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Drawing y = sin(θ) * cos(θ) in C# with a Pen

I want to draw sin(θ)*cos(θ), but it doesn't work.
I can draw sin or cos,
but I want to draw sin(θ)*cos(θ) together.
Here is my code
private void button1_Click(object sender, EventArgs e)
{
Graphics drw = this.CreateGraphics();
Pen pen = new Pen(Brushes.Black, 7.0f);
float x1 = 0;
float y1 = 0;
float xoy = 200;
float ef = 20;
for (double i=0;i<40;i+=1)
{
double radi = (float)(i * 180 / Math.PI);
float temp = (float)Math.Cos(radi)*(float)Math.Sin(radi);
drw.DrawLine(pen, x1 * ef, y1 * ef + xoy, ef * (float)i, temp * ef + xoy);
x1 = (float)i;
y1 = temp;
}
}
And I want this result:

You may find it easier to look at the corresponding Parametric Equations.
private void Form1_Paint(object sender, PaintEventArgs e)
{
var g = e.Graphics;
double pi = Math.PI;
int n = 100;
var t = Enumerable.Range(0, n).Select(p => p * 2 * pi / n).ToArray();
var x = t.Select(p => Math.Sin(2 * p) * Math.Cos(p)).ToArray();
var y = t.Select(p => Math.Sin(2 * p) * Math.Sin(p)).ToArray();
Pen pen = new Pen(Brushes.Black, 3);
int scale = 100;
int shift = 100;
for (int i = 0; i < n - 1; i++)
{
g.DrawLine(pen, scale*(float)x[i] + shift,
scale*(float)y[i] + shift,
scale*(float)x[i + 1] + shift,
scale*(float)y[i + 1] + shift);
}
}

Actually, the real function you are looking for is a little bit different... see an example here. Looking at this article about polar flowers, I'm sure it will get pointed to the right direction, and it also contains a full working source code.
Just an example, supposing you use a panel in your form on which to draw the polar flower:
panel.OnPaint += Panel_Paint;
private void Panel_Paint(Object sender, PaintEventArgs e)
{
Double scale = ((Panel)sender).Width / 2.0d;
Double repetitions = Math.Round(scale, 0);
Double basis = (2.0d * Math.PI) / scale;
Double petals = 2.0d;
using (Graphics g = e.Graphics)
{
using (Pen pen = new Pen(Brushes.Red, 2.0f))
{
for (Double i = 0.0f; i < (repetitions - 1); ++i)
{
Double t0 = i*basis;
Double t1 = (i + 1)*basis;
Double x0 = Math.Sin(petals * t0) * Math.Cos(t0);
Double x1 = Math.Sin(petals * t1) * Math.Cos(t1);
Double y0 = Math.Sin(petals * t0) * Math.Sin(t0);
Double y1 = Math.Sin(petals * t1) * Math.Sin(t1);
g.DrawLine
(
pen,
(Single) ((scale*x0) + scale),
(Single) ((scale*y0) + scale),
(Single) ((scale*x1) + scale),
(Single) ((scale*y1) + scale)
);
}
}
}
}
The basic formulation states that if the petals variable value is:
even, then it represents half the amount of petals of the polar flower
odd, then it represents the amount of petals of the polar flower
so if you define Double petals = 2.0d;, you will obtain 4 petals... and if you define Double petals = 5.0d;, you will obtain 5 petals.

how to detect corners of polygon?

I am making an automated floor plan generation desktop app. In this, at first I draw the polygon on points, using this method
public void DrawPolygonPointF(PaintEventArgs e) {
// Create pen.
Pen blackPen = new Pen(Color.Black, 3);
// Create points that define polygon.
PointF point1 = new PointF(50.0F, 50.0F);
PointF point2 = new PointF(100.0F, 25.0F);
PointF point3 = new PointF(200.0F, 5.0F);
PointF point4 = new PointF(250.0F, 50.0F);
PointF point5 = new PointF(300.0F, 100.0F);
PointF point6 = new PointF(350.0F, 200.0F);
PointF point7 = new PointF(250.0F, 250.0F);
PointF[] curvePoints =
{
point1,
point2,
point3,
point4,
point5,
point6,
point7
};
// Draw polygon curve to screen.
e.Graphics.DrawPolygon(blackPen, curvePoints);
}
NOTE: These points are not actual points, they are for only demo purpose. I am reading the points from a text file.
Now I need to generate a special type of grid.
In generating the grid the first step is detect corners and extend the corner lines.
How do I detect the corners of a polygon so can i move to next step of generating grid?
Corners are marked. I need to extend corner marked with black on horizontally left side and other one is to extend on right side till its touch the line.
A screenshot is attached.
Thanks in advance

In my understanding you are trying to extend edges, not corners.
the procedure could be:
Enumerate edges (each 2 adjacent points define an edge)
For each edge find if it vertical or horizontal (abs(x1-x2) > abs(y1-y2))
Find if edge can be extended, the horizontal (left/right) and the vertical (up/bottom)
bool CheckHorizontalExtensibilityToRight(Point[] curvePoints, Point corner)
{
return curvePoints.Any(cp=>cp.Y < corner.Y && cp.X < corner.X);
}

Try this example and see if you can adapt it to solve your problem...
using System.Drawing;
using System.Windows.Forms;
namespace WindowsFormsApplication
{
public partial class Form1 : Form
{
private struct LineSegment
{
private PointF _a, _b;
public PointF A { get { return _a; } }
public PointF B { get { return _b; } }
public LineSegment(PointF a, PointF b)
{
_a = a; _b = b;
}
public float GetLengthSquared()
{
var dx = _a.X - _b.X;
var dy = _a.Y - _b.Y;
return dx * dx + dy * dy;
}
public bool RectContains(PointF a)
{
var x = a.X;
var y = a.Y;
var x1 = _a.X;
var y1 = _a.Y;
var x2 = _b.X;
var y2 = _b.Y;
return (x1 < x2 ? x1 <= x && x2 >= x : x2 <= x && x1 >= x) && (y1 < y2 ? y1 <= y && y2 >= y : y2 <= y && y1 >= y);
}
public bool ExtendToIntersectWith(LineSegment b)
{
var x1 = _a.X;
var y1 = _a.Y;
var x2 = _b.X;
var y2 = _b.Y;
var x3 = b._a.X;
var y3 = b._a.Y;
var x4 = b._b.X;
var y4 = b._b.Y;
var a1 = y2 - y1;
var b1 = x1 - x2;
var c1 = x1 * y2 - x2 * y1;
var a2 = y4 - y3;
var b2 = x3 - x4;
var c2 = x3 * y4 - x4 * y3;
var d = a1 * b2 - b1 * a2;
if (d == 0)
return false;
var x = (c1 * b2 - b1 * c2) / d;
var y = (a1 * c2 - c1 * a2) / d;
var p = new PointF(x, y);
if (b.RectContains(p) && !RectContains(p))
{
if (new LineSegment(_a, p).GetLengthSquared() < new LineSegment(_b, p).GetLengthSquared())
_a = p;
else
_b = p;
return true;
}
return false;
}
}
public Form1()
{
InitializeComponent();
}
private void Form1_Paint(object sender, PaintEventArgs e)
{
PointF[] curvePoints =
{
/*
new PointF(50.0F, 50.0F),
new PointF(100.0F, 25.0F),
new PointF(200.0F, 5.0F),
new PointF(250.0F, 50.0F),
new PointF(300.0F, 100.0F),
new PointF(350.0F, 200.0F),
new PointF(250.0F, 250.0F)
*/
new PointF(30F, 10F),
new PointF(60F, 10F),
new PointF(60F, 20F),
new PointF(90F, 20F),
new PointF(90F, 60F),
new PointF(10F, 60F),
new PointF(10F, 40F),
new PointF(30F, 40F),
};
int n = curvePoints.Length;
LineSegment[] lineSegments = new LineSegment[n];
int i = 0;
for (; i < n - 1; ++i)
lineSegments[i] = new LineSegment(curvePoints[i], curvePoints[i + 1]);
lineSegments[i] = new LineSegment(curvePoints[i], curvePoints[0]);
for (i = 0; i < n; ++i)
for (int j = 0; j < n; ++j)
lineSegments[i].ExtendToIntersectWith(lineSegments[j]);
for (i = 0; i < n; ++i)
{
var lineSegment = lineSegments[i];
e.Graphics.DrawLine(Pens.Black, lineSegment.A, lineSegment.B);
}
//e.Graphics.DrawPolygon(Pens.Black, curvePoints);
}
}
}

Translating coordinates between two 2d coordinate systems

I'm writing a small graph writing program as a personal project.
Each point is supposed to get drawn on a wpf canvas but I'm having trouble translating the points from the graph's coordinate system, example: x = -8 to 4 y=-4 to 4, to the canvas' coordinate system example: x = 600 to 0 y = 400 to 0.
I'm using the method outlined here to precompute the transformation equations.
However I'm having trouble with the result of the multiplication v=M^-1*u.
My expected result is:
[0.2 ]
[0.0 ]
[-8.0]
[4.0 ]
But the result I'm getting is:
[4.0 ]
[-4.0 ]
[0.01 ]
[-0.006]
I've verified that my transformation matrix is correct and when I do the calculation by hand I'm getting the expected result.
The method to calculate the transformation equations:
private void CalculateTransformationFunctions()
{
// Define the transformation matrix
var transformationMatrix = new Matrix4x4
{
M11 = _destArea.XMin,
M12 = _destArea.YMin,
M13 = 1,
M14 = 0,
M21 = -_destArea.YMin,
M22 = _destArea.XMin,
M23 = 0,
M24 = 1,
M31 = _destArea.XMax,
M32 = _destArea.YMax,
M33 = 1,
M34 = 0,
M41 = -_destArea.YMax,
M42 = _destArea.XMax,
M43 = 0,
M44 = 1
};
// Define the source vector
var srcVector = new Vector4
{
X = _srcArea.XMin,
Y = _srcArea.YMax,
Z = _srcArea.XMax,
W = _srcArea.YMin
};
// Invert the transformationmatrix before the multiplication
Matrix4x4 invertedTransformationMatrix;
if(!Matrix4x4.Invert(transformationMatrix,out invertedTransformationMatrix))
throw new Exception();
// Returns the wrong value
var transformResult = Vector4.Transform(srcVector, invertedTransformationMatrix);
float a = transformResult.X,
b = transformResult.Y,
c = transformResult.Z,
d = transformResult.W;
_xTransformationFunction = (x, y) => (a*x + b*y - b*d - a*c)/(a*a + b*b);
_yTransformationFunction = (x, y) => (b*x - a*y - b*c + a*d)/(a*a + b*b);
}
Which is called in the constructor of its parent class.
My question:
Am I misunderstanding what Vector4.Transform() does here? Or am I completely blind and missing something very obvious?
Full source of the class:
using System;
using System.Numerics;
using System.Windows;
using System.Windows.Media;
using Grapher.Control.Grapher;
namespace Grapher.GraphingMath
{
public class Translator
{
private GraphingArea _srcArea;
private GraphingArea _destArea;
public GraphingArea SourceArea
{
get
{
return _srcArea;
}
set
{
_srcArea = value;
CalculateTransformationFunctions();
}
}
public GraphingArea DestinationArea
{
get { return _destArea; }
set
{
_destArea = value;
CalculateTransformationFunctions();
}
}
private Func<double, double, double> _xTransformationFunction;
private Func<double, double, double> _yTransformationFunction;
public Translator(GraphingArea sourceArea, GraphingArea destArea)
{
_destArea = destArea;
_srcArea = sourceArea;
CalculateTransformationFunctions();
}
public Point TranslatePoint(Point point)
{
var x = point.X;
var y = point.Y;
return new Point
{
X = _xTransformationFunction(x, y),
Y = _yTransformationFunction(x, y)
};
}
/*
x1 y1 1 0
-y1 x1 0 1
M= x2 y2 1 0
-y2 x2 0 1
x1,y1 = dest_min
x2,y2 = dest_max
*/
private void CalculateTransformationFunctions()
{
// Define the transformation matrix
var transformationMatrix = new Matrix4x4
{
M11 = _destArea.XMin,
M12 = _destArea.YMin,
M13 = 1,
M14 = 0,
M21 = -_destArea.YMin,
M22 = _destArea.XMin,
M23 = 0,
M24 = 1,
M31 = _destArea.XMax,
M32 = _destArea.YMax,
M33 = 1,
M34 = 0,
M41 = -_destArea.YMax,
M42 = _destArea.XMax,
M43 = 0,
M44 = 1
};
// Define the source vector
var srcVector = new Vector4
{
X = _srcArea.XMin,
Y = _srcArea.YMax,
Z = _srcArea.XMax,
W = _srcArea.YMin
};
// Invert the transformationmatrix before the multiplication
Matrix4x4 invertedTransformationMatrix;
if(!Matrix4x4.Invert(transformationMatrix,out invertedTransformationMatrix))
throw new Exception();
// Returns the wrong value
var transformResult = Vector4.Transform(srcVector, invertedTransformationMatrix);
float a = transformResult.X,
b = transformResult.Y,
c = transformResult.Z,
d = transformResult.W;
_xTransformationFunction = (x, y) => (a*x + b*y - b*d - a*c)/(a*a + b*b);
_yTransformationFunction = (x, y) => (b*x - a*y - b*c + a*d)/(a*a + b*b);
}
}
}
And for the graphing area struct:
using System;
namespace Grapher.Control.Grapher
{
public struct GraphingArea
{
public float XMin { get; set; }
public float YMin { get; set; }
public float XMax { get; set; }
public float YMax { get; set; }
public float Width => Math.Abs(XMax - XMin);
public float Height => Math.Abs(YMax - YMin);
}
}
In my main method I call the Translator class like this:
Point testPoint = new Point {X = 0, Y = 0};
var srcArea = new GraphingArea
{
XMax = 4,
XMin = -8,
YMax = 4,
YMin = -4
};
var destArea = new GraphingArea
{
XMax = 600,
XMin = 0,
YMax = 400,
YMin = 0
};
var translator = new Translator(srcArea, destArea);
var translatedPoint = translator.TranslatePoint(testPoint);
Edit
Ended up just writing my own matrix multiplication method. I must be misunderstanding what Vector4.Transform() does...
Code here, for anyone interested:
using System.Numerics;
namespace Grapher.GraphingMath.MatrixAndVectorMath
{
public static class Matrix4x4Multiply
{
public static Vector4 Vector4Multiply(Matrix4x4 matrix, Vector4 vector)
{
var mat = new float[4, 4]
{
{matrix.M11, matrix.M12, matrix.M13, matrix.M14},
{matrix.M21, matrix.M22, matrix.M23, matrix.M24},
{matrix.M31, matrix.M32, matrix.M33, matrix.M34},
{matrix.M41, matrix.M42, matrix.M43, matrix.M44}
}; // We'll just wrap the matrix in a float so we can index it.
var vec = new float[4] {vector.X, vector.Y, vector.Z, vector.W}; // And the same with the vector
var result = new float[4] {0, 0, 0, 0};
for (var row = 0; row < mat.GetLength(0); row++)
{
for (var col = 0; col < mat.GetLength(1); col++)
{
result[row] += mat[row, col]*vec[col];
}
}
return new Vector4
{
X = result[0],
Y = result[1],
Z = result[2],
W = result[3]
};
}
}
}

I don't know if this will help but I do something along those lines in one of my projects. I don't use matrices though so it may not be what you are looking for. I simply store the extents of coordinates for the graph and the container (canvas) width and height. I then provide two extensions functions:
public static System.Windows.Point ConvertToScreen(this System.Windows.Point point, CartesianExtents2D extents, double containerWidth, double containerHeight)
{
var x = (point.X - extents.XMinimum) * containerWidth / (extents.XMaximum - extents.XMinimum);
var y = (extents.YMaximum - point.Y) * containerHeight / (extents.YMaximum - extents.YMinimum);
return new System.Windows.Point(x, y);
}
public static System.Windows.Point ConvertToReal(this System.Windows.Point point, CartesianExtents2D extents, double containerWidth, double containerHeight, )
{
var x = extents.XMinimum + (point.X * (extents.XMaximum - extents.XMinimum)) / containerWidth;
var y = extents.YMaximum - (point.Y * (extents.YMaximum - extents.YMinimum)) / containerHeight;
return new System.Windows.Point(x, y);
}
Call thus:
Point p = new Point();
p.ConvertToReal(...);
I'm hoping the contents of CartesianExtents2D is obvious - just min and max for x and y

Mandelbrot Conversion [closed]

Closed. This question needs debugging details. It is not currently accepting answers.
Edit the question to include desired behavior, a specific problem or error, and the shortest code necessary to reproduce the problem. This will help others answer the question.
Closed 8 years ago.
Improve this question
I'm receiving no errors but when I'm running it I'm also unable to see the Mandelbrot it just displays the grey box, I'm currently stuck at this one point thanks for any help, if you see any other parts of my code which contains grammar or coding errors it would be much appreciated if you told me.
using System;
using System.Collections.Generic;
using System.ComponentModel;
using System.Data;
using System.Drawing;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows.Forms;
using System.Drawing.Drawing2D;
namespace SE_Fractal_Assignment
{
public partial class Form1 : Form
{
public Form1()
{
InitializeComponent();
}
public struct HSBColor
{
float h;
float s;
float b;
int a;
public HSBColor(float h, float s, float b)
{
this.a = 0xff;
this.h = Math.Min(Math.Max(h, 0), 255);
this.s = Math.Min(Math.Max(h, 0), 255);
this.b = Math.Min(Math.Max(h, 0), 255);
}
public float H
{
get { return h; }
}
public float S
{
get { return s; }
}
public float B
{
get { return b; }
}
public int A
{
get { return a; }
}
public Color Color
{
get
{
return FromHSB(this);
}
}
public static Color FromHSB(HSBColor hsbColor)
{
float r = hsbColor.b;
float g = hsbColor.b;
float b = hsbColor.b;
if (hsbColor.s != 0)
{
float max = hsbColor.b;
float dif = hsbColor.b * hsbColor.s / 255f;
float min = hsbColor.b - dif;
float h = hsbColor.h * 360f / 255f;
if (h < 60f)
{
r = max;
g = h * dif / 60f + min;
b = min;
}
else if (h < 120f)
{
r = -(h - 120f) * dif / 60f + min;
g = max;
b = min;
}
else if (h < 180f)
{
r = min;
g = max;
b = -(h - 120f) * dif / 60f + min;
}
else if (h < 240f)
{
r = min;
g = -(h - 240f) * dif / 60f + min;
b = max;
}
else if (h < 300f)
{
r = -(h - 240f) * dif / 60f + min;
g = min;
b = max;
}
else if (h <= 360f)
{
r = max;
g = min;
b = -(h - 360f) * dif / 60f + min;
}
else
{
r = 0;
g = 0;
b = 0;
}
}
return Color.FromArgb
(
hsbColor.a,
(int)Math.Round(Math.Min(Math.Max(r, 0), 255)),
(int)Math.Round(Math.Min(Math.Max(g, 0), 255)),
(int)Math.Round(Math.Min(Math.Max(b, 0), 255))
);
}
}
private const int MAX = 256; // max iterations
private const double SX = -2.025; // start value goal
private const double SY = -1.125; // start value imaginary
private const double EX = 0.6; // end value real
private const double EY = 1.125; // end value imaginary
private static int x1, y1, xs, ys, xe, ye;
private static double xstart, ystart, xende, yende, xzoom, yzoom;
private static bool action, rectangle, finished;
private static float xy;
//private Image picture1;
private System.Drawing.Bitmap bitmap;
private Graphics g1;
private Cursor c1, c2;
private HSBColor HSBcol = new HSBColor();
// private HSB HSBcol = new HSB();
private void Form1_Paint(object sender, PaintEventArgs e)
{
g1 = e.Graphics;
g1.DrawImage(bitmap, 0, 0, x1, y1);
g1.Dispose();
}
private void Form1_Load(object sender, EventArgs e)
{
init();
start();
}
public void init()
{
//HSBcol = new HSB();
finished = false;
c1 = Cursors.WaitCursor;
c2 = Cursors.Cross;
x1 = 640;
y1 = 480;
xy = (float)x1 / (float)y1;
bitmap.SetPixel(x1, y1, Color.Blue);
g1 = Graphics.FromImage(bitmap);
finished = true;
// xy = (float)x1 / (float)y1;
//picture = createImage(x1, y1);
//g1 = picture.getGraphics();
}
public void destroy() // delete all instances
{
if (finished)
{
//removeMouseListener(this);
//removeMouseMotionListener(this);
//bitmap = null;
g1 = null;
c1 = null;
c2 = null;
//System.gc(); // garbage collection
GC.Collect();
}
}
public void start()
{
action = false;
rectangle = false;
initvalues();
xzoom = (xende - xstart) / (double)x1;
yzoom = (yende - ystart) / (double)y1;
mandelbrot();
}
public void stop()
{
}
public void paint(Graphics g)
{
update(g);
}
public void update(Graphics g)
{
/* Pen myPen = new Pen(Color.White);
g.DrawImage(bitmap, 0, 0);
if (rectangle)
{
if (xs < xe)
{
if (ys < ye)
{
g.DrawRectangle(myPen, xs, ys, (xe - xs), (ye - ys));
}
}
else
{
g.DrawRectangle(myPen, xs, ys, (xe - xs), (ye - ys));
}
myPen.Dispose();
}*/
}
private void mandelbrot() // calculate all points
{
int x, y;
float h, b, alt = 0.0f;
action = false;
for (x = 0; x < x1; x += 2)
for (y = 0; y < y1; y++)
{
h = pointcolour(xstart + xzoom * (double)x, ystart + yzoom * (double)y);
// color value
if (h != alt)
{
b = 1.0f - h * h; // brightnes
///djm added
///HSBcol.fromHSB(h,0.8f,b);
///
//convert hsb to rgb then make a Java Color
Color color = HSBColor.FromHSB(new HSBColor(h * 255, 0.8f * 255, b * 255));
///g1.setColor(col);
//djm end
//djm added to convert to RGB from HSB
//g1.setColor(Color.getHSBColor(h, 0.8f, b));
//djm test
// Color col = Color.FromArgb(0, 0, 0, 0);
//red = Color.Red;
// green = Color.Green;
// blue = Color.Blue;
//djm
alt = h;
}
Pen pen = new Pen(Color.Aqua);
g1.DrawLine(pen, x, y, x + 1, y);
}
//showStatus("Mandelbrot-Set ready - please select zoom area with pressed mouse.");
//setCursor(c2);
action = true;
}
private float pointcolour(double xwert, double ywert)
// color value from 0.0 to 1.0 by iterations
{
double r = 0.0, i = 0.0, m = 0.0;
int j = 0;
while ((j < MAX) && (m < 4.0))
{
j++;
m = r * r - i * i;
i = 2.0 * r * i + ywert;
r = m + xwert;
}
return (float)j / (float)MAX;
}
private void initvalues() // reset start values
{
xstart = SX;
ystart = SY;
xende = EX;
yende = EY;
if ((float)((xende - xstart) / (yende - ystart)) != xy)
xstart = xende - (yende - ystart) * (double)xy;
}
private void Form1_paint(object sender, PaintEventArgs e)
{
Graphics g = e.Graphics;
g1 = g;
action = false;
rectangle = false;
initvalues();
xzoom = (xende - xstart) / (double)x1;
yzoom = (yende - ystart) / (double)y1;
//picture = g.DrawImage;
//g.DrawImage(picture,0,0);
update(g);
mandelbrot();
}
private void Form1_MouseDown(object sender, MouseEventArgs e)
{
if (action)
{
xs = e.X;
ys = e.Y;
}
}
private void Form1_MouseMove(object sender, MouseEventArgs e)
{
// e.consume();
if (action)
{
xe = e.X;
ye = e.Y;
rectangle = true;
//repaint();
}
}
private void Form1_MouseUp(object sender, MouseEventArgs e)
{
rectangle = false;
}
private void Form1_Click(object sender, MouseEventArgs e)
{
}
public String getAppletInfo()
{
return "fractal.class - Mandelbrot Set a Java Applet by Eckhard Roessel 2000-2001";
}
}
}

Honestly, the code is so cluttered and disorganized, it's hard to know all of what might be wrong with it. Sorry to be so blunt.
That said, a couple of obvious problems I see involving your "g1" Graphics instance member.
First, you are using the same field for two purposes: when computing the original image, you expect this to be a Graphics instance you can use to draw into your bitmap. But in the Paint event, you set it to the Graphics instance for the window, into which the painting should be done.
Second, in that Paint event, you dispose the Graphics instance before you return. But the instance you're disposing isn't yours. It belongs to the Forms system, and the only thing you should be doing with it is drawing into it.
There actually appear to be two different Paint event handlers and it's not clear which one you're using. You only dispose the Graphics instance in one of those places, so that may or may not be the real problem.
Personally, I would break the problem down into different elements. For a relative novice, it can be hard enough just to correctly draw a bitmap. It can also be difficult to really grasp how Paint event handling should be done. And of course, there's the Mandelbrot computations themselves. Trying to implement all three things (and more) at the same time can be overwhelming, and will take a lot longer assuming you can figure it out at all.
I would start by writing a simple program that just has a single PictureBox, which when you click a button, your program creates a new Bitmap object, into which you draw something simple (say, a rectangle, circle, or maybe just some text) and then assigns that Bitmap object to the PictureBox.Image property.
Once you have that working, then you can change the drawing part of the code to draw a Mandelbrot image instead.
Finally, once you have that working, then you can work on using the Paint event to draw the bitmap into your window directly instead of using the PictureBox control (the main reason for wanting to do this would presumably be that you eventually want to update the image as it's being drawn...if you only want to show it at the very end, then IMHO the PictureBox is a better approach).

Random elements (coordinates)

I have to do something like this.
When I click on a node, it expands, and this is OK (I am using Powercharts to do it).
My big problem is creating random coordinates so that when I open the subnode, it doesn't overlap with another node/subnode.
In the Powercharts I have to pass the coordinates, so the big problem is in passing it.
I have to do the random coordinates in C#.
//-------------------------------------------------------
This is what i did so far:
This is what i do, is not overlaping, but i have a problem.
how can i start do the circles from a starting point?
for example, starts in the middle (300,300) and then do circles around it. Is possible?
private void button2_Click(object sender, EventArgs e)
{
g = pictureBox1.CreateGraphics();
g.Clear(pictureBox1.BackColor);
double angle;
Circle item0 = new Circle();
item0.x=200;
item0.y=150;
item0.r=50;
listaCirculos.Add(item0);
Random randomMember = new Random();
g.DrawEllipse(pen1, 200, 150, 50, 50);
while(listaCirculos.Count!=11)
{
int[] it = GenerateNewCircle(600);
Circle item = new Circle();
item.x = it[0];
item.y = it[1];
item.r = 50;
if (circleIsAllowed(listaCirculos, item))
{
listaCirculos.Add(item);
g.DrawEllipse(pen1, Convert.ToInt32(item.x), Convert.ToInt32(item.y), 50, 50);
}
}
}
bool circleIsAllowed(List<Circle> circles, Circle newCircle)
{
foreach(Circle it in circles)
{
//double sumR = it.x + newCircle.r;
//double dx = it.x - newCircle.x;
//double dy = it.y - newCircle.y;
//double squaredDist = dx * dx + dy * dy;
double aX = Math.Pow(it.x - newCircle.x, 2);
double aY = Math.Pow(it.y - newCircle.y, 2);
double Dif = Math.Abs(aX - aY);
double ra1 = it.r / 2;
double ra2 = it.r / 2;
double raDif = Math.Pow(ra1 + ra2, 2);
if ((raDif + 1) > Dif) return false;
//if (squaredDist < sumR*sumR) return false;
}
return true; // no existing circle overlaps
}
public int[] GenerateNewCircle(int maxSize)
{
int x, y;
Random randomMember = new Random();
x = randomMember.Next(0,maxSize);
if (x - 50 < 0)
y = randomMember.Next(x + 50, maxSize);
else if (x + 50 > 600)
y = randomMember.Next(0, x - 50);
else
// in this case, x splits the range 0..n into 2 subranges.
// get a random number and skip the "gap" if necessary
y = randomMember.Next(0, maxSize - 50);
if (y > x - 50)
{
y += 20;
}
int[] abc = new int[2];
abc[0] = x;
abc[1] = y;
return abc;
}

Size sizeShape = new Size("SomeWidth" , "SomeHeight");
List<Retangle> rects = new List<Retangle>();
Random r = new Random();
while(rects.count != "someCount")
{
Point rPoint = new Point(r.Next(500) , r.Next(500))
bool isNew = true;
foreach(Rectangle r in rects)
{
if(r.contains(rPoint))
isNew = false;
}
if(isNew)
rects.Add(GetRect(rPoint , sizeShape));
}
private Rectangle GetRect(Point p , Size s)
{
return new Rectangle(p,s);
}
After than you can use from rects ,
rects has random coordinates.

Here's the link to let you know how to create Random Numbers in C#.
You have to keep in mind that you will generate random numbers, but you also have to make sure the numbers generated will not make objects overlap.

Despite this being pretty poorly worded, I believe what you are looking for is the Random class. It is instantiated as such:
Random thisRandom = new Random();
Then from there, if you want to generate a random coordinate, you need to know the maximum possible X and Y coordinates. Knowing these will allow you to generate X and Y coordinates within the given canvas as such:
int maxX = 500; //This is the maximum X value on your canvas
int maxY = 500; //This is the maximum Y value on your canvas
int newX, newY;
newX = thisRandom.Next(maxX);
newY = thisRandom.Next(maxY);
While its not the absolute best in terms of "true" randomization this should give you what you need.