uses Math;
const
 RectSize = 5;
 MaxN = 128;
var
 n: integer = -1;
 pt: array[0..MaxN] of TPoint;
 C: array[0..MaxN] of single;
 bm: TBitMap;

function GetBinomialCoefficient(m, i: integer): single;
 function Factorial(x: integer): double;
 var
 i: integer;
 begin
 result := 1;
 for i := 2 to x do
 result := result * i;
 end;
begin
 result := Factorial(m) / (Factorial(i) * Factorial(m - i));
end;

procedure DrawBezier(Canvas: TCanvas; Count: integer);
type
 TPointArray = array[word] of TPoint;
 PPointArray = ^TPointArray;
var
 p: PPointArray;
 Step, qx, qy, t, q: single;
 i, j: integer;
begin
 GetMem(p, sizeof(TPoint) * (Count + 1));
 Step := 1.0 / Count;
 for i := 0 to Count do
 begin
 t := i * Step;
 qx := 0;
 qy := 0;
 for j := 0 to n do
 begin
 q := C[j] * IntPower(1 - t, j) * IntPower(t, n - j);
 qx := qx + q * pt[j].x;
 qy := qy + q * pt[j].y;
 end;
 p[i] := Point(round(qx), round(qy));
 end;
 Canvas.Polyline(Slice(p^, Count + 1));
 FreeMem(p);
end;

procedure DrawLines(canvas: TCanvas; const pt: array of TPoint);
var
 i: integer;
begin
 Canvas.Pen.Color := clGreen;
 Canvas.Pen.Width := 1;
 Canvas.MoveTo(pt[0].x, pt[0].y);
 for i := 0 to n do
 begin
 Canvas.Rectangle(Bounds(pt[i].x - RectSize, pt[i].y - RectSize,
 2 * RectSize, 2 * RectSize));
 Canvas.LineTo(pt[i].x, pt[i].y);
 end;
end;

procedure Redraw;
begin
 with Form1.PaintBox1 do
 begin
 bm.Canvas.FillRect(Bounds(0, 0, Width, Height));
 if Form1.CheckBox1.Checked then
 DrawLines(bm.Canvas, pt);
 bm.Canvas.PolyBezier(pt);
 bm.Canvas.Pen.Color := clRed;
 bm.Canvas.pen.Width := Form1.SpinEdit3.Value;
 DrawBezier(bm.Canvas, Form1.SpinEdit2.Value);
 Canvas.Draw(0, 0, bm);
 end;
end;
var
 moving: integer = -1;
 oldr: TRect;

procedure FillRandom(NewN: integer);
var
 i: integer;
begin
 randomize;
 for i := n + 1 to NewN do
 pt[i] := Point(random(Form1.PaintBox1.Width - 20) + 10,
 random(Form1.PaintBox1.Height - 20) + 10);
 n := NewN;
end;

procedure TForm1.FormCreate(Sender: TObject);
begin
 bm := TBitmap.Create;
 bm.Width := PaintBox1.Width;
 bm.Height := PaintBox1.height;
 SpinEdit1.MinValue := 1;
 SpinEdit1.MaxValue := MaxN;
 SpinEdit1.Value := 3;
 SpinEdit2.MinValue := 6;
 SpinEdit2.MaxValue := MaxN * 4;
 SpinEdit2.Value := 50;
 SpinEdit2.OnChange := PaintBox1.OnPaint;
 SpinEdit3.MinValue := 1;
 SpinEdit3.MaxValue := 8;
 SpinEdit3.Value := 3;
 SpinEdit3.OnChange := PaintBox1.OnPaint;
 CheckBox1.Checked := true;
 CheckBox1.OnClick := PaintBox1.OnPaint;
end;

procedure TForm1.PaintBox1MouseDown(Sender: TObject; Button: TMouseButton;
 Shift: TShiftState; X, Y: Integer);
var
 i: integer;
 r: TRect;
begin
 if Button <> mbLeft then
 Exit;
 for i := 0 to n do
 if (abs(X - pt[i].x) <= RectSize) and (abs(Y - pt[i].y) <= RectSize) then
 begin
 moving := i;
 r.TopLeft := Form1.ClientToScreen(PaintBox1.BoundsRect.TopLeft);
 r.BottomRight := Form1.ClientToScreen(PaintBox1.BoundsRect.BottomRight);
 GetClipCursor(oldr);
 ClipCursor(@r);
 Exit;
 end;
 if moving < 0 then
 begin
 SpinEdit1.Value := SpinEdit1.Value + 1;
 pt[n] := Point(X, Y);
 Redraw;
 end;
end;

procedure TForm1.PaintBox1MouseMove(Sender: TObject; Shift: TShiftState; X, Y:
 Integer);
begin
 if moving < 0 then
 Exit;
 pt[moving] := Point(X, Y);
 Redraw;
end;

procedure TForm1.PaintBox1MouseUp(Sender: TObject; Button: TMouseButton;
 Shift: TShiftState; X, Y: Integer);
begin
 if (Button = mbLeft) and (moving >= 0) then
 begin
 moving := -1;
 ClipCursor(@oldr);
 end;
end;

procedure TForm1.SpinEdit1Change(Sender: TObject);
var
 i: integer;
begin
 FillRandom(SpinEdit1.Value);
 SpinEdit2.MinValue := n * 2;
 for i := 0 to n do
 C[i] := GetBinomialCoefficient(n, i);
 Redraw;
end;

procedure TForm1.PaintBox1Paint(Sender: TObject);
begin
 Redraw;
end;