Minkowski Island: Fill color/back color Problems

[Boleeman]

Created a Minkowski Island fractal

Read about it here https://mathworld.wolfram.com/MinkowskiSausage.html 
and here https://en.wikipedia.org/wiki/Minkowski_sausage


It's basically a square Quadratic Koch curve.

Can't figure out why its messing up at recursion levels greater than 3.
Thought it may be a round of error, seeings as it happens at higher recursion level but cant trace it.
Noticed that when I resize the form at level 4 that gap changes. So perhaps its a scaling problem.

Some Graphic GURU advice is needed.

[KodeZwerg]

By watching pictures, the left image does do those strokes to compensate the missing space to paint full element so my guess is that at initial the value for length for each stroke is calculated wrong.

[Martin_fr]

Maybe rounding errors. E.g. you keep summing up floats, and compare against x. But the floats will never reach x.

For example

Code: Pascal  [Select][+][-]

1. program Project1;
2. var
3.   a: double;
4. begin
5.   a := 2.0;
6.   while a <> 0 do begin
7.     a := a - 2 / 5;
8.     writeln(a);
9.   end;
10. end.

will continue running. Because 2/5 is rounded in binary, and not exactly 0.4

So, maybe something like that?

Use "SameValue()" from unit math.

[tetrastes]

Try this

Code: Pascal  [Select][+][-]

1. unit Unit1;
2.  
3. {$mode objfpc}{$H+}
4.  
5. interface
6.  
7. uses
8.   Classes, SysUtils, Forms, Controls, Graphics, Dialogs, ExtCtrls, Buttons,
9.   StdCtrls, Spin, Math;
10.  
11. type
12.  
13.   TPointSingle = record
14.     X: single;
15.     Y: single;
16.   end;
17.  
18.   { TForm1 }
19.  
20.   TForm1 = class(TForm)
21.     Label1: TLabel;
22.     Label2: TLabel;
23.     PaintBox1: TPaintBox;
24.     Panel1: TPanel;
25.     seDepth: TSpinEdit;
26.     seLinewidth: TSpinEdit;
27.     procedure PaintBox1Paint(Sender: TObject);
28.     procedure seDepthChange(Sender: TObject);
29.     procedure seLinewidthChange(Sender: TObject);
30.   private
31.    Initiator: array of TPointSingle;
32.    ScaleFactor: Single;
33.    GeneratorDTheta: array of Single;
34.    procedure DrawSnowflake(ACanvas: TCanvas; depth: Integer);
35.    procedure DrawSnowflakeEdge(ACanvas: TCanvas; depth: Integer; var p1: TPointSingle;
36.   theta: Single; dist: Single);
37.   public
38.  
39.   end;
40.  
41. var
42.   Form1: TForm1;
43.  
44. implementation
45.  
46. {$R *.lfm}
47.  
48. procedure TForm1.DrawSnowflakeEdge(ACanvas: TCanvas; depth: Integer; var p1: TPointSingle;
49.   theta: Single; dist: Single);
50. var
51.   i: Integer;
52.   p2: TPointSingle;
53.   ip1, ip2: TPoint;
54. begin
55.   if depth = 0 then
56.   begin
57.     p2.X := p1.X + (dist * Cos(theta + Pi / 2));
58.     p2.Y := p1.Y + (dist * Sin(theta + Pi / 2));
59.  
60.     ip1.X := Trunc(p1.X + 0.5);      // Because  "banker's rounding" is not what I was taught in University
61.     ip1.Y := Trunc(p1.Y + 0.5);
62.     ip2.X := Trunc(p2.X + 0.5);
63.     ip2.Y := Trunc(p2.Y + 0.5);
64.     ACanvas.Line(ip1, ip2);
65.     p1 := p2;
66.     Exit;
67.   end;
68.  
69.   dist := dist / Sqrt(5.0);
70.   for i := 0 to High(GeneratorDTheta) do
71.   begin
72.     theta := theta + GeneratorDTheta[i];
73.     DrawSnowflakeEdge(ACanvas, depth - 1, p1, theta, dist);
74.   end;
75. end;
76.  
77. procedure TForm1.DrawSnowflake(ACanvas: TCanvas; depth: Integer);
78. var
79.   i: Integer;
80.   p1, p2: TPointSingle;
81.   dx, dy, length, theta: Single;
82. begin
83.   Canvas.Clear;
84.  
85.   for i := 1 to High(Initiator) do
86.   begin
87.     p1 := Initiator[i - 1];
88.     p2 := Initiator[i];
89.  
90.     dx := p2.X - p1.X;
91.     dy := p2.Y - p1.Y;
92.     length := Sqrt(dx * dx + dy * dy);
93.     theta := ArcTan2(dy, dx);
94.     DrawSnowflakeEdge(ACanvas, depth, p1, theta, length);
95.   end;
96. end;
97.  
98. procedure TForm1.PaintBox1Paint(Sender: TObject);
99. var
100.   MinDimension: Integer;
101. begin
102.  
103.   SetLength(Initiator, 0);
104.   SetLength(GeneratorDTheta, 0);
105.  
106.   // Ensure Initiator array is properly initialized
107.   if Length(Initiator) < 5 then
108.     SetLength(Initiator, 5);
109.  
110.  
111.   MinDimension := Min(PaintBox1.Width, PaintBox1.Height);
112.   MinDimension := MinDimension - 300;
113.  
114.   if Length(Initiator) >= 5 then // Ensure Initiator array has enough elements
115.   begin
116.     {Initiator[0] := Point(Round((PaintBox1.Width - MinDimension) / 2), Round((PaintBox1.Height - MinDimension) / 2));
117.     Initiator[1] := Point(Round((PaintBox1.Width + MinDimension) / 2), Round((PaintBox1.Height - MinDimension) / 2));
118.     Initiator[2] := Point(Round((PaintBox1.Width + MinDimension) / 2), Round((PaintBox1.Height + MinDimension) / 2));
119.     Initiator[3] := Point(Round((PaintBox1.Width - MinDimension) / 2), Round((PaintBox1.Height + MinDimension) / 2));
120.     Initiator[4] := Initiator[0];}
121.     Initiator[0].X := (PaintBox1.Width - MinDimension) / 2;
122.     Initiator[0].Y := (PaintBox1.Height - MinDimension) / 2;
123.     Initiator[1].X := (PaintBox1.Width + MinDimension) / 2;
124.     Initiator[1].Y := (PaintBox1.Height - MinDimension) / 2;
125.     Initiator[2].X := (PaintBox1.Width + MinDimension) / 2;
126.     Initiator[2].Y := (PaintBox1.Height + MinDimension) / 2;
127.     Initiator[3].X := (PaintBox1.Width - MinDimension) / 2;
128.     Initiator[3].Y := (PaintBox1.Height + MinDimension) / 2;
129.     Initiator[4] := Initiator[0];
130.   end;
131.  
132.   ScaleFactor := 1 / Sqrt(5);
133.   GeneratorDTheta := nil;
134.   SetLength(GeneratorDTheta, 3);
135.   GeneratorDTheta[0] := -ArcTan(1 / 2);
136.   GeneratorDTheta[1] := Pi / 2;
137.   GeneratorDTheta[2] := -Pi / 2;
138.  
139.   // Redraw the snowflake with the adjusted level
140.   DrawSnowflake(PaintBox1.Canvas, seDepth.Value);
141. end;
142.  
143. procedure TForm1.seDepthChange(Sender: TObject);
144. begin
145.   PaintBox1.Invalidate;
146. end;
147.  
148. procedure TForm1.seLinewidthChange(Sender: TObject);
149. begin
150.   PaintBox1.Canvas.Pen.Width := seLinewidth.Value;
151.   PaintBox1.Invalidate;
152. end;
153.  
154. end.

[Boleeman]

Thanks tetrastes.

Looks like the issue was using point and rounding.
I looked at breaking down into x and y point locations, but I was still rounding to get rid of extended value error.

Using Initiator[0].X  and Initiator[0].Y type syntax without the rounding was what I was completely missing.

Also was good to know about:
    ip1.X := Trunc(p1.X + 0.5);      // Because  "banker's rounding" is not what I was taught in University
    ip1.Y := Trunc(p1.Y + 0.5);
    ip2.X := Trunc(p2.X + 0.5);
    ip2.Y := Trunc(p2.Y + 0.5);

Noticed now when resizing the form, the fractal resizes well.

Thanks once again for your great help.

[Boleeman]

I tried setting the fill color of the fractal with a TColorbutton called cbFillcolor and the backcolor of Paintabox1 with a TColorbutton called cbBackcolor but I could not get that to work.

Usually its simple to do for a TPaintbox, but don't know why it did not change colors.

Here is the complete code:

Code: Pascal  [Select][+][-]

1.  unit Unit1;
2.  
3. {$mode objfpc}{$H+}
4.  
5. interface
6.  
7. uses
8.   Classes, SysUtils, Forms, Controls, Graphics, Dialogs, ExtCtrls, Buttons,
9.   StdCtrls, Spin, Math;
10.  
11. type
12.  
13.   TPointSingle = record
14.     X: single;
15.     Y: single;
16.   end;
17.  
18.   { TForm1 }
19.  
20.   TForm1 = class(TForm)
21.     cbLinecolor: TColorButton;
22.     cbBackcolor: TColorButton;
23.     cbFillcolor: TColorButton;
24.     Label1: TLabel;
25.     Label2: TLabel;
26.     Label3: TLabel;
27.     Label4: TLabel;
28.     Label5: TLabel;
29.     PaintBox1: TPaintBox;
30.     Panel1: TPanel;
31.     seDepth: TSpinEdit;
32.     seLinewidth: TSpinEdit;
33.     procedure cbFillcolorColorChanged(Sender: TObject);
34.     procedure cbLinecolorColorChanged(Sender: TObject);
35.     procedure cbBackcolorColorChanged(Sender: TObject);
36.     procedure PaintBox1Paint(Sender: TObject);
37.     procedure seDepthChange(Sender: TObject);
38.     procedure seLinewidthChange(Sender: TObject);
39.   private
40.    Initiator: array of TPointSingle;
41.    ScaleFactor: Single;
42.    GeneratorDTheta: array of Single;
43.    procedure DrawSnowflake(ACanvas: TCanvas; depth: Integer);
44.    procedure DrawSnowflakeEdge(ACanvas: TCanvas; depth: Integer; var p1: TPointSingle;
45.   theta: Single; dist: Single);
46.    procedure FillSnowflake(ACanvas: TCanvas; CenterPoint: TPoint; depth: Integer; p1: TPointSingle;
47.   theta: Single; dist: Single);
48.   public
49.  
50.   end;
51.  
52. var
53.   Form1: TForm1;
54.  
55. implementation
56.  
57. {$R *.lfm}
58.  
59. procedure TForm1.FillSnowflake(ACanvas: TCanvas; CenterPoint: TPoint; depth: Integer; p1: TPointSingle;
60.   theta: Single; dist: Single);
61. var
62.   i: Integer;
63.   p2: TPointSingle;
64.   Points: array of TPoint;
65. begin
66.   if depth = 0 then
67.     Exit;
68.  
69.   SetLength(Points, 5);
70.  
71.   for i := 1 to High(Initiator) do
72.   begin
73.     p1 := Initiator[i - 1];
74.     p2 := Initiator[i];
75.  
76.     Points[0] := CenterPoint;
77.     Points[1].X := Round(p1.X);
78.     Points[1].Y := Round(p1.Y);
79.     Points[2].X := Round(p2.X);
80.     Points[2].Y := Round(p2.Y);
81.     Points[3] := CenterPoint;
82.     Points[4] := Points[1];
83.  
84.     ACanvas.Polygon(Points);
85.   end;
86. end;
87.  
88. procedure TForm1.DrawSnowflakeEdge(ACanvas: TCanvas; depth: Integer; var p1: TPointSingle;
89.   theta: Single; dist: Single);
90. var
91.   i: Integer;
92.   p2: TPointSingle;
93.   ip1, ip2: TPoint;
94. begin
95.   if depth = 0 then
96.   begin
97.     p2.X := p1.X + (dist * Cos(theta + Pi / 2));
98.     p2.Y := p1.Y + (dist * Sin(theta + Pi / 2));
99.  
100.     ip1.X := Trunc(p1.X + 0.5);      // Because  "banker's rounding" is not what I was taught in University
101.     ip1.Y := Trunc(p1.Y + 0.5);
102.     ip2.X := Trunc(p2.X + 0.5);
103.     ip2.Y := Trunc(p2.Y + 0.5);
104.     ACanvas.Line(ip1, ip2);
105.     p1 := p2;
106.     Exit;
107.   end;
108.  
109.   dist := dist / Sqrt(5.0);
110.   for i := 0 to High(GeneratorDTheta) do
111.   begin
112.     theta := theta + GeneratorDTheta[i];
113.     DrawSnowflakeEdge(ACanvas, depth - 1, p1, theta, dist);
114.   end;
115. end;
116.  
117. procedure TForm1.DrawSnowflake(ACanvas: TCanvas; depth: Integer);
118. var
119.   i: Integer;
120.   p1, p2: TPointSingle;
121.   dx, dy, length, theta: Single;
122. begin
123.   ACanvas.Pen.Color := cbLinecolor.ButtonColor;
124.  
125.   for i := 1 to High(Initiator) do
126.   begin
127.     p1 := Initiator[i - 1];
128.     p2 := Initiator[i];
129.  
130.     dx := p2.X - p1.X;
131.     dy := p2.Y - p1.Y;
132.     length := Sqrt(dx * dx + dy * dy);
133.     theta := ArcTan2(dy, dx);
134.     DrawSnowflakeEdge(ACanvas, depth, p1, theta, length);
135.   end;
136. end;
137.  
138. procedure TForm1.PaintBox1Paint(Sender: TObject);
139. var
140.   MinDimension: Integer;
141. begin
142.  
143.   SetLength(Initiator, 0);
144.   SetLength(GeneratorDTheta, 0);
145.  
146.   // Ensure Initiator array is properly initialized
147.   if Length(Initiator) < 5 then
148.     SetLength(Initiator, 5);
149.  
150.  
151.   MinDimension := Min(PaintBox1.Width, PaintBox1.Height);
152.   MinDimension := MinDimension - 300;
153.  
154.   if Length(Initiator) >= 5 then // Ensure Initiator array has enough elements
155.   begin
156.  
157.     Initiator[0].X := (PaintBox1.Width - MinDimension) / 2;
158.     Initiator[0].Y := (PaintBox1.Height - MinDimension) / 2;
159.     Initiator[1].X := (PaintBox1.Width + MinDimension) / 2;
160.     Initiator[1].Y := (PaintBox1.Height - MinDimension) / 2;
161.     Initiator[2].X := (PaintBox1.Width + MinDimension) / 2;
162.     Initiator[2].Y := (PaintBox1.Height + MinDimension) / 2;
163.     Initiator[3].X := (PaintBox1.Width - MinDimension) / 2;
164.     Initiator[3].Y := (PaintBox1.Height + MinDimension) / 2;
165.     Initiator[4] := Initiator[0];
166.   end;
167.  
168.   ScaleFactor := 1 / Sqrt(5);
169.   GeneratorDTheta := nil;
170.   SetLength(GeneratorDTheta, 3);
171.   GeneratorDTheta[0] := -ArcTan(1 / 2);
172.   GeneratorDTheta[1] := Pi / 2;
173.   GeneratorDTheta[2] := -Pi / 2;
174.  
175.   DrawSnowflake(PaintBox1.Canvas, seDepth.Value);
176. end;
177.  
178. procedure TForm1.cbLinecolorColorChanged(Sender: TObject);
179. begin
180. PaintBox1.Canvas.Pen.Color := cbLinecolor.ButtonColor;
181. PaintBox1.Invalidate;
182. end;
183.  
184. procedure TForm1.cbFillcolorColorChanged(Sender: TObject);
185. begin
186.  PaintBox1.Invalidate;
187. end;
188.  
189. procedure TForm1.cbBackcolorColorChanged(Sender: TObject);
190. begin
191.   PaintBox1.Invalidate;
192. end;
193.  
194. procedure TForm1.seDepthChange(Sender: TObject);
195. begin
196.   PaintBox1.Invalidate;
197. end;
198.  
199. procedure TForm1.seLinewidthChange(Sender: TObject);
200. begin
201.   PaintBox1.Canvas.Pen.Width := seLinewidth.Value;
202.   PaintBox1.Invalidate;
203. end;
204.  
205. end.

I could change the line color of the fractal.

[tetrastes]

You call FillSnowflake nowhere, and don't use cbFillcolor.ButtonColor and cbBackcolor.ButtonColor.

[Boleeman]

I made fillsnowflake like this:

Code: Pascal  [Select][+][-]

1. procedure TForm1.FillSnowflake(ACanvas: TCanvas; CenterPoint: TPoint; depth: Integer; p1: TPointSingle;
2.   theta: Single; dist: Single);
3. var
4.   i: Integer;
5.   p2: TPointSingle;
6.   Points: array of TPoint;
7. begin
8.   if depth = 0 then
9.     Exit;
10.  
11.   SetLength(Points, 5);
12.  
13.   for i := 1 to High(Initiator) do
14.   begin
15.     p1 := Initiator[i - 1];
16.     p2 := Initiator[i];
17.  
18.     Points[0] := CenterPoint;
19.     Points[1].X := Round(p1.X);
20.     Points[1].Y := Round(p1.Y);
21.     Points[2].X := Round(p2.X);
22.     Points[2].Y := Round(p2.Y);
23.     Points[3] := CenterPoint;
24.     Points[4] := Points[1];
25.  
26.     ACanvas.Brush.Color := cbFillcolor.ButtonColor;
27.     ACanvas.Polygon(Points);
28.   end;
29. end;      

and then I called it in procedure TForm1.PaintBox1Paint(Sender: TObject);

Code: Pascal  [Select][+][-]

1. procedure TForm1.PaintBox1Paint(Sender: TObject);
2. var
3.   MinDimension: Integer;
4. begin
5.  
6.   SetLength(Initiator, 0);
7.   SetLength(GeneratorDTheta, 0);
8.  
9.   // Ensure Initiator array is properly initialized
10.   if Length(Initiator) < 5 then
11.     SetLength(Initiator, 5);
12.  
13.  
14.   MinDimension := Min(PaintBox1.Width, PaintBox1.Height);
15.   MinDimension := MinDimension - 300;
16.  
17.   if Length(Initiator) >= 5 then // Ensure Initiator array has enough elements
18.   begin
19.  
20.     Initiator[0].X := (PaintBox1.Width - MinDimension) / 2;
21.     Initiator[0].Y := (PaintBox1.Height - MinDimension) / 2;
22.     Initiator[1].X := (PaintBox1.Width + MinDimension) / 2;
23.     Initiator[1].Y := (PaintBox1.Height - MinDimension) / 2;
24.     Initiator[2].X := (PaintBox1.Width + MinDimension) / 2;
25.     Initiator[2].Y := (PaintBox1.Height + MinDimension) / 2;
26.     Initiator[3].X := (PaintBox1.Width - MinDimension) / 2;
27.     Initiator[3].Y := (PaintBox1.Height + MinDimension) / 2;
28.     Initiator[4] := Initiator[0];
29.   end;
30.  
31.   ScaleFactor := 1 / Sqrt(5);
32.   GeneratorDTheta := nil;
33.   SetLength(GeneratorDTheta, 3);
34.   GeneratorDTheta[0] := -ArcTan(1 / 2);
35.   GeneratorDTheta[1] := Pi / 2;
36.   GeneratorDTheta[2] := -Pi / 2;
37.  
38.   DrawSnowflake(PaintBox1.Canvas, seDepth.Value);
39.   FillSnowflake(PaintBox1.Canvas, Point(PaintBox1.Width div 2, PaintBox1.Height div 2), seDepth.Value, Initiator[0], 0, 0);
40. end;                        

I also tried calling it recursively:

Code: Pascal  [Select][+][-]

1. procedure TForm1.FillSnowflake(ACanvas: TCanvas; CenterPoint: TPoint; depth: Integer;
2.   p1: TPointSingle; theta: Single; dist: Single);
3. var
4.   i: Integer;
5.   p2: TPointSingle;
6.   Points: array of TPoint;
7. begin
8.   if depth = 0 then
9.     Exit;
10.  
11.   SetLength(Points, 5);
12.  
13.   for i := 1 to High(Initiator) do
14.   begin
15.     p1 := Initiator[i - 1];
16.     p2 := Initiator[i];
17.  
18.     Points[0] := CenterPoint;
19.     Points[1].X := Round(p1.X);
20.     Points[1].Y := Round(p1.Y);
21.     Points[2].X := Round(p2.X);
22.     Points[2].Y := Round(p2.Y);
23.     Points[3] := CenterPoint;
24.     Points[4] := Points[1];
25.  
26.     // Fill the region with the fill color
27.     ACanvas.Brush.Color := cbFillcolor.ButtonColor; // Use the color from cbFillcolor
28.     ACanvas.Polygon(Points);
29.  
30.     // Recursively fill the inner segments
31.     p1.X := CenterPoint.X + (p1.X - CenterPoint.X) * ScaleFactor;
32.     p1.Y := CenterPoint.Y + (p1.Y - CenterPoint.Y) * ScaleFactor;
33.     p2.X := CenterPoint.X + (p2.X - CenterPoint.X) * ScaleFactor;
34.     p2.Y := CenterPoint.Y + (p2.Y - CenterPoint.Y) * ScaleFactor;
35.     FillSnowflake(ACanvas, CenterPoint, depth - 1, p1, theta, dist);
36.   end;
37. end;

Still get 1 square filled only.