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[[Category:Geometry]] [[Category:Recursion]]

{{task}} The '''Ramer–Douglas–Peucker''' algorithm is a line simplification algorithm for reducing the number of points used to define its shape.

;Task: Using the '''Ramer–Douglas–Peucker''' algorithm, simplify the 2D line defined by the points: (0,0) (1,0.1) (2,-0.1) (3,5) (4,6) (5,7) (6,8.1) (7,9) (8,9) (9,9)

The error threshold to be used is: '''1.0'''.

Display the remaining points here.

;Reference: :* the Wikipedia article: [https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm Ramer-Douglas-Peucker algorithm].

## C++

```#include <iostream>
#include <cmath>
#include <utility>
#include <vector>
#include <stdexcept>
using namespace std;

typedef std::pair<double, double> Point;

double PerpendicularDistance(const Point &pt, const Point &lineStart, const Point &lineEnd)
{
double dx = lineEnd.first - lineStart.first;
double dy = lineEnd.second - lineStart.second;

//Normalise
double mag = pow(pow(dx,2.0)+pow(dy,2.0),0.5);
if(mag > 0.0)
{
dx /= mag; dy /= mag;
}

double pvx = pt.first - lineStart.first;
double pvy = pt.second - lineStart.second;

//Get dot product (project pv onto normalized direction)
double pvdot = dx * pvx + dy * pvy;

//Scale line direction vector
double dsx = pvdot * dx;
double dsy = pvdot * dy;

//Subtract this from pv
double ax = pvx - dsx;
double ay = pvy - dsy;

return pow(pow(ax,2.0)+pow(ay,2.0),0.5);
}

void RamerDouglasPeucker(const vector<Point> &pointList, double epsilon, vector<Point> &out)
{
if(pointList.size()<2)
throw invalid_argument("Not enough points to simplify");

// Find the point with the maximum distance from line between start and end
double dmax = 0.0;
size_t index = 0;
size_t end = pointList.size()-1;
for(size_t i = 1; i < end; i++)
{
double d = PerpendicularDistance(pointList[i], pointList[0], pointList[end]);
if (d > dmax)
{
index = i;
dmax = d;
}
}

// If max distance is greater than epsilon, recursively simplify
if(dmax > epsilon)
{
// Recursive call
vector<Point> recResults1;
vector<Point> recResults2;
vector<Point> firstLine(pointList.begin(), pointList.begin()+index+1);
vector<Point> lastLine(pointList.begin()+index, pointList.end());
RamerDouglasPeucker(firstLine, epsilon, recResults1);
RamerDouglasPeucker(lastLine, epsilon, recResults2);

// Build the result list
out.assign(recResults1.begin(), recResults1.end()-1);
out.insert(out.end(), recResults2.begin(), recResults2.end());
if(out.size()<2)
throw runtime_error("Problem assembling output");
}
else
{
//Just return start and end points
out.clear();
out.push_back(pointList[0]);
out.push_back(pointList[end]);
}
}

int main()
{
vector<Point> pointList;
vector<Point> pointListOut;

pointList.push_back(Point(0.0, 0.0));
pointList.push_back(Point(1.0, 0.1));
pointList.push_back(Point(2.0, -0.1));
pointList.push_back(Point(3.0, 5.0));
pointList.push_back(Point(4.0, 6.0));
pointList.push_back(Point(5.0, 7.0));
pointList.push_back(Point(6.0, 8.1));
pointList.push_back(Point(7.0, 9.0));
pointList.push_back(Point(8.0, 9.0));
pointList.push_back(Point(9.0, 9.0));

RamerDouglasPeucker(pointList, 1.0, pointListOut);

cout << "result" << endl;
for(size_t i=0;i< pointListOut.size();i++)
{
cout << pointListOut[i].first << "," << pointListOut[i].second << endl;
}

return 0;
}
```

{{out}}

```result
0,0
2,-0.1
3,5
7,9
9,9
```

## C#

{{trans|Java}}

```using System;
using System.Collections.Generic;
using System.Linq;

namespace LineSimplification {
using Point = Tuple<double, double>;

class Program {
static double PerpendicularDistance(Point pt, Point lineStart, Point lineEnd) {
double dx = lineEnd.Item1 - lineStart.Item1;
double dy = lineEnd.Item2 - lineStart.Item2;

// Normalize
double mag = Math.Sqrt(dx * dx + dy * dy);
if (mag > 0.0) {
dx /= mag;
dy /= mag;
}
double pvx = pt.Item1 - lineStart.Item1;
double pvy = pt.Item2 - lineStart.Item2;

// Get dot product (project pv onto normalized direction)
double pvdot = dx * pvx + dy * pvy;

// Scale line direction vector and subtract it from pv
double ax = pvx - pvdot * dx;
double ay = pvy - pvdot * dy;

return Math.Sqrt(ax * ax + ay * ay);
}

static void RamerDouglasPeucker(List<Point> pointList, double epsilon, List<Point> output) {
if (pointList.Count < 2) {
throw new ArgumentOutOfRangeException("Not enough points to simplify");
}

// Find the point with the maximum distance from line between the start and end
double dmax = 0.0;
int index = 0;
int end = pointList.Count - 1;
for (int i = 1; i < end; ++i) {
double d = PerpendicularDistance(pointList[i], pointList[0], pointList[end]);
if (d > dmax) {
index = i;
dmax = d;
}
}

// If max distance is greater than epsilon, recursively simplify
if (dmax > epsilon) {
List<Point> recResults1 = new List<Point>();
List<Point> recResults2 = new List<Point>();
List<Point> firstLine = pointList.Take(index + 1).ToList();
List<Point> lastLine = pointList.Skip(index).ToList();
RamerDouglasPeucker(firstLine, epsilon, recResults1);
RamerDouglasPeucker(lastLine, epsilon, recResults2);

// build the result list
if (output.Count < 2) throw new Exception("Problem assembling output");
}
else {
// Just return start and end points
output.Clear();
}
}

static void Main(string[] args) {
List<Point> pointList = new List<Point>() {
new Point(0.0,0.0),
new Point(1.0,0.1),
new Point(2.0,-0.1),
new Point(3.0,5.0),
new Point(4.0,6.0),
new Point(5.0,7.0),
new Point(6.0,8.1),
new Point(7.0,9.0),
new Point(8.0,9.0),
new Point(9.0,9.0),
};
List<Point> pointListOut = new List<Point>();
RamerDouglasPeucker(pointList, 1.0, pointListOut);
Console.WriteLine("Points remaining after simplification:");
pointListOut.ForEach(p => Console.WriteLine(p));
}
}
}
```

{{out}}

```Points remaining after simplification:
(0, 0)
(2, -0.1)
(3, 5)
(7, 9)
(9, 9)
```

## D

{{trans|C++}}

```import std.algorithm;
import std.exception : enforce;
import std.math;
import std.stdio;

void main() {
creal[] pointList = [
0.0 +  0.0i,
1.0 +  0.1i,
2.0 + -0.1i,
3.0 +  5.0i,
4.0 +  6.0i,
5.0 +  7.0i,
6.0 +  8.1i,
7.0 +  9.0i,
8.0 +  9.0i,
9.0 +  9.0i
];
creal[] pointListOut;

ramerDouglasPeucker(pointList, 1.0, pointListOut);

writeln("result");
for (size_t i=0; i< pointListOut.length; i++) {
writeln(pointListOut[i].re, ",", pointListOut[i].im);
}
}

real perpendicularDistance(const creal pt, const creal lineStart, const creal lineEnd) {
creal d = lineEnd - lineStart;

//Normalise
real mag =  hypot(d.re, d.im);
if (mag > 0.0) {
d /= mag;
}

creal pv = pt - lineStart;

//Get dot product (project pv onto normalized direction)
real pvdot = d.re * pv.re + d.im * pv.im;

//Scale line direction vector
creal ds = pvdot * d;

//Subtract this from pv
creal a = pv - ds;

return hypot(a.re, a.im);
}

void ramerDouglasPeucker(const creal[] pointList, real epsilon, ref creal[] output) {
enforce(pointList.length >= 2, "Not enough points to simplify");

// Find the point with the maximum distance from line between start and end
real dmax = 0.0;
size_t index = 0;
size_t end = pointList.length-1;
for (size_t i=1; i<end; i++) {
real d = perpendicularDistance(pointList[i], pointList[0], pointList[end]);
if (d > dmax) {
index = i;
dmax = d;
}
}

// If max distance is greater than epsilon, recursively simplify
if (dmax > epsilon) {
// Recursive call
creal[] firstLine = pointList[0..index+1].dup;
creal[] lastLine = pointList[index+1..\$].dup;

creal[] recResults1;
ramerDouglasPeucker(firstLine, epsilon, recResults1);

creal[] recResults2;
ramerDouglasPeucker(lastLine, epsilon, recResults2);

// Build the result list
output = recResults1 ~ recResults2;

enforce(output.length>=2, "Problem assembling output");
} else {
//Just return start and end points
output.length = 0;
output ~= pointList[0];
output ~= pointList[end];
}
}
```

{{out}}

```result
0,0
2,-0.1
3,5
7,9
9,9
```

## Go

```package main

import (
"fmt"
"math"
)

type point struct{ x, y float64 }

func RDP(l []point, ε float64) []point {
x := 0
dMax := -1.
last := len(l) - 1
p1 := l[0]
p2 := l[last]
x21 := p2.x - p1.x
y21 := p2.y - p1.y
for i, p := range l[1:last] {
if d := math.Abs(y21*p.x - x21*p.y + p2.x*p1.y - p2.y*p1.x); d > dMax {
x = i + 1
dMax = d
}
}
if dMax > ε {
return append(RDP(l[:x+1], ε), RDP(l[x:], ε)[1:]...)
}
return []point{l[0], l[len(l)-1]}
}

func main() {
fmt.Println(RDP([]point{{0, 0}, {1, 0.1}, {2, -0.1},
{3, 5}, {4, 6}, {5, 7}, {6, 8.1}, {7, 9}, {8, 9}, {9, 9}}, 1))
}
```

{{out}}

```
[{0 0} {2 -0.1} {3 5} {7 9} {9 9}]

```

## J

'''Solution:'''

```mp=: +/ .*           NB. matrix product
norm=: +/&.:*:       NB. vector norm
normalize=: (% norm)^:(0 < norm)

dxy=. normalize@({: - {.)
pv=. -"1 {.
NB.*perpDist v Calculate perpendicular distance of points from a line
perpDist=: norm"1@(pv ([ -"1 mp"1~ */ ]) dxy) f.

rdp=: verb define
1 rdp y
:
points=. ,:^:(2 > #@\$) y
epsilon=. x
if. 2 > # points do. points return. end.

NB. point with the maximum distance from line between start and end
'imax dmax'=. ((i. , ]) >./) perpDist points
if. dmax > epsilon do.
epsilon ((}:@rdp (1+imax)&{.) , (rdp imax&}.)) points
else.
({. ,: {:) points
end.
)
```

'''Example Usage:'''

```   Points=: 0 0,1 0.1,2 _0.1,3 5,4 6,5 7,6 8.1,7 9,8 9,:9 9
1.0 rdp Points
0    0
2 _0.1
3    5
7    9
9    9
```

## Java

{{trans|Kotlin}} {{works with|Java|9}}

```import javafx.util.Pair;

import java.util.ArrayList;
import java.util.List;

public class LineSimplification {
private static class Point extends Pair<Double, Double> {
Point(Double key, Double value) {
super(key, value);
}

@Override
public String toString() {
return String.format("(%f, %f)", getKey(), getValue());
}
}

private static double perpendicularDistance(Point pt, Point lineStart, Point lineEnd) {
double dx = lineEnd.getKey() - lineStart.getKey();
double dy = lineEnd.getValue() - lineStart.getValue();

// Normalize
double mag = Math.hypot(dx, dy);
if (mag > 0.0) {
dx /= mag;
dy /= mag;
}
double pvx = pt.getKey() - lineStart.getKey();
double pvy = pt.getValue() - lineStart.getValue();

// Get dot product (project pv onto normalized direction)
double pvdot = dx * pvx + dy * pvy;

// Scale line direction vector and subtract it from pv
double ax = pvx - pvdot * dx;
double ay = pvy - pvdot * dy;

return Math.hypot(ax, ay);
}

private static void ramerDouglasPeucker(List<Point> pointList, double epsilon, List<Point> out) {
if (pointList.size() < 2) throw new IllegalArgumentException("Not enough points to simplify");

// Find the point with the maximum distance from line between the start and end
double dmax = 0.0;
int index = 0;
int end = pointList.size() - 1;
for (int i = 1; i < end; ++i) {
double d = perpendicularDistance(pointList.get(i), pointList.get(0), pointList.get(end));
if (d > dmax) {
index = i;
dmax = d;
}
}

// If max distance is greater than epsilon, recursively simplify
if (dmax > epsilon) {
List<Point> recResults1 = new ArrayList<>();
List<Point> recResults2 = new ArrayList<>();
List<Point> firstLine = pointList.subList(0, index + 1);
List<Point> lastLine = pointList.subList(index, pointList.size());
ramerDouglasPeucker(firstLine, epsilon, recResults1);
ramerDouglasPeucker(lastLine, epsilon, recResults2);

// build the result list
out.addAll(recResults1.subList(0, recResults1.size() - 1));
if (out.size() < 2) throw new RuntimeException("Problem assembling output");
} else {
// Just return start and end points
out.clear();
}
}

public static void main(String[] args) {
List<Point> pointList = List.of(
new Point(0.0, 0.0),
new Point(1.0, 0.1),
new Point(2.0, -0.1),
new Point(3.0, 5.0),
new Point(4.0, 6.0),
new Point(5.0, 7.0),
new Point(6.0, 8.1),
new Point(7.0, 9.0),
new Point(8.0, 9.0),
new Point(9.0, 9.0)
);
List<Point> pointListOut = new ArrayList<>();
ramerDouglasPeucker(pointList, 1.0, pointListOut);
System.out.println("Points remaining after simplification:");
pointListOut.forEach(System.out::println);
}
}
```

{{out}}

```Points remaining after simplification:
(0.000000, 0.000000)
(2.000000, -0.100000)
(3.000000, 5.000000)
(7.000000, 9.000000)
(9.000000, 9.000000)
```

## JavaScript

{{trans|Go}}

```/**
* @typedef {{
*    x: (!number),
*    y: (!number)
* }}
*/
let pointType;

/**
* @param {!Array<pointType>} l
* @param {number} eps
*/
const RDP = (l, eps) => {
const last = l.length - 1;
const p1 = l[0];
const p2 = l[last];
const x21 = p2.x - p1.x;
const y21 = p2.y - p1.y;

const [dMax, x] = l.slice(1, last)
.map(p => Math.abs(y21 * p.x - x21 * p.y + p2.x * p1.y - p2.y * p1.x))
.reduce((p, c, i) => {
const v = Math.max(p[0], c);
return [v, v === p[0] ? p[1] : i + 1];
}, [-1, 0]);

if (dMax > eps) {
return [...RDP(l.slice(0, x + 1), eps), ...RDP(l.slice(x), eps).slice(1)];
}
return [l[0], l[last]]
};

const points = [
{x: 0, y: 0},
{x: 1, y: 0.1},
{x: 2, y: -0.1},
{x: 3, y: 5},
{x: 4, y: 6},
{x: 5, y: 7},
{x: 6, y: 8.1},
{x: 7, y: 9},
{x: 8, y: 9},
{x: 9, y: 9}];

console.log(RDP(points, 1));
```

{{out}}

```[{x: 0, y: 0},
{x: 2, y: -0.1},
{x: 3, y: 5},
{x: 7, y: 9},
{x: 9, y: 9}]
```

## Julia

{{works with|Julia|0.6}} {{trans|C++}}

```const Point = Vector{Float64}

function perpdist(pt::Point, lnstart::Point, lnend::Point)
d = normalize!(lnend .- lnstart)

pv = pt .- lnstart
# Get dot product (project pv onto normalized direction)
pvdot = dot(d, pv)
# Scale line direction vector
ds = pvdot .* d
# Subtract this from pv
return norm(pv .- ds)
end

function rdp(plist::Vector{Point}, ϵ::Float64 = 1.0)
if length(plist) < 2
throw(ArgumentError("not enough points to simplify"))
end

# Find the point with the maximum distance from line between start and end
distances  = collect(perpdist(pt, plist[1], plist[end]) for pt in plist)
dmax, imax = findmax(distances)

# If max distance is greater than epsilon, recursively simplify
if dmax > ϵ
fstline = plist[1:imax]
lstline = plist[imax:end]

recrst1 = rdp(fstline, ϵ)
recrst2 = rdp(lstline, ϵ)

out = vcat(recrst1, recrst2)
else
out = [plist[1], plist[end]]
end

return out
end

plist = Point[[0.0, 0.0], [1.0, 0.1], [2.0, -0.1], [3.0, 5.0], [4.0, 6.0], [5.0, 7.0], [6.0, 8.1], [7.0, 9.0], [8.0, 9.0], [9.0, 9.0]]
@show plist
@show rdp(plist)
```

{{out}}

```plist = Array{Float64,1}[[0.0, 0.0], [1.0, 0.1], [2.0, -0.1], [3.0, 5.0], [4.0, 6.0], [5.0, 7.0], [6.0, 8.1], [7.0, 9.0], [8.0, 9.0], [9.0, 9.0]]
rdp(plist) = Array{Float64,1}[[0.0, 0.0], [2.0, -0.1], [2.0, -0.1], [3.0, 5.0], [3.0, 5.0], [7.0, 9.0], [7.0, 9.0], [9.0, 9.0]]
```

## Kotlin

{{trans|C++}}

```// version 1.1.0

typealias Point = Pair<Double, Double>

fun perpendicularDistance(pt: Point, lineStart: Point, lineEnd: Point): Double {
var dx = lineEnd.first - lineStart.first
var dy = lineEnd.second - lineStart.second

// Normalize
val mag = Math.hypot(dx, dy)
if (mag > 0.0) { dx /= mag; dy /= mag }
val pvx = pt.first - lineStart.first
val pvy = pt.second - lineStart.second

// Get dot product (project pv onto normalized direction)
val pvdot = dx * pvx + dy * pvy

// Scale line direction vector and substract it from pv
val ax = pvx - pvdot * dx
val ay = pvy - pvdot * dy

return Math.hypot(ax, ay)
}

fun RamerDouglasPeucker(pointList: List<Point>, epsilon: Double, out: MutableList<Point>) {
if (pointList.size < 2) throw IllegalArgumentException("Not enough points to simplify")

// Find the point with the maximum distance from line between start and end
var dmax = 0.0
var index = 0
val end = pointList.size - 1
for (i in 1 until end) {
val d = perpendicularDistance(pointList[i], pointList[0], pointList[end])
if (d > dmax) { index = i; dmax = d }
}

// If max distance is greater than epsilon, recursively simplify
if (dmax > epsilon) {
val recResults1 = mutableListOf<Point>()
val recResults2 = mutableListOf<Point>()
val firstLine = pointList.take(index + 1)
val lastLine  = pointList.drop(index)
RamerDouglasPeucker(firstLine, epsilon, recResults1)
RamerDouglasPeucker(lastLine, epsilon, recResults2)

// build the result list
if (out.size < 2) throw RuntimeException("Problem assembling output")
}
else {
// Just return start and end points
out.clear()
}
}

fun main(args: Array<String>) {
val pointList = listOf(
Point(0.0, 0.0),
Point(1.0, 0.1),
Point(2.0, -0.1),
Point(3.0, 5.0),
Point(4.0, 6.0),
Point(5.0, 7.0),
Point(6.0, 8.1),
Point(7.0, 9.0),
Point(8.0, 9.0),
Point(9.0, 9.0)
)
val pointListOut = mutableListOf<Point>()
RamerDouglasPeucker(pointList, 1.0, pointListOut)
println("Points remaining after simplification:")
for (p in pointListOut) println(p)
}
```

{{out}}

```
Points remaining after simplification:
(0.0, 0.0)
(2.0, -0.1)
(3.0, 5.0)
(7.0, 9.0)
(9.0, 9.0)

```

## Nim

```import math

type
Point = tuple[x, y: float64]

proc pointLineDistance(pt, lineStart, lineEnd: Point): float64 =
var n, d, dx, dy: float64
dx = lineEnd.x - lineStart.x
dy = lineEnd.y - lineStart.y
n = abs(dx * (lineStart.y - pt.y) - (lineStart.x - pt.x) * dy)
d = sqrt(dx * dx + dy * dy)
n / d

proc rdp(points: seq[Point], startIndex, lastIndex: int, ε: float64 = 1.0): seq[Point] =
var dmax = 0.0
var index = startIndex

for i in index+1..<lastIndex:
var d = pointLineDistance(points[i], points[startIndex], points[lastIndex])
if d > dmax:
index = i
dmax = d

if dmax > ε:
var res1 = rdp(points, startIndex, index, ε)
var res2 = rdp(points, index, lastIndex, ε)

var finalRes: seq[Point] = @[]

result = finalRes
else:
result = @[points[startIndex], points[lastIndex]]

var line: seq[Point] = @[(0.0, 0.0), (1.0, 0.1), (2.0, -0.1), (3.0, 5.0), (4.0, 6.0),
(5.0, 7.0), (6.0, 8.1), (7.0,  9.0), (8.0, 9.0), (9.0, 9.0)]
echo rdp(line, line.low, line.high)
```

{{out}}

```@[(x: 0.0, y: 0.0), (x: 2.0, y: -0.1), (x: 3.0, y: 5.0), (x: 7.0, y: 9.0), (x: 9.0, y: 9.0)]
```

## Perl

{{trans|Perl 6}}

```use strict;
use warnings;
use feature 'say';
use List::MoreUtils qw(firstidx minmax);

my \$epsilon = 1;

sub norm {
my(@list) = @_;
my \$sum;
\$sum += \$_**2 for @list;
sqrt(\$sum)
}

sub perpendicular_distance {
our(@start,@end,@point);
local(*start,*end,*point) = (shift, shift, shift);
return 0 if \$start[0]==\$point[0] && \$start[1]==\$point[1]
or   \$end[0]==\$point[0] &&   \$end[1]==\$point[1];
my ( \$dx,  \$dy)  = (  \$end[0]-\$start[0],  \$end[1]-\$start[1]);
my (\$dpx, \$dpy)  = (\$point[0]-\$start[0],\$point[1]-\$start[1]);
my \$t = norm(\$dx, \$dy);
\$dx /= \$t;
\$dy /= \$t;
norm(\$dpx - \$dx*(\$dx*\$dpx + \$dy*\$dpy), \$dpy - \$dy*(\$dx*\$dpx + \$dy*\$dpy));
}

sub Ramer_Douglas_Peucker {
my(@points) = @_;
return @points if @points == 2;
my @d;
push @d, perpendicular_distance(@points[0, -1, \$_]) for 0..@points-1;
my(undef,\$dmax) = minmax @d;
my \$index = firstidx { \$_ == \$dmax } @d;
if (\$dmax > \$epsilon) {
my @lo = Ramer_Douglas_Peucker( @points[0..\$index]);
my @hi = Ramer_Douglas_Peucker( @points[\$index..\$#points]);
return  @lo[0..@lo-2], @hi;
}
@points[0, -1];
}

say '(' . join(' ', @\$_) . ') '
for Ramer_Douglas_Peucker( [0,0],[1,0.1],[2,-0.1],[3,5],[4,6],[5,7],[6,8.1],[7,9],[8,9],[9,9] )
```

{{out}}

```(0 0)
(2 -0.1)
(3 5)
(7 9)
(9 9)
```

## Perl 6

{{works with|Rakudo|2017.05}} {{trans|C++}}

```sub norm (*@list) { @list»².sum.sqrt }

sub perpendicular-distance (@start, @end where @end !eqv @start, @point) {
return 0 if @point eqv any(@start, @end);
my ( \$Δx, \$Δy ) =   @end «-» @start;
my (\$Δpx, \$Δpy) = @point «-» @start;
(\$Δx, \$Δy) «/=» norm \$Δx, \$Δy;
norm (\$Δpx, \$Δpy) «-» (\$Δx, \$Δy) «*» (\$Δx*\$Δpx + \$Δy*\$Δpy);
}

sub Ramer-Douglas-Peucker(@points where * > 1, \ε = 1) {
return @points if @points == 2;
my @d = (^@points).map: { perpendicular-distance |@points[0, *-1, \$_] };
my (\$index, \$dmax) = @d.first: @d.max, :kv;
return flat
Ramer-Douglas-Peucker( @points[0..\$index], ε )[^(*-1)],
Ramer-Douglas-Peucker( @points[\$index..*], ε )
if \$dmax > ε;
@points[0, *-1];
}

# TESTING
say Ramer-Douglas-Peucker(
[(0,0),(1,0.1),(2,-0.1),(3,5),(4,6),(5,7),(6,8.1),(7,9),(8,9),(9,9)]
);
```

{{out}}

```((0 0) (2 -0.1) (3 5) (7 9) (9 9))
```

## Phix

{{trans|Go}}

```function rdp(sequence l, atom e)
if length(l)<2 then crash("not enough points to simplify") end if
integer idx := 0
atom dMax := -1,
{p1x,p1y} := l[1],
{p2x,p2y} := l[\$],
x21 := p2x - p1x,
y21 := p2y - p1y
for i=1 to length(l) do
atom {px,py} = l[i],
d = abs(y21*px-x21*py + p2x*p1y-p2y*p1x)
if d>dMax then
idx = i
dMax = d
end if
end for
if dMax>e then
return rdp(l[1..idx], e) & rdp(l[idx..\$], e)[2..\$]
end if
return {l[1], l[\$]}
end function

sequence points = {{0, 0}, {1, 0.1}, {2, -0.1}, {3, 5}, {4, 6},
{5, 7}, {6, 8.1}, {7,    9}, {8, 9}, {9, 9}}
?rdp(points, 1)
```

{{out}}

```
{{0,0},{2,-0.1},{3,5},{7,9},{9,9}}

```

## PHP

{{trans|C++}}

```function perpendicular_distance(array \$pt, array \$line) {
// Calculate the normalized delta x and y of the line.
\$dx = \$line[1][0] - \$line[0][0];
\$dy = \$line[1][1] - \$line[0][1];
\$mag = sqrt(\$dx * \$dx + \$dy * \$dy);
if (\$mag > 0) {
\$dx /= \$mag;
\$dy /= \$mag;
}

// Calculate dot product, projecting onto normalized direction.
\$pvx = \$pt[0] - \$line[0][0];
\$pvy = \$pt[1] - \$line[0][1];
\$pvdot = \$dx * \$pvx + \$dy * \$pvy;

// Scale line direction vector and subtract from pv.
\$dsx = \$pvdot * \$dx;
\$dsy = \$pvdot * \$dy;
\$ax = \$pvx - \$dsx;
\$ay = \$pvy - \$dsy;

return sqrt(\$ax * \$ax + \$ay * \$ay);
}

function ramer_douglas_peucker(array \$points, \$epsilon) {
if (count(\$points) < 2) {
throw new InvalidArgumentException('Not enough points to simplify');
}

// Find the point with the maximum distance from the line between start/end.
\$dmax = 0;
\$index = 0;
\$end = count(\$points) - 1;
\$start_end_line = [\$points[0], \$points[\$end]];
for (\$i = 1; \$i < \$end; \$i++) {
\$dist = perpendicular_distance(\$points[\$i], \$start_end_line);
if (\$dist > \$dmax) {
\$index = \$i;
\$dmax = \$dist;
}
}

// If max distance is larger than epsilon, recursively simplify.
if (\$dmax > \$epsilon) {
\$new_start = ramer_douglas_peucker(array_slice(\$points, 0, \$index + 1), \$epsilon);
\$new_end = ramer_douglas_peucker(array_slice(\$points, \$index), \$epsilon);
array_pop(\$new_start);
return array_merge(\$new_start, \$new_end);
}

// Max distance is below epsilon, so return a line from with just the
// start and end points.
return [ \$points[0], \$points[\$end]];
}

\$polyline = [
[0,0],
[1,0.1],
[2,-0.1],
[3,5],
[4,6],
[5,7],
[6,8.1],
[7,9],
[8,9],
[9,9],
];

\$result = ramer_douglas_peucker(\$polyline, 1.0);
print "Result:\n";
foreach (\$result as \$point) {
print \$point[0] . ',' . \$point[1] . "\n";
}
```

{{out}}

```
Result:
0,0
2,-0.1
3,5
7,9
9,9

```

## Python

An approach using the shapely library:

```from __future__ import print_function
from shapely.geometry import LineString

if __name__=="__main__":
line = LineString([(0,0),(1,0.1),(2,-0.1),(3,5),(4,6),(5,7),(6,8.1),(7,9),(8,9),(9,9)])
print (line.simplify(1.0, preserve_topology=False))
```

{{out}}

```LINESTRING (0 0, 2 -0.1, 3 5, 7 9, 9 9)
```

## Racket

```#lang racket
(require math/flonum)
;; points are lists of x y (maybe extensible to z)
;; x+y gets both parts as values
(define (x+y p) (values (first p) (second p)))

;; https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line
(define (⊥-distance P1 P2)
(let*-values
([(x1 y1) (x+y P1)]
[(x2 y2) (x+y P2)]
[(dx dy) (values (- x2 x1) (- y2 y1))]
[(h) (sqrt (+ (sqr dy) (sqr dx)))])
(λ (P0)
(let-values (((x0 y0) (x+y P0)))
(/ (abs (+ (* dy x0) (* -1 dx y0) (* x2 y1) (* -1 y2 x1))) h)))))

(define (douglas-peucker points-in ϵ)
(let recur ((ps points-in))
;; curried distance function which will be applicable to all points
(let*-values
([(p0) (first ps)]
[(pz) (last ps)]
[(p-d) (⊥-distance p0 pz)]
;; Find the point with the maximum distance
[(dmax index)
(for/fold ((dmax 0) (index 0))
((i (in-range 1 (sub1 (length ps))))) ; skips the first, stops before the last
(define d (p-d (list-ref ps i)))
(if (> d dmax) (values d i) (values dmax index)))])
;; If max distance is greater than epsilon, recursively simplify
(if (> dmax ϵ)
;; recursive call
(let-values ([(l r) (split-at ps index)])
(append (drop-right (recur l) 1) (recur r)))
(list p0 pz))))) ;; else we can return this simplification

(module+ main
(douglas-peucker
'((0 0) (1 0.1) (2 -0.1) (3 5) (4 6) (5 7) (6 8.1) (7 9) (8 9) (9 9))
1.0))

(module+ test
(require rackunit)
(check-= ((⊥-distance '(0 0) '(0 1)) '(1 0)) 1 epsilon.0))
```

{{out}}

```'((0 0) (2 -0.1) (3 5) (7 9) (9 9))
```

## REXX

The computation for the ''perpendicular distance'' was taken from the '''GO''' example.

```/*REXX program uses the  Ramer─Douglas─Peucker (RDP)  line simplification algorithm  for*/
/*───────────────────────────── reducing the number of points used to define its shape. */
parse arg epsilon pts                            /*obtain optional arguments from the CL*/
if epsilon='' | epsilon=","   then epsilon= 1    /*Not specified?  Then use the default.*/
if pts=''  then pts= '(0,0) (1,0.1) (2,-0.1) (3,5) (4,6) (5,7) (6,8.1) (7,9) (8,9) (9,9)'
pts= space(pts)                                  /*elide all superfluous blanks.        */
say '  error threshold: '   epsilon              /*echo the error threshold to the term.*/
say ' points specified: '   pts                  /*  "   "    shape points   "  "    "  */
\$= RDP(pts)                                      /*invoke Ramer─Douglas─Peucker function*/
say 'points simplified: '   rez(\$)               /*display points with () ───► terminal.*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
bld: parse arg _; #= words(_); dMax=-#; idx=1; do j=1  for #; @.j= word(_, j); end; return
px:  parse arg _;          return word( translate(_, , ','),  1)   /*obtain the X coörd.*/
py:  parse arg _;          return word( translate(_, , ','),  2)   /*   "    "  Y   "   */
reb: parse arg a,b,,_;                  do k=a  to b;   _=_ @.k;    end;   return strip(_)
rez: parse arg z,_;   do k=1  for words(z); _= _ '('word(z, k)") "; end;   return strip(_)
/*──────────────────────────────────────────────────────────────────────────────────────*/
RDP: procedure expose epsilon; parse arg PT;  call bld  space( translate(PT, , ')(][}{') )
L= px(@.#)-px(@.1)
H= py(@.#)-py(@.1)                          /* [↓] find point IDX with max distance*/
do i=2  to #-1
d= abs(H*px(@.i) - L*py(@.i) + px(@.#)*py(@.1) - py(@.#)*px(@.1) )
if d>dMax  then do;   idx= i;   dMax= d
end
end   /*i*/              /* [↑]  D is the perpendicular distance*/

if dMax>epsilon  then do;   r= RDP( reb(1, idx) )
return subword(r, 1, words(r) - 1)     RDP( reb(idx, #) )
end
return @.1  @.#
```

{{out|output|text= when using the default inputs:}}

```
error threshold:  1
points specified:  (0,0) (1,0.1) (2,-0.1) (3,5) (4,6) (5,7) (6,8.1) (7,9) (8,9) (9,9)
points simplified:  (0,0)  (2,-0.1)  (3,5)  (7,9)  (9,9)

```

## Sidef

{{trans|Perl 6}}

```func perpendicular_distance(Arr start, Arr end, Arr point) {
((point == start) || (point == end)) && return 0
var (Δx,  Δy ) = (  end »-« start)...
var (Δpx, Δpy) = (point »-« start)...
var h = hypot(Δx, Δy)
[\Δx, \Δy].map { *_ /= h }
(([Δpx, Δpy] »-« ([Δx, Δy] »*» (Δx*Δpx + Δy*Δpy))) »**» 2).sum.sqrt
}

func Ramer_Douglas_Peucker(Arr points { .all { .len > 1 } }, ε = 1) {
points.len == 2 && return points

var d = (^points -> map {
perpendicular_distance(points[0], points[-1], points[_])
})

if (d.max > ε) {
var i = d.index(d.max)
return [Ramer_Douglas_Peucker(points.ft(0, i), ε).ft(0, -2)...,
Ramer_Douglas_Peucker(points.ft(i),    ε)...]
}

return [points[0,-1]]
}

say Ramer_Douglas_Peucker(
[[0,0],[1,0.1],[2,-0.1],[3,5],[4,6],[5,7],[6,8.1],[7,9],[8,9],[9,9]]
)
```

{{out}}

```
[[0, 0], [2, -1/10], [3, 5], [7, 9], [9, 9]]

```

## Yabasic

```sub perpendicularDistance(tabla(), i, ini, fin)
local dx, cy, mag, pvx, pvy, pvdot, dsx, dsy, ax, ay

dx = tabla(fin, 1) - tabla(ini, 1)
dy = tabla(fin, 2) - tabla(ini, 2)

//Normalise
mag = (dx^2 + dy^2)^0.5
if mag > 0 dx = dx / mag : dy = dy / mag

pvx = tabla(i, 1) - tabla(ini, 1)
pvy = tabla(i, 2) - tabla(ini, 2)

//Get dot product (project pv onto normalized direction)
pvdot = dx * pvx + dy * pvy

//Scale line direction vector
dsx = pvdot * dx
dsy = pvdot * dy

//Subtract this from pv
ax = pvx - dsx
ay = pvy - dsy

return (ax^2 + ay^2)^0.5
end sub

sub DouglasPeucker(PointList(), ini, fin, epsilon)
local dmax, index, i, d
// Find the point with the maximum distance

for i = ini + 1 to fin
d = perpendicularDistance(PointList(), i, ini, fin)
if d > dmax index = i : dmax = d
next

// If max distance is greater than epsilon, recursively simplify
if dmax > epsilon then
PointList(index, 3) = true
// Recursive call
DouglasPeucker(PointList(), ini, index, epsilon)
DouglasPeucker(PointList(), index, fin, epsilon)
end if
end sub

data 0,0, 1,0.1,  2,-0.1,  3,5,  4,6,  5,7,  6,8.1,  7,9,  8,9,  9,9

dim matriz(10, 3)

for i = 1 to 10
read matriz(i, 1), matriz(i, 2)
next

DouglasPeucker(matriz(), 1, 10, 1)

matriz(1, 3) = true : matriz(10, 3) = true
for i = 1 to 10
if matriz(i, 3) print matriz(i, 1), matriz(i, 2)
next
```

## zkl

{{trans|Perl 6}}

```fcn perpendicularDistance(start,end, point){  // all are tuples: (x,y) -->|d|
dx,dy   := end  .zipWith('-,start);	// deltas
dpx,dpy := point.zipWith('-,start);
mag     := (dx*dx + dy*dy).sqrt();
if(mag>0.0){ dx/=mag; dy/=mag; }
p,dsx,dsy := dx*dpx + dy*dpy, p*dx, p*dy;
((dpx - dsx).pow(2) + (dpy - dsy).pow(2)).sqrt()
}

fcn RamerDouglasPeucker(points,epsilon=1.0){  // list of tuples --> same
if(points.len()==2) return(points);  // but we'll do one point
d:=points.pump(List,  // first result/element is always zero
fcn(p, s,e){ perpendicularDistance(s,e,p) }.fp1(points[0],points[-1]));
index,dmax := (0.0).minMaxNs(d)[1], d[index]; // minMaxNs-->index of min & max
if(dmax>epsilon){
return(RamerDouglasPeucker(points[0,index],epsilon)[0,-1].extend(
RamerDouglasPeucker(points[index,*],epsilon)))
} else return(points[0],points[-1]);
}
```
```RamerDouglasPeucker(
T( T(0.0, 0.0), T(1.0, 0.1), T(2.0, -0.1), T(3.0, 5.0), T(4.0, 6.0),
T(5.0, 7.0), T(6.0, 8.1), T(7.0,  9.0), T(8.0, 9.0), T(9.0, 9.0) ))
.println();
```

{{out}}

```
L(L(0,0),L(2,-0.1),L(3,5),L(7,9),L(9,9))

```

'''References'''