndAlso Not Ccw(h(h.Count - 2), h(h.Count - 1), pt)
h.RemoveAt(h.Count - 1)
End While
h.Add(pt)
Next
h.RemoveAt(h.Count - 1)
Return h
End Function
Function Ccw(a As Point, b As Point, c As Point) As Boolean
Return ((b.X - a.X) * (c.Y - a.Y)) > ((b.Y - a.Y) * (c.X - a.X))
End Function
Sub Main()
Dim points As New List(Of Point) From {
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)
}
Dim hull = ConvexHull(points)
Dim it = hull.GetEnumerator()
Console.Write("Convex Hull: [")
If it.MoveNext() Then
Console.Write(it.Current)
End If
While it.MoveNext()
Console.Write(", {0}", it.Current)
End While
Console.WriteLine("]")
End Sub
End Module</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
Wren
<lang ecmascript>import "/sort" for Sort import "/trait" for Comparable, Stepped
class Point is Comparable {
construct new(x, y) {
_x = x
_y = y
}
x { _x }
y { _y }
compare(other) { (_x != other.x) ? (_x - other.x).sign : (_y - other.y).sign }
toString { "(%(_x), %(_y))" }
}
/* ccw returns true if the three points make a counter-clockwise turn */ var ccw = Fn.new { |a, b, c| ((b.x - a.x) * (c.y - a.y)) > ((b.y - a.y) * (c.x - a.x)) }
var convexHull = Fn.new { |pts|
if (pts.isEmpty) return [] Sort.quick(pts) var h = []
// lower hull
for (pt in pts) {
while (h.count >= 2 && !ccw.call(h[-2], h[-1], pt)) h.removeAt(-1)
h.add(pt)
}
// upper hull
var t = h.count + 1
for (i in Stepped.descend(pts.count-2..0, 1)) {
var pt = pts[i]
while (h.count >= t && !ccw.call(h[-2], h[-1], pt)) h.removeAt(-1)
h.add(pt)
}
h.removeAt(-1) return h
}
var pts = [
Point.new(16, 3), Point.new(12, 17), Point.new( 0, 6), Point.new(-4, -6), Point.new(16, 6), Point.new(16, -7), Point.new(16, -3), Point.new(17, -4), Point.new( 5, 19), Point.new(19, -8), Point.new( 3, 16), Point.new(12, 13), Point.new( 3, -4), Point.new(17, 5), Point.new(-3, 15), Point.new(-3, -9), Point.new( 0, 11), Point.new(-9, -3), Point.new(-4, -2), Point.new(12, 10)
] System.print("Convex Hull: %(convexHull.call(pts))")</lang>
- Output:
Convex Hull: [(-9, -3), (-3, -9), (19, -8), (17, 5), (12, 17), (5, 19), (-3, 15)]
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))
- Programming Tasks
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