// Vector from FromP to ToP
function TForm1.Vektor(FromP, : TPoint): TPoint;
 begin
  Result.x := .x - FromP.x;
  Result.y := .y - FromP.y;
 end;
 // neue x Komponente des Verktors
// new x-component of the vector
function TForm1.xComp(Vektor: TPoint; Angle: Extended): Integer;
 begin
  Result := Round(Vektor.x * cos(Angle) - (Vektor.y) * sin(Angle));
 end;
 // neue Y-Komponente des Vektors
// new y-component of the vector
function TForm1.yComp(Vektor: TPoint; Angle: Extended): Integer;
 begin
  Result := Round((Vektor.x) * (sin(Angle)) + (vektor.y) * cos(Angle));
 end;

 function TForm1.RotImage(srcbit: TBitmap; Angle: Extended; FPoint: TPoint;
  Background: TColor): TBitmap;
 {
 srcbit: TBitmap; // Bitmap dass gedreht werden soll ; Bitmap to be rotated
 Angle: extended; // Winkel in Bogenma?, angle
 FPoint: TPoint; // Punkt um den gedreht wird ; Point to be rotated around
 Background: TColor): TBitmap; // Hintergrundfarbe des neuen Bitmaps ;
  // Backgroundcolor of the new bitmap
}
 var
  highest, lowest, mostleft, mostright: TPoint;
  topoverh, leftoverh: integer;
  x, y, newx, newy: integer;
 begin
  Result := TBitmap.Create;
  // Drehwinkel runterrechnen auf eine Umdrehung, wenn notig
 // Calculate angle down on one rotation, if necessary
 while Angle >= (2 * pi) do
  begin
  angle := Angle - (2 * pi);
  end;
  // neue Ausma?e festlegen
 // specify new size
 if (angle <= (pi / 2)) then
  begin
  highest := Point(0,0); //OL
  Lowest := Point(Srcbit.Width, Srcbit.Height); //UR
  mostleft := Point(0,Srcbit.Height); //UL
  mostright := Point(Srcbit.Width, 0); //OR
 end
  else if (angle <= pi) then
  begin
  highest := Point(0,Srcbit.Height);
  Lowest := Point(Srcbit.Width, 0);
  mostleft := Point(Srcbit.Width, Srcbit.Height);
  mostright := Point(0,0);
  end
  else if (Angle <= (pi * 3 / 2)) then
  begin
  highest := Point(Srcbit.Width, Srcbit.Height);
  Lowest := Point(0,0);
  mostleft := Point(Srcbit.Width, 0);
  mostright := Point(0,Srcbit.Height);
  end
  else
  begin
  highest := Point(Srcbit.Width, 0);
  Lowest := Point(0,Srcbit.Height);
  mostleft := Point(0,0);
  mostright := Point(Srcbit.Width, Srcbit.Height);
  end;
  topoverh := yComp(Vektor(FPoint, highest), Angle);
  leftoverh := xComp(Vektor(FPoint, mostleft), Angle);
  Result.Height := Abs(yComp(Vektor(FPoint, lowest), Angle)) + Abs(topOverh);
  Result.Width := Abs(xComp(Vektor(FPoint, mostright), Angle)) + Abs(leftoverh);
  // Verschiebung des FPoint im neuen Bild gegenuber srcbit
 // change of FPoint in the new picture in relation on srcbit
 overh := Overh + FPoint.y;
  Leftoverh := LeftOverh + FPoint.x;
  // erstmal mit Hintergrundfarbe fullen
 // at first fill with background color
 Result.Canvas.Brush.Color := Background;
  Result.Canvas.pen.Color := background;
  Result.Canvas.Fillrect(Rect(0,0,Result.Width, Result.Height));
  // Start des eigentlichen Rotierens
 // Start of actual rotation
 for y := 0 to srcbit.Height - 1 do
  begin  // Zeilen ; Rows
  for x := 0 to srcbit.Width - 1 do
  begin  // Spalten ; Columns
  newX := xComp(Vektor(FPoint, Point(x, y)), Angle);
  newY := yComp(Vektor(FPoint, Point(x, y)), Angle);
  newX := FPoint.x + newx - leftoverh;
  // Verschieben wegen der neuen Ausma?e
  newy := FPoint.y + newy - topoverh;
  // Move beacause of new size
  Result.Canvas.Pixels[newx, newy] := srcbit.Canvas.Pixels[x, y];
  // auch das Pixel daneben fullen um Leerpixel bei Drehungen zu verhindern
  // also fil lthe pixel beside to prevent empty pixels
  if ((angle < (pi / 2)) or
  ((angle > pi) and
  (angle < (pi * 3 / 2)))) then
  begin
  Result.Canvas.Pixels[newx, newy + 1] := srcbit.Canvas.Pixels[x, y];
  end
  else
  begin
  Result.Canvas.Pixels[newx + 1,newy] := srcbit.Canvas.Pixels[x, y];
  end;
  end;
  end;
 end;

 procedure TForm1.Button1Click(Sender: TObject);
 var
  mybitmap, newbit: TBitMap;
 begin
  if OpenDialog1.Execute then
  begin
  mybitmap := TBitmap.Create;
  mybitmap.LoadFromFile(OpenDialog1.FileName);
  newbit := RotImage(mybitmap, DegToRad(45),
  Point(mybitmap.Width div 2, mybitmap.Height div 2), clBlack);
  Image1.Canvas.Draw(0,0, newBit);
  end;
 end;
 end;