asked on

3D Cube

Hi,

I made a small program that draws a 3D cube. I am very pleased with the result. But now I want to rotate the Cube like a globe. Could you take a look at the code and tell me if it is easy to implement? You can download the project here:

http://home.quicknet.nl/qn/prive/mpoots/downloads/3D.zip

If 300 points is not enough just tell me.

Marcel

Or look here:

unit uObjects;

interface

uses
Graphics, Contnrs, Dialogs;

type

TVertice = class {Vertice is a corner of a polygon}
public
x,y,z: Integer;
constructor Create(aX, aY, aZ: Integer);
procedure Adjust(aX, aY, aZ : Integer);
end;

TPolygon = Class(TObjectList)
private
fCanvas : TCanvas;
function GetVertice(i: Integer): TVertice;
function Calc3DPoint(aPoint, aZ : Integer): Integer;
public
Constructor Create(aCanvas : TCanvas; aSquare, aKind : Integer);
property Vertices[i: Integer]: TVertice read GetVertice; default;
procedure Draw;
procedure DrawLine(aFrX, aFrY, aToX, aToY : Integer; aColor: TColor);
procedure AddjustCoords(aSquare : Integer);
end;

TBaseObject = Class(TObjectList)
private
fCanvas : TCanvas;
function GetPolygon(i: Integer): TPolygon;
public
fWidth : Integer;
fHeight : Integer;
Constructor Create(aCanvas : TCanvas; aWidth, aHeight : Integer);
property PolyGons[i: Integer]: TPolygon read GetPolygon; default;
procedure Initialise;
procedure DrawLine(aFrX, aFrY, aToX, aToY : Integer; aColor: TColor);
procedure DrawPolygons;
procedure AdjustCoords(aX, aY, aZ, aH: Integer);
end;

var
gX, gY, gZ, gH, CentreX, CentreY : Integer;

const
CUBE = 1;
TRIANGLE = 2;

implementation

uses
uMain, Classes;

{ TBaseObject }

procedure TBaseObject.AdjustCoords(aX, aY, aZ, aH: Integer);
begin
gX := aX;
gY := aY;
gZ := aZ;
gH := aH; // Horizon for perspective
Initialise;
end;

constructor TBaseObject.Create(aCanvas : TCanvas; aWidth, aHeight : Integer);
begin
gX := 50;
gY := 50;
gZ := 25;
gH := 50;
fCanvas := aCanvas;
fWidth := aWidth;
fHeight := aHeight;
Add(TPolygon.Create(aCanvas, 0, CUBE));
Add(TPolygon.Create(aCanvas, 1, CUBE));
Add(TPolygon.Create(aCanvas, 2, CUBE));
Add(TPolygon.Create(aCanvas, 3, CUBE));
Add(TPolygon.Create(aCanvas, 4, CUBE));
Add(TPolygon.Create(aCanvas, 5, CUBE));
end;

procedure TBaseObject.DrawLine(aFrX, aFrY, aToX, aToY: Integer; aColor: TColor);
begin
fCanvas.Pen.Color := aColor;
fCanvas.MoveTo(aFrX, aFrY);
fCanvas.LineTo(aToX, aToY);
end;

procedure TBaseObject.DrawPolygons;
var i : Integer;
begin
for i := 0 to Count - 1 do
PolyGons[i].Draw;
end;

function TBaseObject.GetPolygon(i: Integer): TPolygon;
begin
Result := TPolygon(Items[i]);
end;

procedure TBaseObject.Initialise;
var i : Integer;
begin
fCanvas.Pen.Color := clBlack;
frmMain.pbScreen.Canvas.Brush.Color := clBlack;
fCanvas.Rectangle(0, 0, fWidth, fheight);
CentreX := fWidth div 2;
CentreY := fHeight div 2;
if frmMain.cbAxis.Checked then
begin
DrawLine(fWidth div 2, 0, fWidth div 2, fHeight, clGreen);
DrawLine(0, fHeight div 2, fWidth, fHeight div 2, clGreen);
end;
for i := 0 to count - 1 do
Polygons[i].AddjustCoords(i);
DrawPolygons;
fCanvas.Refresh;
end;

{ TPolygon }

procedure TPolygon.AddjustCoords(aSquare : Integer);
begin
case aSquare of {Cube has 8 squares but you draw only 6}
0: begin
Vertices[0].Adjust(-gX, gY, gZ); //0
Vertices[1].Adjust(-gX, gY, -gZ); //1
Vertices[2].Adjust( gX, gY, -gZ); //2
Vertices[3].Adjust( gX, gY, gZ); //3
end;
1: begin
Vertices[0].Adjust(-gX, -gY, -gZ); //5
Vertices[1].Adjust( gX, -gY, -gZ); //6
Vertices[2].Adjust( gX, -gY, gZ); //7
Vertices[3].Adjust(-gX, -gY, gZ); //4
end;
2: begin
Vertices[0].Adjust(-gX, gY, gZ); //0
Vertices[1].Adjust(-gX, gY, -gZ); //1
Vertices[2].Adjust(-gX, -gY, -gZ); //5
Vertices[3].Adjust(-gX, -gY, gZ); //4
end;
3: begin
Vertices[0].Adjust(-gX, gY, -gZ); //1
Vertices[1].Adjust( gX, gY, -gZ); //2
Vertices[2].Adjust( gX, -gY, -gZ); //6
Vertices[3].Adjust(-gX, -gY, -gZ); //5
end;
4: begin
Vertices[0].Adjust( gX, gY, -gZ); //2
Vertices[1].Adjust( gX, gY, gZ); //3
Vertices[2].Adjust( gX, -gY, gZ); //7
Vertices[3].Adjust( gX, -gY, -gZ); //6
end;
5: begin
Vertices[0].Adjust(-gX, gY, gZ); //0
Vertices[1].Adjust( gX, gY, gZ); //3
Vertices[2].Adjust( gX, -gY, gZ); //7
Vertices[3].Adjust(-gX, -gY, gZ); //4
end;
end;
end;

function TPolygon.Calc3DPoint(aPoint, aZ: Integer): Integer;
begin
Result := Round(aPoint * ((aZ + gH) / gH));
end;

constructor TPolygon.Create(aCanvas: TCanvas; aSquare, aKind : Integer);
begin
fCanvas := aCanvas;
case aKind of
CUBE : Begin
case aSquare of {Cube has 8 squares}
0: begin
Add(TVertice.Create(-gX, gY, gZ)); //0
Add(TVertice.Create(-gX, gY, -gZ)); //1
Add(TVertice.Create( gX, gY, -gZ)); //2
Add(TVertice.Create( gX, gY, gZ)); //3
end;
1: begin
Add(TVertice.Create(-gX, -gY, -gZ)); //5
Add(TVertice.Create( gX, -gY, -gZ)); //6
Add(TVertice.Create( gX, -gY, gZ)); //7
Add(TVertice.Create(-gX, -gY, gZ)); //4
end;
2: begin
Add(TVertice.Create(-gX, gY, gZ)); //0
Add(TVertice.Create(-gX, gY, -gZ)); //1
Add(TVertice.Create(-gX, -gY, -gZ)); //5
Add(TVertice.Create(-gX, -gY, gZ)); //4
end;
3: begin
Add(TVertice.Create(-gX, gY, -gZ)); //1
Add(TVertice.Create( gX, gY, -gZ)); //2
Add(TVertice.Create( gX, -gY, -gZ)); //6
Add(TVertice.Create(-gX, -gY, -gZ)); //5
end;
4: begin
Add(TVertice.Create( gX, gY, -gZ)); //2
Add(TVertice.Create( gX, gY, gZ)); //3
Add(TVertice.Create( gX, -gY, gZ)); //7
Add(TVertice.Create( gX, -gY, -gZ)); //6
end;
5: begin
Add(TVertice.Create(-gX, gY, gZ)); //0
Add(TVertice.Create( gX, gY, gZ)); //3
Add(TVertice.Create( gX, -gY, gZ)); //7
Add(TVertice.Create(-gX, -gY, gZ)); //4
end;
end;
end;
TRIANGLE : Begin // Open for Update
end;
end;
end;

procedure TPolygon.Draw;
var i, x, y, first, last, aFrX, aFrY, aToX, aToY : Integer;
begin
{simply connect the one point to the next and
finallythe first one to the last}
x := CentreX; // Centre of the Paintbox
y := CentreY;
last := Count -1;
first := 0;
for i:= 0 to Count - 2 do
begin
aFrX := Calc3DPoint(Vertices[i].x , Vertices[i].z ) + X;
aFrY := Calc3DPoint(Vertices[i].y , Vertices[i].z ) + Y;
aToX := Calc3DPoint(Vertices[i+1].x , Vertices[i+1].z) + X;
aToY := Calc3DPoint(Vertices[i+1].y , Vertices[i+1].z) + Y;
DrawLine(aFrX, aFrY, aToX, aToY, clRed);
end;
aFrX := Calc3DPoint(Vertices[first].x , Vertices[first].z) + X;
aFrY := Calc3DPoint(Vertices[first].y , Vertices[first].z) + Y;
aToX := Calc3DPoint(Vertices[last ].x , Vertices[last ].z) + X;
aToY := Calc3DPoint(Vertices[last ].y , Vertices[last ].z) + Y;
DrawLine(aFrX, aFrY, aToX, aToY, clRed);
end;

procedure TPolygon.DrawLine(aFrX, aFrY, aToX, aToY: Integer; aColor: TColor);
begin
fCanvas.Pen.Color := aColor;
fCanvas.MoveTo(aFrX, aFrY);
fCanvas.LineTo(aToX, aToY);
end;

function TPolygon.GetVertice(i: Integer): TVertice;
begin
Result := TVertice(Items[i]);
end;

{ TVertice }

procedure TVertice.Adjust(aX, aY, aZ: Integer);
begin
x := aX;
y := aY;
z := aZ;
end;

constructor TVertice.Create(aX, aY, aZ: Integer);
begin
x := aX;
y := aY;
z := aZ;
end;

end.

_____________________________________________________________
unit uMain;

interface

uses
Windows, Messages, SysUtils, Variants, Classes, Graphics, Controls, Forms,
Dialogs, StdCtrls, ExtCtrls, uObjects;

type
TfrmMain = class(TForm)
pnMain: TPanel;
pbScreen: TPaintBox;
pnRight: TPanel;
cbAxis: TCheckBox;
pnBottom: TPanel;
sbX: TScrollBar;
sbY: TScrollBar;
sbZ: TScrollBar;
Label1: TLabel;
sbH: TScrollBar;
Label2: TLabel;
procedure FormCreate(Sender: TObject);
procedure FormDestroy(Sender: TObject);
procedure pbScreenPaint(Sender: TObject);
procedure cbAxisClick(Sender: TObject);
procedure sbXChange(Sender: TObject);
private
{ Private declarations }
MyObject : TBaseObject;
public
{ Public declarations }
end;

var
frmMain: TfrmMain;

implementation

{$R *.dfm}

procedure TfrmMain.FormCreate(Sender: TObject);
begin
DoubleBuffered := True;
if not Assigned(MyObject) then
MyObject := TBaseObject.Create(pbScreen.Canvas, pbScreen.Width, pbScreen.Height);
end;

procedure TfrmMain.FormDestroy(Sender: TObject);
begin
if Assigned(MyObject) then
MyObject.Free;
end;

procedure TfrmMain.pbScreenPaint(Sender: TObject);
begin
if Assigned(MyObject) then
begin
MyObject.fWidth := pbScreen.Width;
MyObject.fHeight := pbScreen.Height;
MyObject.Initialise;
end;
end;

procedure TfrmMain.cbAxisClick(Sender: TObject);
begin
MyObject.Initialise;
end;

procedure TfrmMain.sbXChange(Sender: TObject);
begin
MyObject.AdjustCoords(sbX.Position, sbY.Position, sbZ.Position, sbH.Position);
frmMain.Caption := '3D basics X : ' + IntToStr(sbX.Position) + ' Y : ' + IntToStr(sbY.Position) +
' Z : ' + IntToStr(sbZ.Position) + ' H : ' + IntToStr(sbH.Position);
end;

end.
______________________________________________
object frmMain: TfrmMain
Left = 319
Top = 124
Width = 513
Height = 477
Caption = '3D Basics '
Color = clBtnFace
Font.Charset = DEFAULT_CHARSET
Font.Color = clWindowText
Font.Height = -11
Font.Name = 'MS Sans Serif'
Font.Style = []
OldCreateOrder = False
OnCreate = FormCreate
OnDestroy = FormDestroy
OnResize = pbScreenPaint
PixelsPerInch = 96
TextHeight = 13
object pnMain: TPanel
Left = 0
Top = 0
Width = 420
Height = 430
Align = alClient
BevelOuter = bvLowered
TabOrder = 0
object pbScreen: TPaintBox
Left = 1
Top = 1
Width = 418
Height = 428
Align = alClient
Color = clBtnFace
ParentColor = False
OnPaint = pbScreenPaint
end
end
object pnRight: TPanel
Left = 420
Top = 0
Width = 85
Height = 430
Align = alRight
BevelOuter = bvNone
TabOrder = 1
object Label1: TLabel
Left = 7
Top = 24
Width = 69
Height = 13
Caption = 'X Y Z'
end
object Label2: TLabel
Left = 8
Top = 145
Width = 8
Height = 13
Caption = 'H'
end
object cbAxis: TCheckBox
Left = 1
Top = 0
Width = 89
Height = 17
Caption = 'X and Y axis'
TabOrder = 0
OnClick = cbAxisClick
end
object sbX: TScrollBar
Left = 5
Top = 40
Width = 16
Height = 101
Kind = sbVertical
PageSize = 0
Position = 50
TabOrder = 1
OnChange = sbXChange
end
object sbY: TScrollBar
Left = 35
Top = 40
Width = 16
Height = 101
Kind = sbVertical
PageSize = 0
Position = 50
TabOrder = 2
OnChange = sbXChange
end
object sbZ: TScrollBar
Left = 64
Top = 40
Width = 16
Height = 101
Kind = sbVertical
PageSize = 0
Position = 25
TabOrder = 3
OnChange = sbXChange
end
object sbH: TScrollBar
Left = 5
Top = 160
Width = 16
Height = 101
Kind = sbVertical
Max = 300
Min = 26
PageSize = 0
Position = 50
TabOrder = 4
OnChange = sbXChange
end
end
object pnBottom: TPanel
Left = 0
Top = 430
Width = 505
Height = 20
Align = alBottom
BevelOuter = bvNone
TabOrder = 2
end
end
____________________________________________________________

sftweng

You might like to use this Math3D library:

unit Math3D;

// Vector and matrix math library for OpenGL
// by LI Qingrui (lqr@ustc.edu)
// Feel free to use and modify.

// Note: Matrice used here is row-majored. This means a left-to-right rule
// of matrix concatenation. And there is no need to transpose before calling
// glLoadMatrixf or glMultMatrixf.

interface

// TVector2x

type
TVector2i = array[0..1] of Integer;
TVector2f = array[0..1] of Single;
TVector2d = array[0..1] of Double;
TVector2p = array[0..1] of Pointer;
const
NullVector2i: TVector2i = (0,0);
NullVector2f: TVector2f = (0,0);
NullVector2d: TVector2d = (0,0);
NullVector2p: TVector2p = (nil,nil);
XVector2i: TVector2i = (1,0);
XVector2f: TVector2f = (1,0);
XVector2d: TVector2d = (1,0);
YVector2i: TVector2i = (0,1);
YVector2f: TVector2f = (0,1);
YVector2d: TVector2d = (0,1);

// TVector3x

type
TVector3i = array[0..2] of Integer;
TVector3f = array[0..2] of Single;
TVector3d = array[0..2] of Double;
TVector3p = array[0..2] of Pointer;
PVector3i = ^TVector3i;
PVector3f = ^TVector3f;
PVector3d = ^TVector3d;

TVector3fDynArray = array of TVector3f;
const
NullVector3i: TVector3i = (0,0,0);
NullVector3f: TVector3f = (0,0,0);
NullVector3d: TVector3d = (0,0,0);
NullVector3p: TVector3p = (nil,nil,nil);
XVector3i: TVector3i = (1,0,0);
XVector3f: TVector3f = (1,0,0);
XVector3d: TVector3d = (1,0,0);
YVector3i: TVector3i = (0,1,0);
YVector3f: TVector3f = (0,1,0);
YVector3d: TVector3d = (0,1,0);
ZVector3i: TVector3i = (0,0,1);
ZVector3f: TVector3f = (0,0,1);
ZVector3d: TVector3d = (0,0,1);

// TVector4x

type
TVector4i = array[0..3] of Integer;
TVector4f = array[0..3] of Single;
TVector4d = array[0..3] of Double;
TVector4p = array[0..3] of Pointer;
const
NullVector4i: TVector4i = (0,0,0,0);
NullVector4f: TVector4f = (0,0,0,0);
NullVector4d: TVector4d = (0,0,0,0);
NullVector4p: TVector4p = (nil,nil,nil,nil);

// Common used

type
TQuaternion = TVector4f;

// Matrix Types

type
TMatrix3i = array[0..2] of TVector3i;
TMatrix3f = array[0..2] of TVector3f;
TMatrix3d = array[0..2] of TVector3d;
PMatrix3i = ^TMatrix3i;
PMatrix3f = ^TMatrix3f;
PMatrix3d = ^TMatrix3d;
TMatrix4i = array[0..3] of TVector4i;
TMatrix4f = array[0..3] of TVector4f;
TMatrix4d = array[0..3] of TVector4d;
PMatrix4i = ^TMatrix4i;
PMatrix4f = ^TMatrix4f;
PMatrix4d = ^TMatrix4d;

const
NullMatrix4f: TMatrix4f = (
(0,0,0,0),
(0,0,0,0),
(0,0,0,0),
(0,0,0,0)
);
IdentityMatrix4f: TMatrix4f = (
(1,0,0,0),
(0,1,0,0),
(0,0,1,0),
(0,0,0,1)
);

function Vector4f(x, y, z, w: Single): TVector4f; overload;
function Vector4f(const v3: TVector3f; w: Single = 1.0): TVector4f; overload;
function Vector3f(x, y, z: Single): TVector3f; overload;
function Vector3f(const v4: TVector4f): TVector3f; overload;
function VectorAdd(const V1, V2: TVector4f): TVector4f; overload;
function VectorAdd(const V1, V2: TVector3f): TVector3f; overload;
function VectorSubtract(const V1, V2: TVector4f): TVector4f; overload;
function VectorSubtract(const V1, V2: TVector3f): TVector3f; overload;
function VectorScale(const V: TVector3f; Factor: Single): TVector3f;
function VectorLength(const V: TVector3f): Single;
function VectorNormalize(var V: TVector3f): Single;
function VectorDot(const V1, V2: TVector3f): Single;
function VectorCross(const V1, V2: TVector3f): TVector3f;
function VectorLerp(const V1, V2: TVector3f; t: Single): TVector3f;
function NormalLerp(const n1, n2: TVector3f; t: single): TVector3f;
function VectorTransform(const V: TVector4f; M: TMatrix4f): TVector4f; overload;
function VectorTransform(const V: TVector3f; const M: TMatrix4f): TVector3f; overload;
function NormalTransform(const V: TVector3f; const M: TMatrix4f): TVector3f;
function VectorMaximize(const V1, V2: TVector3f): TVector3f;
function VectorMinimize(const V1, V2: TVector3f): TVector3f;
function VectorPerpendicular(const V, N: TVector3f): TVector3f;
function VectorReflect(const V, N: TVector3f): TVector3f;
function Quaternion(Angle: Single; Axis: TVector3f): TQuaternion;
function QuaternionConjugate(const Q: TQuaternion): TQuaternion;
function QuaternionImagePart(const Q: TQuaternion): TVector3f;
function QuaternionMultiply(const qL, qR: TQuaternion): TQuaternion;
function QuaternionSlerp(const QStart, QEnd: TQuaternion; t: Single): TQuaternion;
function QuaternionToMatrix(const Q: TQuaternion): TMatrix4f;
function MatrixMultiply(const M1, M2: TMatrix4f): TMatrix4f;
procedure MatrixTranspose(var M: TMatrix4f);
//procedure MatrixInvert(var M: TMatrix4f);
function MatrixInvert(const M: TMatrix4f): TMatrix4f;
procedure MatrixAdjoint(var M: TMatrix4f);
function MatrixDeterminant(const M: TMatrix4f): Single;
procedure MatrixScale(var M: TMatrix4f; s: Single);
function MatrixScaling(sx, sy, sz: single): TMatrix4f;
function MatrixTranslation(sx, sy, sz: single): TMatrix4f;
function MatrixRotation(Angle: Single; Axis: TVector3f): TMatrix4f;
function MatrixRotationX(Angle: Single): TMatrix4f;
function MatrixRotationY(Angle: Single): TMatrix4f;
function MatrixRotationZ(Angle: Single): TMatrix4f;
function MatrixShearingZ(dx, dy: single): TMatrix4f;
function MatrixShadow(const plane: TVector4f; const lightpos: TVector3f): TMatrix4f; overload;
function MatrixShadow(const plane, lightpos: TVector4f): TMatrix4f; overload;
function MatrixProjection(fov, aspect, nearplane, farplane: single): TMatrix4f;
function MatrixView(const From, At, Worldup: TVector3f): TMatrix4f;

procedure VectorArrayTranslate(var va: array of TVector3f; x, y, z: single);
function VectorArrayLerp(const v1, v2: TVector3fDynArray; t: single): TVector3fDynArray;
function NormalArrayLerp(const n1, n2: TVector3fDynArray; t: single): TVector3fDynArray;
procedure GenerateNormals(out normals: array of TVector3f;
const vertices: array of TVector3f; const indices: array of TVector3i);

implementation

uses Math;

const
// some very small numbers
EPSILON = 1e-100;
EPSILON2 = 1e-50;

function Vector4f(x, y, z, w: Single): TVector4f;
begin
Result[0] := x;
Result[1] := y;
Result[2] := z;
Result[3] := w;
end;

function Vector4f(const v3: TVector3f; w: Single): TVector4f;
begin
result[0] := v3[0];
result[1] := v3[1];
result[2] := v3[2];
result[3] := w;
end;

function Vector3f(x, y, z: Single): TVector3f;
begin
Result[0] := x;
Result[1] := y;
Result[2] := z;
end;

function Vector3f(const v4: TVector4f): TVector3f;
begin
result[0] := v4[0] / v4[3];
result[1] := v4[1] / v4[3];
result[2] := v4[2] / v4[3];
end;

function VectorAdd(const V1, V2: TVector4f): TVector4f;
begin
Result[0] := V1[0] + V2[0];
Result[1] := V1[1] + V2[1];
Result[2] := V1[2] + V2[2];
Result[3] := V1[3] + V2[3];
end;

function VectorAdd(const V1, V2: TVector3f): TVector3f;
begin
Result[0] := V1[0] + V2[0];
Result[1] := V1[1] + V2[1];
Result[2] := V1[2] + V2[2];
end;

function VectorSubtract(const V1, V2: TVector4f): TVector4f;
begin
Result[0] := V1[0] - V2[0];
Result[1] := V1[1] - V2[1];
Result[2] := V1[2] - V2[2];
Result[3] := V1[3] - V2[3];
end;

function VectorSubtract(const V1, V2: TVector3f): TVector3f;
begin
Result[0] := V1[0] - V2[0];
Result[1] := V1[1] - V2[1];
Result[2] := V1[2] - V2[2];
end;

function VectorScale(const V: TVector3f; Factor: Single): TVector3f;
begin
result[0] := V[0] * Factor;
result[1] := V[1] * Factor;
result[2] := V[2] * Factor;
end;

function VectorLength(const V: TVector3f): Single;
begin
Result := Sqrt(Sqr(V[0]) + Sqr(V[1]) + Sqr(V[2]));
end;

function VectorNormalize(var V: TVector3f): Single;
begin
Result := Sqrt(Sqr(V[0]) + Sqr(V[1]) + Sqr(V[2]));
V[0] := V[0] / Result;
V[1] := V[1] / Result;
V[2] := V[2] / Result;
end;

function VectorDot(const V1, V2: TVector3f): Single;
begin
result := V1[0]*V2[0] + V1[1]*V2[1] + V1[2]*V2[2];
end;

function VectorCross(const V1, V2: TVector3f): TVector3f;
begin
Result[0] := V1[1] * V2[2]-V1[2] * V2[1];
Result[1] := V1[2] * V2[0]-V1[0] * V2[2];
Result[2] := V1[0] * V2[1]-V1[1] * V2[0];
end;

function VectorLerp(const V1, V2: TVector3f; t: Single): TVector3f;
begin
Result[0] := V1[0] + (V2[0] - V1[0]) * t;
Result[1] := V1[1] + (V2[1] - V1[1]) * t;
Result[2] := V1[2] + (V2[2] - V1[2]) * t;
end;

function VectorTransform(const V: TVector3f; const M: TMatrix4f): TVector3f;
// Transforms a vector3 by a given matrix, projecting the result back into w = 1.
var W: Single;
begin
W := V[0] * M[0, 3] + V[1] * M[1, 3] + V[2] * M[2, 3] + M[3, 3];
Result[0] := (V[0] * M[0, 0] + V[1] * M[1, 0] + V[2] * M[2, 0] + M[3, 0]) / W;
Result[1] := (V[0] * M[0, 1] + V[1] * M[1, 1] + V[2] * M[2, 1] + M[3, 1]) / W;
Result[2] := (V[0] * M[0, 2] + V[1] * M[1, 2] + V[2] * M[2, 2] + M[3, 2]) / W;
end;

function NormalTransform(const V: TVector3f; const M: TMatrix4f): TVector3f;
begin
Result[0] := V[0] * M[0, 0] + V[1] * M[1, 0] + V[2] * M[2, 0];
Result[1] := V[0] * M[0, 1] + V[1] * M[1, 1] + V[2] * M[2, 1];
Result[2] := V[0] * M[0, 2] + V[1] * M[1, 2] + V[2] * M[2, 2];
end;

function VectorTransform(const V: TVector4f; M: TMatrix4f): TVector4f;
// transforms a homogeneous vector by multiplying it with a matrix
begin
Result[0] := V[0] * M[0, 0] + V[1] * M[1, 0] + V[2] * M[2, 0] + V[3] * M[3, 0];
Result[1] := V[0] * M[0, 1] + V[1] * M[1, 1] + V[2] * M[2, 1] + V[3] * M[3, 1];
Result[2] := V[0] * M[0, 2] + V[1] * M[1, 2] + V[2] * M[2, 2] + V[3] * M[3, 2];
Result[3] := V[0] * M[0, 3] + V[1] * M[1, 3] + V[2] * M[2, 3] + V[3] * M[3, 3];
end;

function VectorMaximize(const V1, V2: TVector3f): TVector3f;
begin
Result[0] := Max(V1[0], V2[0]);
Result[1] := Max(V1[1], V2[1]);
Result[2] := Max(V1[2], V2[2]);
end;

function VectorMinimize(const V1, V2: TVector3f): TVector3f;
begin
Result[0] := Min(V1[0], V2[0]);
Result[1] := Min(V1[1], V2[1]);
Result[2] := Min(V1[2], V2[2]);
end;

function VectorPerpendicular(const V, N: TVector3f): TVector3f;
// calculates a vector perpendicular to N (N is assumed to be of unit length)
// subtract out any component parallel to N
var Dot: Single;
begin
Dot := VectorDot(V, N);
Result[0] := V[0]-Dot * N[0];
Result[1] := V[1]-Dot * N[1];
Result[2] := V[2]-Dot * N[2];
end;

function VectorReflect(const V, N: TVector3f): TVector3f;
// reflects vector V against N (assumes N is normalized)
var Dot: Single;
begin
Dot := VectorDot(V, N);
Result[0] := V[0]-2 * Dot * N[0];
Result[1] := V[1]-2 * Dot * N[1];
Result[2] := V[2]-2 * Dot * N[2];
end;

function VectorCosAngle(const V1, V2: TVector3f): Single;
// calculates the cosine of the angle between Vector1 and Vector2
begin
Result := VectorDot(V1, V2) / (VectorLength(V1) * VectorLength(V2));
end;

function BezierCVCurvePoint(t: single; p0, p1, p2, p3: single): single;
begin
result := (1-t)*(1-t)*(1-t)*p0 + (1-t)*(1-t)*t*p1 + (1-t)*t*t*p2 + t*t*t*p3;
end;

function Quaternion(Angle: Single; Axis: TVector3f): TQuaternion;
begin
VectorNormalize(Axis);
result[0] := sin(Angle/2) * Axis[0];
result[1] := sin(Angle/2) * Axis[1];
result[2] := sin(Angle/2) * Axis[2];
result[3] := cos(Angle/2);
end;

function QuaternionConjugate(const Q: TQuaternion): TQuaternion;
begin
result[0] := -Q[0];
result[1] := -Q[1];
result[2] := -Q[2];
result[3] := Q[3];
end;

function QuaternionMultiply(const qL, qR: TQuaternion): TQuaternion;
begin
Result[3] := qL[3] * qR[3] - qL[0] * qR[0] - qL[1] * qR[1] - qL[2] * qR[2];
Result[0] := qL[3] * qR[0] + qL[0] * qR[3] + qL[1] * qR[2] - qL[2] * qR[1];
Result[1] := qL[3] * qR[1] + qL[1] * qR[3] + qL[2] * qR[0] - qL[0] * qR[2];
Result[2] := qL[3] * qR[2] + qL[2] * qR[3] + qL[0] * qR[1] - qL[1] * qR[0];
end;

function QuaternionImagePart(const Q: TQuaternion): TVector3f;
begin
Result[0] := Q[0];
Result[1] := Q[1];
Result[2] := Q[2];
end;

function QuaternionSlerp(const QStart, QEnd: TQuaternion; t: Single): TQuaternion;
// spherical linear interpolation of unit quaternions with spins
// QStart, QEnd - start and end unit quaternions
// t - interpolation parameter (0 to 1)
var beta, // complementary interp parameter
theta, // Angle between A and B
sint, cost: Single; // sine, cosine of theta
bflip: Boolean; // use negativ t?
begin
// cosine theta
cost := VectorCosAngle(QuaternionImagePart(QStart), QuaternionImagePart(QEnd));

// if QEnd is on opposite hemisphere from QStart, use -QEnd instead
if cost < 0 then
begin
cost := -cost;
bflip := True;
end
else bflip := False;

// if QEnd is (within precision limits) the same as QStart,
// just linear interpolate between QStart and QEnd.
// Can't do spins, since we don't know what direction to spin.
if (1 - cost) < EPSILON then beta := 1 - t
else begin // normal case
theta := arccos(cost);
sint := sin(theta);
beta := sin(theta - t * theta) / sint;
t := sin(t * theta) / sint;
end;

if bflip then t := -t;
// interpolate
Result[0] := beta * QStart[0] + t * QEnd[0];
Result[1] := beta * QStart[1] + t * QEnd[1];
Result[2] := beta * QStart[2] + t * QEnd[2];
Result[3] := beta * QStart[3] + t * QEnd[3];
end;

function QuaternionToMatrix(const Q: TQuaternion): TMatrix4f;
// Constructs rotation matrix from (possibly non-unit) quaternion.
// Assumes matrix is used to multiply column vector on the left:
// vnew = mat vold. Works correctly for right-handed coordinate system
// and right-handed rotations.
var
V: TVector3f;
SinA, CosA, A, B, C: Extended;
begin
V := QuaternionImagePart(Q);
VectorNormalize(V);
SinCos(Q[3] / 2, SinA, CosA);
A := V[0] * SinA;
B := V[1] * SinA;
C := V[2] * SinA;

Result := IdentityMatrix4f;
Result[0, 0] := 1 - 2 * B * B - 2 * C * C;
Result[0, 1] := 2 * A * B - 2 * CosA * C;
Result[0, 2] := 2 * A * C + 2 * CosA * B;

Result[1, 0] := 2 * A * B + 2 * CosA * C;
Result[1, 1] := 1 - 2 * A * A - 2 * C * C;
Result[1, 2] := 2 * B * C - 2 * CosA * A;

Result[2, 0] := 2 * A * C - 2 * CosA * B;
Result[2, 1] := 2 * B * C + 2 * CosA * A;
Result[2, 2] := 1 - 2 * A * A - 2 * B * B;
end;

function MatrixMultiply(const M1, M2: TMatrix4f): TMatrix4f;
var I, J: Integer;
begin
for I := 0 to 3 do
for J := 0 to 3 do
Result[I, J] := M1[I, 0] * M2[0, J] +
M1[I, 1] * M2[1, J] +
M1[I, 2] * M2[2, J] +
M1[I, 3] * M2[3, J];
end;

function MatrixScaling(sx, sy, sz: single): TMatrix4f;
begin
Result := IdentityMatrix4f;
Result[0, 0] := sx;
Result[1, 1] := sy;
Result[2, 2] := sz;
end;

function MatrixTranslation(sx, sy, sz: single): TMatrix4f;
begin
Result := IdentityMatrix4f;
Result[3, 0] := sx;
Result[3, 1] := sy;
Result[3, 2] := sz;
end;

function MatrixRotation(Angle: Single; Axis: TVector3f): TMatrix4f;
var
cosine, sine, Len, one_minus_cosine: Extended;
begin
SinCos(Angle, Sine, Cosine);
one_minus_cosine := 1 - cosine;
Len := VectorNormalize(Axis);
if Len = 0 then Result := IdentityMatrix4f
else begin
Result[0, 0] := (one_minus_cosine * Sqr(Axis[0])) + Cosine;
Result[0, 1] := (one_minus_cosine * Axis[0] * Axis[1]) - (Axis[2] * Sine);
Result[0, 2] := (one_minus_cosine * Axis[2] * Axis[0]) + (Axis[1] * Sine);
Result[0, 3] := 0;

Result[1, 0] := (one_minus_cosine * Axis[0] * Axis[1]) + (Axis[2] * Sine);
Result[1, 1] := (one_minus_cosine * Sqr(Axis[1])) + Cosine;
Result[1, 2] := (one_minus_cosine * Axis[1] * Axis[2]) - (Axis[0] * Sine);
Result[1, 3] := 0;

Result[2, 0] := (one_minus_cosine * Axis[2] * Axis[0]) - (Axis[1] * Sine);
Result[2, 1] := (one_minus_cosine * Axis[1] * Axis[2]) + (Axis[0] * Sine);
Result[2, 2] := (one_minus_cosine * Sqr(Axis[2])) + Cosine;
Result[2, 3] := 0;

Result[3, 0] := 0;
Result[3, 1] := 0;
Result[3, 2] := 0;
Result[3, 3] := 1;
end;
end;

function MatrixRotationX(Angle: Single): TMatrix4f;
// creates matrix for rotation about x-axis
var Sine, Cosine: Extended;
begin
SinCos(Angle, Sine, Cosine);
Result := IdentityMatrix4f;
Result[1, 1] := Cosine;
Result[1, 2] := Sine;
Result[2, 1] := -Sine;
Result[2, 2] := Cosine;
end;

function MatrixRotationY(Angle: Single): TMatrix4f;
// creates matrix for rotation about y-axis
var Sine, Cosine: Extended;
begin
SinCos(Angle, Sine, Cosine);
Result := IdentityMatrix4f;
Result[0, 0] := Cosine;
Result[0, 2] := -Sine;
Result[2, 0] := Sine;
Result[2, 2] := Cosine;
end;

function MatrixRotationZ(Angle: Single): TMatrix4f;
// creates matrix for rotation about z-axis
var Sine, Cosine: Extended;
begin
SinCos(Angle, Sine, Cosine);
Result := IdentityMatrix4f;
Result[0, 0] := Cosine;
Result[0, 1] := Sine;
Result[1, 0] := -Sine;
Result[1, 1] := Cosine;
end;

function MatrixDetInternal(a1, a2, a3, b1, b2, b3, c1, c2, c3: Single): Single;
// internal version for the determinant of a 3x3 matrix
begin
Result := a1 * (b2 * c3 - b3 * c2) -
b1 * (a2 * c3 - a3 * c2) +
c1 * (a2 * b3 - a3 * b2);
end;

function MatrixDeterminant(const M: TMatrix4f): Single;
var a1, a2, a3, a4,
b1, b2, b3, b4,
c1, c2, c3, c4,
d1, d2, d3, d4 : Single;
begin
a1 := M[0, 0]; b1 := M[0, 1]; c1 := M[0, 2]; d1 := M[0, 3];
a2 := M[1, 0]; b2 := M[1, 1]; c2 := M[1, 2]; d2 := M[1, 3];
a3 := M[2, 0]; b3 := M[2, 1]; c3 := M[2, 2]; d3 := M[2, 3];
a4 := M[3, 0]; b4 := M[3, 1]; c4 := M[3, 2]; d4 := M[3, 3];

Result := a1 * MatrixDetInternal(b2, b3, b4, c2, c3, c4, d2, d3, d4) -
b1 * MatrixDetInternal(a2, a3, a4, c2, c3, c4, d2, d3, d4) +
c1 * MatrixDetInternal(a2, a3, a4, b2, b3, b4, d2, d3, d4) -
d1 * MatrixDetInternal(a2, a3, a4, b2, b3, b4, c2, c3, c4);
end;

procedure MatrixScale(var M: TMatrix4f; s: Single);
var I, J: Integer;
begin
for I := 0 to 3 do
for J := 0 to 3 do M[I, J] := M[I, J] * s;
end;

procedure MatrixAdjoint(var M: TMatrix4f); register;
// Adjoint of a 4x4 matrix - used in the computation of the inverse of matrix
var a1, a2, a3, a4,
b1, b2, b3, b4,
c1, c2, c3, c4,
d1, d2, d3, d4: Single;
begin
a1 := M[0, 0]; b1 := M[0, 1];
c1 := M[0, 2]; d1 := M[0, 3];
a2 := M[1, 0]; b2 := M[1, 1];
c2 := M[1, 2]; d2 := M[1, 3];
a3 := M[2, 0]; b3 := M[2, 1];
c3 := M[2, 2]; d3 := M[2, 3];
a4 := M[3, 0]; b4 := M[3, 1];
c4 := M[3, 2]; d4 := M[3, 3];

// row column labeling reversed since we transpose rows & columns
M[0, 0] := MatrixDetInternal(b2, b3, b4, c2, c3, c4, d2, d3, d4);
M[1, 0] := -MatrixDetInternal(a2, a3, a4, c2, c3, c4, d2, d3, d4);
M[2, 0] := MatrixDetInternal(a2, a3, a4, b2, b3, b4, d2, d3, d4);
M[3, 0] := -MatrixDetInternal(a2, a3, a4, b2, b3, b4, c2, c3, c4);

M[0, 1] := -MatrixDetInternal(b1, b3, b4, c1, c3, c4, d1, d3, d4);
M[1, 1] := MatrixDetInternal(a1, a3, a4, c1, c3, c4, d1, d3, d4);
M[2, 1] := -MatrixDetInternal(a1, a3, a4, b1, b3, b4, d1, d3, d4);
M[3, 1] := MatrixDetInternal(a1, a3, a4, b1, b3, b4, c1, c3, c4);

M[0, 2] := MatrixDetInternal(b1, b2, b4, c1, c2, c4, d1, d2, d4);
M[1, 2] := -MatrixDetInternal(a1, a2, a4, c1, c2, c4, d1, d2, d4);
M[2, 2] := MatrixDetInternal(a1, a2, a4, b1, b2, b4, d1, d2, d4);
M[3, 2] := -MatrixDetInternal(a1, a2, a4, b1, b2, b4, c1, c2, c4);

M[0, 3] := -MatrixDetInternal(b1, b2, b3, c1, c2, c3, d1, d2, d3);
M[1, 3] := MatrixDetInternal(a1, a2, a3, c1, c2, c3, d1, d2, d3);
M[2, 3] := -MatrixDetInternal(a1, a2, a3, b1, b2, b3, d1, d2, d3);
M[3, 3] := MatrixDetInternal(a1, a2, a3, b1, b2, b3, c1, c2, c3);
end;
{
procedure MatrixInvert(var M: TMatrix4f);
var Det: Single;
begin
Det := MatrixDeterminant(M);
if Abs(Det) < EPSILON then M := IdentityMatrix4f
else
begin
MatrixAdjoint(M);
MatrixScale(M, 1 / Det);
end;
end;
}
function MatrixInvert(const M: TMatrix4f): TMatrix4f;
var DetInv: Single;
begin
if (abs(M[3, 3] - 1) > 0.001) or
(abs(M[0, 3]) > 0.001) or
(abs(M[1, 3]) > 0.001) or
(abs(M[2, 3]) > 0.001)
then raise EInvalidArgument.Create('arg not valid for this MatrixInvert function');
DetInv := 1 /( M[0, 0] * (M[1, 1] * M[2, 2] - M[1, 2] * M[2, 1]) -
M[0, 1] * (M[1, 0] * M[2, 2] - M[1, 2] * M[2, 0]) +
M[0, 2] * (M[1, 0] * M[2, 1] - M[1, 1] * M[2, 0]) );

result[0, 0] := DetInv * (M[1, 1] * M[2, 2] - M[1, 2] * M[2, 1]);
result[0, 1] := -DetInv * (M[0, 1] * M[2, 2] - M[0, 2] * M[2, 1]);
result[0, 2] := DetInv * (M[0, 1] * M[1, 2] - M[0, 2] * M[1, 1]);
result[0, 3] := 0;

result[1, 0] := -DetInv * (M[1, 0] * M[2, 2] - M[1, 2] * M[2, 0]);
result[1, 1] := DetInv * (M[0, 0] * M[2, 2] - M[0, 2] * M[2, 0]);
result[1, 2] := -DetInv * (M[0, 0] * M[1, 2] - M[0, 2] * M[1, 0]);
result[1, 3] := 0;

result[2, 0] := DetInv * (M[1, 0] * M[2, 1] - M[1, 1] * M[2, 0]);
result[2, 1] := -DetInv * (M[0, 0] * M[2, 1] - M[0, 1] * M[2, 0]);
result[2, 2] := DetInv * (M[0, 0] * M[1, 1] - M[0, 1] * M[1, 0]);
result[2, 3] := 0;

result[3, 0] := -(M[3, 0] * result[0, 0] + M[3, 1] * result[1, 0] + M[3, 2] * result[2, 0]);
result[3, 1] := -(M[3, 0] * result[0, 1] + M[3, 1] * result[1, 1] + M[3, 2] * result[2, 1]);
result[3, 2] := -(M[3, 0] * result[0, 2] + M[3, 1] * result[1, 2] + M[3, 2] * result[2, 2]);
result[3, 3] := 1;
end;

procedure MatrixTranspose(var M: TMatrix4f);
var
I, J: Integer;
t: Single;
begin
for I := 0 to 2 do
for J := I+1 to 3 do
begin
t := M[I, J];
M[I, J] := M[J, I];
M[J, I] := t;
end;
end;

function MatrixShearingZ(dx, dy: single): TMatrix4f;
begin
result := IdentityMatrix4f;
result[0, 2] := dx;
result[1, 2] := dy;
end;

procedure VectorArrayTranslate(var va: array of TVector3f; x, y, z: single);
var
i: integer;
t: TVector3f;
begin
t := Vector3f(x, y, z);
for i := 0 to High(va) do
va[i] := VectorAdd(va[i], t);
end;

function VectorArrayLerp(const v1, v2: TVector3fDynArray; t: single): TVector3fDynArray;
var i, len: integer;
begin
Assert(Length(v1) = Length(v2));
len := Length(v1);
if t = 0 then result := v1
else if t = 1 then result := v2
else begin
SetLength(result, len);
for i := 0 to len-1 do
result[i] := VectorLerp(v1[i], v2[i], t);
end;
end;

function NormalLerp(const n1, n2: TVector3f; t: single): TVector3f;
begin
result := VectorLerp(n1, n2, t);
VectorNormalize(result);
end;

function NormalArrayLerp(const n1, n2: TVector3fDynArray; t: single): TVector3fDynArray;
var i, len: integer;
begin
Assert(Length(n1) = Length(n2));
if t = 0 then result := n1
else if t = 1 then result := n2
else begin
len := Length(n1);
SetLength(result, len);
for i := 0 to len-1 do
begin
result[i] := VectorLerp(n1[i], n2[i], t);
VectorNormalize(result[i]);
end;
end;
end;

function MatrixShadow(const plane, lightpos: TVector4f): TMatrix4f;
var dot: single;
begin
dot := plane[0] * lightpos[0] + plane[1] * lightpos[1] +
plane[2] * lightpos[2] + plane[3] * lightpos[3];
result[0, 0] := dot - lightpos[0] * plane[0];
result[1, 0] := - lightpos[0] * plane[1];
result[2, 0] := - lightpos[0] * plane[2];
result[3, 0] := - lightpos[0] * plane[3];

result[0, 1] := - lightpos[1] * plane[0];
result[1, 1] := dot - lightpos[1] * plane[1];
result[2, 1] := - lightpos[1] * plane[2];
result[3, 1] := - lightpos[1] * plane[3];

result[0, 2] := - lightpos[2] * plane[0];
result[1, 2] := - lightpos[2] * plane[1];
result[2, 2] := dot - lightpos[2] * plane[2];
result[3, 2] := - lightpos[2] * plane[3];

result[0, 3] := - lightpos[3] * plane[0];
result[1, 3] := - lightpos[3] * plane[1];
result[2, 3] := - lightpos[3] * plane[2];
result[3, 3] := dot - lightpos[3] * plane[3];
end;

function MatrixShadow(const plane: TVector4f; const lightpos: TVector3f): TMatrix4f;
begin
result := MatrixShadow(plane, Vector4f(lightpos));
end;

function MatrixProjection(fov, aspect, nearplane, farplane: single): TMatrix4f;
var
w, h, q: Single;
begin
h := (cos(FOV/2)/sin(FOV/2));
w := Aspect * h;
Q := FarPlane / (FarPlane - NearPlane);
result := NullMatrix4f;
result[0, 0] := w;
result[1, 1] := h;
result[2, 2] := Q;
result[2, 3] := 1.0;
result[3, 2] := -Q * NearPlane;
end;

function MatrixView(const From, At, Worldup: TVector3f): TMatrix4f;
var
View: TVector3f;
DotProduct: Single;
Up, Right: TVector3f;
begin
// Get the z basis vector, which points straight ahead. This is the
// difference from the eyepoint to the lookat point.
View := VectorSubtract(At, From);
// Normalize the z basis vector
VectorNormalize(View);
// Get the dot product, and calculate the projection of the z basis
// vector onto the up vector. The projection is the y basis vector.
DotProduct := VectorDot(WorldUp, View);
Up := VectorSubtract(WorldUp, VectorScale(View, DotProduct));
// Normalize the y basis vector
VectorNormalize(Up);
// The x basis vector is found simply with the cross product of the y
// and z basis vectors
Right := VectorCross(Up, View);
// Start building the matrix. The first three rows contains the basis
// vectors used to rotate the view to point at the lookat point
Result[0, 0] := Right[0]; Result[0, 1] := Up[0]; Result[0, 2] := View[0]; Result[0, 3] := 0;
Result[1, 0] := Right[1]; Result[1, 1] := Up[1]; Result[1, 2] := View[1]; Result[1, 3] := 0;
Result[2, 0] := Right[2]; Result[2, 1] := Up[2]; Result[2, 2] := View[2]; Result[2, 3] := 0;
// Do the translation values (rotations are still about the eyepoint)
Result[3, 0] := - VectorDot(From, Right);
Result[3, 1] := - VectorDot(From, Up);
Result[3, 2] := - VectorDot(From, View);
Result[3, 3] := 1;
end;

procedure GenerateNormals(out normals: array of TVector3f;
const vertices: array of TVector3f; const indices: array of TVector3i);
var
i: integer;
v: TVector3f;
begin
FillChar(normals, Length(normals) * sizeof(TVector3f), 0);
for i := 0 to Length(indices) - 1 do
begin
v := VectorCross(VectorSubtract(vertices[indices[i, 2]], vertices[indices[i, 1]]),
VectorSubtract(vertices[indices[i, 0]], vertices[indices[i, 1]]));
VectorNormalize(v);
normals[indices[i, 0]] := VectorAdd(normals[indices[i, 0]], v);
normals[indices[i, 1]] := VectorAdd(normals[indices[i, 1]], v);
normals[indices[i, 2]] := VectorAdd(normals[indices[i, 2]], v);
end;
for i := 0 to length(normals) - 1 do
VectorNormalize(normals[i]);
end;

end.

sftweng

For a complete project that uses the library, download the following:

http://members.rogers.com/alan.bu/SBSIBalls.zip

mpoots

ASKER

No I am sorry but that is not what I was looking for. I downloaded the project and it's too big for me to find what I need and I have no clue what the project does. I doesn't look like 3D effect to me at all. I hope your answer doesn't discourage other people from responding. Thanks for your answer.

Marcel

sftweng

I must have misunderstood your requirement then. The SBSIBalls application is essentially the rotation in three dimensions of a tetrahedron which contains spheres at each vertex and at the midpoints between each pair of vertices. The key is in the Rotatxxx procedures, which use a transformation matrix from the 3D library.

To see it in operation, click one of the radio buttons for X, Y, or Z rotation and then press the "Step" or "Rotate" button. What you will see is the entire structure of spheres rotation around the chosen axis, with appropriate resizing of the sphere.

Yes, it's probably overkill, but I think you'll find everything you need there to rotate your cube.

sftweng

If my hints aren't enough, here's an offer. For an additional 200 points, I'll make the changes to your program for 3D rotation and send it back modified.

mpoots

ASKER

That is cool. I will give you 500 point if you do that.

Marcel

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sftweng

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mpoots

ASKER

sftweng,

I am very pleased with the result. Kuddos to you and I will award you some extra points.

The project can be downloaded here:

http://home.quicknet.nl/qn/prive/mpoots/downloads/3D.zip

Marcel