7. 3D Viewing. Projection: why is projection necessary? CS Dept, Univ of Kentucky

Transcription

1 7. 3D Viewing Projection: why is projection necessary? 1

2 7. 3D Viewing Projection: why is projection necessary? Because the display surface is 2D 2

3 7.1 Projections Perspective projection 3

4 7.1 Projections Orthographic (parallel) projection 4

5 Perspective projection 5 Vanishing point Perspective projections of parallel lines not parallel to the projection plane will converge to a vanishing point Principal vanishing point vanishing point of parallel lines that are parallel to one of the three principal axes

6 Orthographic projection Three orthographic projections: 6

7 Orthographic projection Isometric projection of unit cube along direction (1, -1, -1): 7

8 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) Projection plane is normal to the z axis 8

9 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) 9

10 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) Orthographic (parallel) projection: 10

11 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) 11

12 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) Perspective projection: (COP is not at the origin) 12

13 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) 13

14 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) 14

15 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) Where are the vanishing points? Parallel lines after perspective projection are still parallel lines if they are also parallel to the projection plane. Why? 15 Parallel lines after perspective projection are no longer parallel lines if they are not parallel to the projection plane. Why?

16 Parallel lines after perspective projection are still parallel lines if they are also parallel to the projection plane. Why? 16

17 Parallel lines after perspective projection are no longer parallel lines if they are not parallel to the projection plane. Why? 17

18 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) Principal vanishing point : vanishing point generated by lines parallel to one of the principal axes (at most three PVPs). Two-point perspective projection is popular 18

19 7.2 Mathematics of Projections (projections can be defined by 4x4 matrices) How to find vanishing points? 19 Construct a line parallel to AB that passes thru the view point (eye). The intersection of this line with the projection plane is the vanishing point of AB.

20 7.3 Camera Model for Projective View How to create a perspective view of a scene in OpenGL? How to control the camera s position and orientation in OpenGL? 20

21 7.3 Camera Model for Projective View Conceptual model of 3D viewing: 21

22 7.3 Camera Model for Projective View Define Viewing (Eye, or Camera) Coordinate System: (specification of a 3D view) (Positioning and pointing the camera) 22 glmatrixmode ( GL_MODELVIEW ); glloadidentity ( ); glulookat ( eye.x, eye.y, eye.z, look.x, look.y, look.z, up.x, up.y, up.z);

23 7.3 Camera Model for Projective View 23

24 7.3 Camera Model for Projective View Define the view volume: (create a camera model) glmatrixmode ( GL_PROJECTION ); glloadidentity ( ); gluperspective ( viewangle, aspectratio, N, F ); 24

25 7.3 Camera Model for Projective View (w, h, -F) (w1, h1, -N) 25

26 7.4 Building Viewing Matrix 26

27 7.4 Building Viewing Matrix 27

28 7.4 Building Viewing Matrix 28 Modelview Matrix( ): M m Modeling part ( ): embodies all the modeling transformations for the object M v M v M m accumulate all the modeling transformations into a single matrix translation + rotation Viewing part ( ): accounts for the WC to VC transformation set by the camera s position and orientation

29 7.4 Building Viewing Matrix ( dx, d y, dz) ( ( eye) x, ( eye) y, ( eye) z ) 29

30 7.4 Building Viewing Matrix 30

31 7.4 Building Viewing Matrix Shearing: so that window center would coincide with (0, 0, - N ) 31

32 7.4 Building Viewing Matrix Scaling1: so user defined truncated view volume would coincide with the canonical view volume for perspective projection 32

33 also called a a homography 7.4 Building Viewing Matrix Perspective Transformation: convert CVV for perspective projection to a quasi-cvv for parallel projection 33

34 7.4 Building Viewing Matrix Translation: translate center of the quasi-cvv to the origin (0,0,0) 34

35 7.4 Building Viewing Matrix Scaling2: scale z-direction by 2 to get the CVV for parallel projection 35

36 7.4 Building Viewing Matrix M p? 36

37 7.5 GL_PROJECTION & GL_MODELVIEW GL_PROJECTION: - applied to every point that comes after it GL_MODELVIEW: - applied to every point in a particular model 37

38 GL_PROJECTION glmatrixmode(gl_projection); glloadidentity(); glortho(-1, 1, -1, 1, -1.0, 1.0); gltranslate(100, 100, 100); glrotatef(45,1, 0, 0); Gl_Projection_Matrix= identity_matrix * orthographic_matrix * Translation_matrix * Rotation_matrix 38

39 GL_MODELVIEW 39 glmatrixmode(gl_projection); glloadidentity(); glortho(-1, 1, -1, 1, -1.0, 1.0); gltranslate(camera_x, camera_y, camera_z); glmatrixmode(gl_modelview); glloadidentity(); gltranslate(box_x, box_y, box_z); // draw box here glloadidentity(); gltranslate(bottle_x, bottle_y, bottle_z); // draw bottle here

40 GL Matrices 40

41 GL Matrices 41

42 GL_MODELVIEW For the box: Box_Vertices_Matrix = Projection_Ortho_Matrix * Projection_Translation_Matrix * Box_Translation_Matrix For the bottle: Bottle_Vertices_Matrix = Projection_Ortho_Matrix * Projection_Translation_Matrix * Bottle_Translation_Matrix 42

43 7.6 Clipping in Homogeneous Coordinates What does M pt do? A G B E C F D G 43

44 7.6 Clipping in Homogeneous Coordinates 44

45 7.6 Clipping in Homogeneous Coordinates Now consider the following example: 45

46 7.6 Clipping in Homogeneous Coordinates If P Q is clipped against CVV after the perspective division, the clipping algorithm would have the line segment discarded. But R S is actually inside CVV. The reason that this happens is because the division performed for P changes the sign of z-component from positive to negative. Remedy: perform clipping before perspective division, i.e., clip in homogeneous coordinates, then 46 perform perspective division.

47 7.6 Clipping in Homogeneous Coordinates Basic idea: Instead of using X/W, Y/W, Z/W, we use W+X, W-X, W+Y, W-Y, W+Z, W-Z for the clipping process 47

48 7.6 Clipping in Homogeneous Coordinates Why? a point (after the perspective division) is inside the CVV for parallel projection if 48

49 Clipping in Homogeneous Coordinates W+X, W-X, W+Y, W-Y, W+Z, W-Z are called Boundary Coordinates (BC s) 49 W+X x=-1 W-X x=1 W+Y y=-1 W-Y y=1 W+Z z=-1 W-Z z=1

50 7.6 Clipping in Homogeneous Coordinates 50

51 7.6 Clipping in Homogeneous Coordinates 51

52 52 Clip a Line Segment in Homogeneous Coordinates Use Cyrus-Beck clipper: Input: t A B A t L B B B B B A A A A A w z y x w z y x )* ( ) (,,,,,, How to clip a line segment in homogeneous coordinates?

53 Clip a Line Segment in Homogeneous Coordinates Compute BC s for A and B Compute outcodes for A and B Perform trivial rejection test Perform trivial acceptance test 53

54 Clip a Line Segment in Homogeneous Coordinates If A and B are on different sides of x=1, then compute parameter of the intersection point as follows: t A w A ( A A ) ( B B w x w x x ) 54 Then update related items values

55 Clip a Line Segment in Homogeneous Coordinates For x=1, we consider: with W X Solving it to get W X 0 A ( B A ) t W W W A ( B A ) X X X t 55 t Aw Ax ( Aw Ax ) ( Bw Bx)

56 End of 3D Viewing 56