C#-Code

using System;
using System.Collections.Generic;
using System.Text;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Media;
using System.Windows.Shapes;

using System.Text.RegularExpressions;

namespace convex_line_of_point_set  {

    public partial class MainWindow : Window    {

        public static string NL = Environment.NewLine;

        public MainWindow() {
            InitializeComponent();
            this.MyCanvas.ClipToBounds = true;
            this.tb_anzahl.Text = anzahl_default.ToString();

            this.tb_1.Text = MyTestTool.test_CalcPolygonArea();
        }

        private point_set ps;
        private point_set.xyr[] line_points;

        private double rw = 8.0;
        private bool bo_data_available = false;
        private bool bo_more_info = false;

        private void draw_pointes(int i, point_set.xyr[] set)        {

            Rectangle myRect = new Rectangle();
            myRect.Width = rw; myRect.Height = rw;
            myRect.Fill = new SolidColorBrush(Colors.Blue);
            Canvas.SetLeft(myRect, set[i].x);
            Canvas.SetTop(myRect,  set[i].y);
            MyCanvas.Children.Add(myRect);
        }

        private void draw_lines(int i)        {
            Line line = new Line();
            line.Stroke = Brushes.Red;
            line.StrokeThickness = rw / 4;
            line.X1 = line_points[i].x + rw / 2;
            line.X2 = line_points[i + 1].x + rw / 2;
            line.Y1 = line_points[i].y + rw / 2;
            line.Y2 = line_points[i + 1].y + rw / 2;

            MyCanvas.Children.Add(line);

            Rectangle myRect = new Rectangle();
            myRect.Width = rw; myRect.Height = rw;
            myRect.Fill = new SolidColorBrush(Colors.Yellow);
            Canvas.SetLeft(myRect, line_points[i].x);
            Canvas.SetTop(myRect, line_points[i].y);
            MyCanvas.Children.Add(myRect);
        }

        private static uint anzahl_default = 50, anzahl = anzahl_default;

        private void bt_punkte_Click(object sender, RoutedEventArgs e)
        //create and draw point set
        {
            uint anzahl_out;
            if (!uint.TryParse(this.tb_anzahl.Text, out anzahl_out))
            {
                MessageBox.Show("no number in textbox");
                this.tb_anzahl.Text = anzahl_default.ToString();
                return;
            }
            else
                if (anzahl_out <= 0)
                    return;
                else
                    anzahl_default = anzahl_out;

            anzahl = anzahl_default;

            MyCanvas.Children.Clear();

            int bo_xy = 1;
            if ((bool)this.rb_rphi_zyl.IsChecked) bo_xy = 2;
            if ((bool)this.rb_rphi_euc.IsChecked) bo_xy = 3;

            int laenge = (int)(Math.Min(size_values[2], size_values[3]) - 2 * rw);
            ps = new point_set(anzahl: (int)anzahl, laenge: laenge, bo_xy: bo_xy);
            this.tb_1.Text = ps.point_set_as_string_org;

            point_set.xyr[] set = ps.set;


            line_points = new point_set.xyr[ps.linepoints + 1];
            for (var i = 0; i < set.Length; i++)
                if (set[i].rank != -1)
                    line_points[set[i].rank] = set[i];
            line_points[line_points.Length - 1] = line_points[0];

            
            for (var i = 0; i < set.Length; i++)
                draw_pointes(i, set);

            this.bo_data_available = true;
        }

        private void bt_clear_Click(object sender, RoutedEventArgs e)
        {
            this.MyCanvas.Children.Clear();
            this.tb_1.Clear();
            this.bo_data_available = false;
        }

        private void bt_run_Click(object sender, RoutedEventArgs e)
        //draw convex line in one step
        {
            if (!this.bo_data_available)
                return;
                   
            for (var i = 0; i < line_points.Length - 1; i++)
                draw_lines(i);

            double verhaeltnis = 1.0 * (line_points.Length - 1) / ps.set.Length;
            double verhaeltnis_p = 1.0 * (line_points.Length - 1) / (int)(Math.Min(size_values[2], size_values[3]));

            this.tb_1.Text = 
                (line_points.Length - 1).ToString("n0") + 
                NL + ps.set.Length.ToString("n0") + 
                NL + verhaeltnis.ToString("p2") + 

                NL + ps.area.ToString("n2") +
                
                NL + ps.point_set_as_string ;
        }

        private int bt_by_step_counter = 0;
        private void bt_by_step_Click(object sender, RoutedEventArgs e)
        //draw convex line step by step
        {
            if (!this.bo_data_available)
                return;

            draw_lines(bt_by_step_counter);
            this.tb_1.Text = ps.strings_step_by_step[bt_by_step_counter];

            bt_by_step_counter++;
            bt_by_step_counter = bt_by_step_counter % (line_points.Length - 1);
        }

        private double[] size_values;

        private void tb_anzahl_TextChanged(object sender, TextChangedEventArgs e)
        {
            TextBox a_textBox = (TextBox)sender;
            string a_newText = string.Empty;

            for (int i = 0; i < a_textBox.Text.Length; i++)
            {
                if (Regex.IsMatch(a_textBox.Text[i].ToString(), "[0-9]"))//  "^[a-zA-Z0-9-]+$"))
                {
                    a_newText += a_textBox.Text[i];
                }
            }

            a_textBox.Text = a_newText;
            a_textBox.SelectionStart = a_textBox.Text.Length;
        }

        private void main_window_SizeChanged(object sender, SizeChangedEventArgs e)        {

            double set_w = 0, set_h = 0, minx = -1, miny = -1, maxx = -1, maxy = -1;

            if(bo_data_available)            {

                double x0 = double.MaxValue, y0 = double.MaxValue, x1 = double.MinValue, y1 = double.MinValue;

                for(var i = 0; i < ps.set.Length; i++)  {

                    if (ps.set[i].x < x0) x0 = ps.set[i].x;
                    if (ps.set[i].y < y0) y0 = ps.set[i].y;
                    if (ps.set[i].x > x1) x1 = ps.set[i].x;
                    if (ps.set[i].y > y1) y1 = ps.set[i].y;
                }

                minx = x0; miny = y0; maxx = x1; maxy = y1;
                set_w = x1 - x0;
                set_h = y1 - y0;
            }

            this.size_values = new double[]             {
                this.main_window.ActualWidth, this.main_window.ActualHeight,
                this.MyCanvas.ActualWidth, this.MyCanvas.ActualHeight,
                this.main_border.ActualWidth, this.main_border.ActualHeight,
                set_w, set_h, minx, miny, maxx, maxy
            };

            if(bo_more_info)
                this.tb_1.Text = this.size_values.ToMyString(false);     
        }
    }

    public class point_set    {

        public class xyr
        {
            public double x;
            public double y;
            public int rank;
        }

        public string   point_set_as_string = String.Empty,
                        point_set_copy_as_string = String.Empty,
                        point_set_as_string_org = String.Empty;

        public xyr[] set, set_copy, set_temp;

        public List<string> strings_step_by_step = new List<string>();

        public int linepoints = 0;
        public double area = 0;

        private int most_south = -1;

        public point_set(int anzahl = 50, int laenge = 10, int bo_xy = 1)
        {
            this.set = new xyr[anzahl]; this.set_copy = new xyr[anzahl]; this.set_temp = new xyr[anzahl];

            Random rand = new Random();
            double dmost_south = double.MaxValue;

            for(var i= 0; i < this.set.Length; i++)
            {
                set[i] = new xyr();

                double x = 0, y = 0;

                if (bo_xy == 1)
                {
                    x = laenge * rand.NextDouble();
                    y = laenge * rand.NextDouble();
                }

                if (bo_xy == 2)
                {
                    double r = laenge * rand.NextDouble() / 2; ; double ph = 2.0 * Math.PI * rand.NextDouble();
                    y = r * Math.Sin(ph) + laenge / 2;
                    x = r * Math.Cos(ph) + laenge / 2;
                }

                if (bo_xy == 3) { 
                    bool bo_okay = false;
                    while (!bo_okay)
                    {
                        x = laenge * rand.NextDouble();
                        y = laenge * rand.NextDouble();
                        double lh = laenge / 2;
                        bo_okay = Math.Sqrt((x - lh) * (x - lh) + (y - lh) * (y - lh)) <= lh;
                    }
                }

                set[i].x = x;
                set[i].y = y;

                set[i].rank = -1;
                if (set[i].y < dmost_south)
                {
                    dmost_south = set[i].y;
                    this.most_south = i;
                }
            }

            this.point_set_as_string_org = set_as_string(set);

            set[most_south].rank = 0;

            this.strings_step_by_step.Add(set_as_string(set));

            this.copy_and_adjust(this.set, out this.set_copy, this.set[this.most_south], 0.0);

            double phi = 0.0;
            int counter = 1;

            while (true)    {
                int merker_i = -1;
                double phi_temp = double.MaxValue;
                for (var i = 0; i < this.set_copy.Length; i++)  {

                    if ((this.set_copy[i].rank < 1) && !((this.set_copy[i].y == 0) && (this.set_copy[i].x == 0)))   {
                        double phi_i = Math.Atan2(this.set_copy[i].y, this.set_copy[i].x);
                        if (phi_i < 0.0)
                            phi_i = 2.0 * Math.PI - Math.Abs(phi_i);

                        if (phi_i < phi_temp)   {
                            phi_temp = phi_i;
                            merker_i = i;
                        }
                    }
                }
                phi += phi_temp;
                if (phi > 2 * Math.PI)
                    break;

                if (this.set_copy[merker_i].rank != 0)  {
                    this.set_copy[merker_i].rank = counter;
                    this.set[merker_i].rank = counter;
                    this.strings_step_by_step.Add(set_as_string(set));
                }
                else
                    break;

                counter++;
                copy_and_adjust(this.set_copy, out this.set_copy, this.set_copy[merker_i], phi_temp);
            }

            for (var i = 0; i < this.set_copy.Length; i++)
                if (this.set[i].rank != -1)
                    linepoints++;

            double[] xp = new double[linepoints], yp = new double[linepoints];
            for (var i = 0; i < this.set_copy.Length; i++)
                if (this.set[i].rank != -1) {
                    xp[this.set[i].rank] = this.set[i].x;
                    yp[this.set[i].rank] = this.set[i].y;
                }

            this.area = MyTestTool.CalcPolygonArea(xp, yp);
            this.point_set_as_string = set_as_string(set);
        }

        private void copy_and_adjust(xyr[] from, out xyr[] to, xyr adjust, double w )
        {
            to = new xyr[from.Length];
            for (var i = 0; i < from.Length; i++)
            {
                to[i] = new xyr();
                to[i].x = from[i].x - adjust.x;
                to[i].y = from[i].y - adjust.y;
                to[i].rank = from[i].rank;
                to[i] = drehe(to[i], w);
            }
            point_set_copy_as_string = set_as_string(to);
        }

        private string set_as_string(xyr[] set)
        {
            StringBuilder sb = new StringBuilder();
            sb.AppendLine();
            for (var i = 0; i < this.set.Length; i++)
            {
                sb.Append(" [" + set[i].x.ToString("0.00") + "; ");
                sb.Append(set[i].y.ToString("0.00") + "] ");
                sb.AppendLine(set[i].rank.ToString());
            }            
            return sb.ToString();
        }

        private xyr drehe(xyr p, double w)
        {
            xyr result = new xyr();
            result.rank = p.rank;
            double cos = Math.Cos(w), sin = Math.Sin(w);
            result.x = p.x * cos + p.y * sin;
            result.y = p.y * cos - p.x * sin;
            return result;
        }
    }

    public static class MyTestTool    {

        public static string test_atan2()
        {
            String LZ = " ";
            StringBuilder sb = new StringBuilder();
            double[] x = new double[] { 1, 1, 0, -1, -1, -1,  0,  1,  1        };
            double[] y = new double[] { 0, 1, 1,  1,  0, -1, -1, -1, -0.000001 };
            for(var i = 0; i < y.Length; i++)
            {
                double phi = Math.Atan2(y[i], x[i]);
                if (phi < 0.0)
                    phi = 2.0 * Math.PI - Math.Abs(phi);
                sb.AppendLine(i.ToString() + LZ + phi.ToString("0.0000") + 
                    LZ + (180 * phi / Math.PI).ToString("0.00"));
            }
            return sb.ToString();
        }

        public static string ToMyString(this double[] x, bool bo_horizontal = true)
        {
            StringBuilder sb = new StringBuilder();
            string dir = " ";
            if (!bo_horizontal)
                dir = Environment.NewLine;
            for (var i = 0; i < x.Length; i++)
                sb.Append(x[i].ToString("0.0000") + dir);
            return sb.ToString();
        }

        public static double CalcPolygonArea(double[] x, double[] y)
        {
            if ((x == null) || (y == null)) return 0.0;  
            int n = Math.Min(x.Length, y.Length);        
            if (n < 3) return 0.0;                       
            double area = 0.0;                              
            for (int i = 0; i < n; i++)
                area += (y[i] + y[(i + 1) % n]) * (x[i] - x[(i + 1) % n]);
            return Math.Abs(area / 2.0);                    
        }

        public static string test_CalcPolygonArea()
        {            
            double[] y = new double[] { 0, 10, 10,  0, -10, 0 };
            double[] x = new double[] { 0, 10, 20, 20,  10, 0 };
            return CalcPolygonArea(x, y).ToString();
        }
    }
}

C#-Code

<Window x:Name="main_window" x:Class="convex_line_of_point_set.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:d="http://schemas.microsoft.com/expression/blend/2008"
        xmlns:mc="http://schemas.openxmlformats.org/markup-compatibility/2006"
        xmlns:local="clr-namespace:convex_line_of_point_set"
        mc:Ignorable="d"
        Title="convex line" Height="555.419" Width="641.85" WindowStartupLocation="CenterScreen" Background="{DynamicResource {x:Static SystemColors.ActiveCaptionBrushKey}}" SizeChanged="main_window_SizeChanged" >
    <Grid Margin="0">
        <Grid.ColumnDefinitions>
            <ColumnDefinition/>
            <ColumnDefinition Width="150"/>
        </Grid.ColumnDefinitions>
        <Grid.RowDefinitions>
            <RowDefinition/>
            <RowDefinition Height="70"/>
        </Grid.RowDefinitions>
        <StackPanel Margin="5" Grid.Row="1" Orientation="Horizontal" Grid.ColumnSpan="3">
            <Button x:Name="bt_clear" Content=" clear canvas " Margin="5,0,0,0" Click="bt_clear_Click" VerticalAlignment="Top"/>
            <Button x:Name="bt_punkte" Content=" draw new points " Margin="5,0,0,0" Click="bt_punkte_Click" VerticalAlignment="Top"/>
            <Button x:Name="bt_run" Content=" draw connected convex line " Margin="5,0,0,0" Click="bt_run_Click" VerticalAlignment="Top"/>
            <Button x:Name="bt_by_step" Content=" step by step " Width="75" Click="bt_by_step_Click" VerticalAlignment="Top"/>
            <StackPanel VerticalAlignment="Center">
                <Label Content="0    1     2" Margin="0" HorizontalAlignment="Center" RenderTransformOrigin="0.5,0.5" ToolTip="xy(1) rphi_zyl(2) rph_euclid(3)" />
                <StackPanel Orientation="Horizontal">
                    <RadioButton x:Name="rb_xy" VerticalAlignment="Center" IsChecked="True" HorizontalAlignment="Center" Margin="5" />
                    <RadioButton x:Name="rb_rphi_zyl" HorizontalAlignment="Center" VerticalAlignment="Center" Margin="5"/>
                    <RadioButton x:Name="rb_rphi_euc" HorizontalAlignment="Center" VerticalAlignment="Center" Margin="5"/>
                </StackPanel>
            </StackPanel>

            <TextBox x:Name="tb_anzahl" TextWrapping="Wrap" Width="50" Margin="15,0,0,0" Text="50" TextChanged="tb_anzahl_TextChanged" VerticalAlignment="Top"/>
        </StackPanel>
        <Border x:Name="main_border" BorderBrush="Black" BorderThickness="5" Margin="5,5,0,4">
            <Canvas x:Name="MyCanvas" Background="#FF11F58E"/>
        </Border>
        <TextBox x:Name="tb_1" Grid.Column="1" Margin="5" TextWrapping="Wrap" VerticalScrollBarVisibility="Auto" Background="Blue" Foreground="#FFE6E608" BorderThickness="1"/>
    </Grid>
</Window>