Task: Archimedean spiral
C#: using System;
using System.Linq;
using System.Drawing;
using System.Diagnostics;
using System.Drawing.Drawing2D;

class Program
{
    const int width = 380;
    const int height = 380;
    static PointF archimedeanPoint(int degrees)
    {
        const double a = 1;
        const double b = 9;
        double t = degrees * Math.PI / 180;
        double r = a + b * t;
        return new PointF { X = (float)(width / 2 + r * Math.Cos(t)), Y = (float)(height / 2 + r * Math.Sin(t)) };
    }

    static void Main(string[] args)
    {
        var bm = new Bitmap(width, height);
        var g = Graphics.FromImage(bm);
        g.SmoothingMode = SmoothingMode.AntiAlias;
        g.FillRectangle(new SolidBrush(Color.White), new Rectangle { X = 0, Y = 0, Width = width, Height = height });
        var pen = new Pen(Color.OrangeRed, 1.5f);

        var spiral = Enumerable.Range(0, 360 * 3).AsParallel().AsOrdered().Select(archimedeanPoint);
        var p0 = new PointF(width / 2, height / 2);
        foreach (var p1 in spiral)
        {
            g.DrawLine(pen, p0, p1);
            p0 = p1;
        }
        g.Save(); 
        bm.Save("archimedes-csharp.png");
        Process.Start("archimedes-csharp.png"); 
    }
}

Java: import java.awt.*;
import static java.lang.Math.*;
import javax.swing.*;

public class ArchimedeanSpiral extends JPanel {

    public ArchimedeanSpiral() {
        setPreferredSize(new Dimension(640, 640));
        setBackground(Color.white);
    }

    void drawGrid(Graphics2D g) {
        g.setColor(new Color(0xEEEEEE));
        g.setStroke(new BasicStroke(2));

        double angle = toRadians(45);

        int w = getWidth();
        int center = w / 2;
        int margin = 10;
        int numRings = 8;

        int spacing = (w - 2 * margin) / (numRings * 2);

        for (int i = 0; i < numRings; i++) {
            int pos = margin + i * spacing;
            int size = w - (2 * margin + i * 2 * spacing);
            g.drawOval(pos, pos, size, size);

            double ia = i * angle;
            int x2 = center + (int) (cos(ia) * (w - 2 * margin) / 2);
            int y2 = center - (int) (sin(ia) * (w - 2 * margin) / 2);

            g.drawLine(center, center, x2, y2);
        }
    }

    void drawSpiral(Graphics2D g) {
        g.setStroke(new BasicStroke(2));
        g.setColor(Color.orange);

        double degrees = toRadians(0.1);
        double center = getWidth() / 2;
        double end = 360 * 2 * 10 * degrees;
        double a = 0;
        double b = 20;
        double c = 1;

        for (double theta = 0; theta < end; theta += degrees) {
            double r = a + b * pow(theta, 1 / c);
            double x = r * cos(theta);
            double y = r * sin(theta);
            plot(g, (int) (center + x), (int) (center - y));
        }
    }

    void plot(Graphics2D g, int x, int y) {
        g.drawOval(x, y, 1, 1);
    }

    @Override
    public void paintComponent(Graphics gg) {
        super.paintComponent(gg);
        Graphics2D g = (Graphics2D) gg;
        g.setRenderingHint(RenderingHints.KEY_ANTIALIASING,
                RenderingHints.VALUE_ANTIALIAS_ON);

        drawGrid(g);
        drawSpiral(g);
    }

    public static void main(String[] args) {
        SwingUtilities.invokeLater(() -> {
            JFrame f = new JFrame();
            f.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
            f.setTitle("Archimedean Spiral");
            f.setResizable(false);
            f.add(new ArchimedeanSpiral(), BorderLayout.CENTER);
            f.pack();
            f.setLocationRelativeTo(null);
            f.setVisible(true);
        });
    }
}