3D Transformation. In 3D, we have x, y, and z. We will continue use column vectors:. Homogenous systems:. x y z. x y z. glvertex3f(x, y,z);

Transcription

1 3D Transformation

2 In 3D, we have x, y, and z. We will continue use column vectors:. Homogenous systems:. 3D Transformation glvertex3f(x, y,z); x y z x y z

3 A Right Handle Coordinate System x y z; y z x; z x y;

4 Transformation: 3D Translation Given a position (x, y, z) and an offset vector (tx, ty, tz): x' x + tx y' y + ty z' z + tz x' y' z' tx ty tz x y z

5 Transformation: 3D Scaling Change the object size: x' sx x y' sy y z' sz z x' y' z' sx sy sz x y z

6 Transformation: 3D Rotation Rotation needs an angle and an axis. Rotation is defined according to the right hand rule (our convention).

7 Transformation: 3D Rotation Axis Matrix Rotate around X: glrotatef(θ,,, ); Rotate around Y: glrotatef(θ,,, ); Rotate around Z: glrotatef(θ,,, ); x' y' z' cosθ sinθ sinθ cosθ x y z x' y' z' cosθ sinθ sinθ cosθ x y z x' y' z' cosθ sinθ sinθ cosθ x y z

8 How to quickly remember them The axis coordinate is unchanged. For the other two coordinates: Diagonals are filled with cos. Off diagonals are filled with sin. There is a sign

9 How to quickly remember them Here is an easier way to think about it: x' y' z' The vector after rotation a b c d e f g h i x y z The vector before rotation

10 How to quickly remember them a d g a b c d e f g h i X axis What X becomes After rotation b e h a b c d e f g h i Y axis What Y becomes After rotation c f i a b c d e f g h i Z axis What Z becomes After rotation

11 For example X Y Z cosθ sinθ sinθ cosθ θ cosθ sinθ sinθ cosθ θ X Y Z

12 Generic Rotation Use the rotation function: glrotatef(theta, x, y, z); In degree No need to be normalized x 2 ( c) + c xy( c) zs xz( c) + ys yx( c) + zs y 2 ( c) + c yz( c) xs xz( c) ys yz( c) + xs z 2 ( c) + c (x, y, z) c cosθ s sinθ Don t need to remember it, just for your reference. θ

13 What if I don t like a transformation, how do I get back? How to reverse a transformation x' y' z' M x y z x y z? x' y' z'?m x y z :? M M M MM I

14 Inverse Translation x' x + tx y' y + ty z' z + tz x' y' z' tx ty tz x y z x x' tx y y' ty z z' tz x y z tx ty tz x' y' z' gltranslatef(tx, ty, tz); gltranslatef(-tx, -ty, -tz);

15 Inverse Scaling x' x sx y' y sy z' z sz x' y' z' sx sy sz x y z x x'/ sx y y'/ sy z z'/ sz x y z sx sy sz x' y' z' glscalef(sx, sy, sz); glscalef(/sx, /sy, /sz);

16 Inverse Rotation x' y' z' cosθ sinθ sinθ cosθ x y z glrotatef(theta,,, ); Rotate_X by θ x' y' z' cosθ sinθ sinθ cosθ x y z glrotatef(-theta,,, ); Rotate_X by θ

17 Inverse Transformation The transformation has multiple matrices: v v' Mv (M M 2 M 3 M 4 )v Scaling Translation Rotation Translation v M M 2 M 3 M 4 Its inverse: v v' M v (M 4 M 3 M 2 M )v Translation Rotation Translation Scaling v M 4 M 3 M 2 M

18 OpenGL handles multiple matrices M I glloadidentity(); glrotatef( ); gltranslatef( ); glscalef( ); gltranslatef( ); glbegin(gl_points); glvertex3fv(v); glend(); M IM R M IM M R T M IM M M R T S M IM M M M R T S T v'mv IM R M T M S M T v

19 OpenGL handles multiple matrices Reverse Order glloadidentity(); glrotatef( ); gltranslatef( ); glscalef( ); gltranslatef( ); glbegin(gl_points); glvertex3fv(v); glend(); Define v Translation Scaling Translation Rotation v'mv IM R M T M S M T v

20 OpenGL has a reason. glrotatef( ); //A gltranslatef( );//B glscalef( ); //C glvertex3fv(v); Define v vcv vbv vav We think: v is transformed in a world coordinate system. Given a world locala(world) local2b(local) local3c(local2) Define vin local3 OpenGL think: world is moved into local. Each transformation is defined respect to the local.

21 For example gltranslatef(3,2,); glvertex3f(2,2,); We think (2, 2) moves by an offset (3, 2). OpenGL thinks the coordinate System moves by an offset (3, 2). The vertex defines at (2, 2) locally.

22 For example glrotatef(45,,,); glvertex3f(4,,); We think (4, ) rotates 45 degree. OpenGL thinks the coordinate System rotates 45 degree. The vertex defines at (4, ) locally.

23 For example glrotatef(45,,,); gltranslatef(4,,); glvertex3f(,,); We think (, ) first Translates, then rotates. OpenGL thinks the coordinate System first rotates, then translates.

24 For example glrotatef(45,,,); gltranslatef(4,,); glbegin( ); glvertex3f(,,); glend(); L2 Important Note: The second translation is defined respect to L, not W! L W OpenGL thinks the coordinate System first rotates, then translates.

25 A quiz L2 glrotatef(45,,,); gltranslatef(4,,); glvertex3f(,,); L W L2 gltranslatef(2.8,2.8,); glrotatef(45,,,); glvertex3f(,,); L (2.8, 2.8) W

26 What if we change the order? L2 glrotatef(45,,,); gltranslatef(2.8,2.8,); glvertex3f(,,); Important Note: The second transformation is defined respect to L, not W! L W

27 Order is important. L2 L2 L L (2.8, 2.8) W W gltranslatef(2.8,2.8,); glrotatef(45,,,); glvertex3f(,,); glrotatef(45,,,); gltranslatef(2.8,2.8,); glvertex3f(,,);

28 A quiz gltranslatef(3,4,); L W (3., 4.)

29 A quiz L2 gltranslatef(3,4,); glrotatef(45,,,); L W (3., 4.)

30 A quiz L3 gltranslatef(3,4,); glrotatef(45,,,); L glscalef(,2,); W (3., 4.)

31 A quiz L4 L3 gltranslatef(3,4,); glrotatef(45,,,); L glscalef(,2,); gltranslatef(3,,); W (3., 4.)

32 A quiz L4 L3 gltranslatef(3,4,); glrotatef(45,,,); L glscalef(,2,); gltranslatef(3,,); W glvertex3f(3,,); (3., 4.)

33 How do I know the coordinate system? W L3 OpenGL only use a ModelView matrix M. M is transformed in various ways. M s effect is to update each vertex as: x' y' z' M x y z a b c d e f g h i j k l x y z

34 How do I know the coordinate system? W L3 Case : d h l a b c d e f g h i j k l If we draw a dot at (,, ) in L3, we actually get (d, h, l) in W.

35 How do I know the coordinate system? W L3 Case 2: a + d e + h i + l a b c d e f g h i j k l If we draw a dot at (,, ) in L3, we actually get (a+d, e+h, i+l) in W.

36 How do I know the coordinate system? W L3 Case 2: a + d e + h i + l d h l a e i In other words, a unit xvector in L3 is (a, e, i) in W. a + d e + h i + l a b c d e f g h i j k l

37 How do I know the coordinate system? W L3 Case 2: b + d f + h j + l d h l b f j In other words, a unit yvector in L3 is (b, f, j) in W. b + d f + h j + l a b c d e f g h i j k l

38 How do I know the coordinate system? ModelView matrix is the coordinate system L3: L3 L3 s X in W L3 s Z in W x' y' z' a b c d e f g h i j k l x y z W L3 s Y in W L3 s Origin in W

39 Summary If we think: Each transformation updates the vertex in an absolute world. Then, the code is arranged in a reverse order. Alternatively, OpenGL thinks: A transformation updates the coordinate system. For each change, the transformation is defined in the current (local) coordinate system. Transformation matrix and coordinate system are the same. The code is in the forward order!