1. 3D Viewing

2. 3D Viewing Transformation PipelineMC
Modeling Transformation xw yw zw xv yv zv xw yw zw xv yv zv V P0 N WC
Viewing Transformation VC
clipping
window
view volume xv yv zv PC
Projection Transformation
Normalization Transformation and Clipping NC
znorm
ynorm
(1,1,1)
(-1,-1,-1)
viewport
screen
(xvmin, yvmin, 0)
(xvmax, yvmax, 0)
Viewport Transformation DC

3. 3D Viewing Transformation PipelineMC
Modeling Transformation WC
Viewing Transformation VC
Projection Transformation PC
Normalization Transformation and Clipping NC
Viewport Transformation DC

4. 3D Viewing Transformation PipelineMC
Modeling Transformation WC
Viewing Transformation VC
Projection Transformation PC
Normalization Transformation and Clipping NC
Viewport Transformation DC

5. 3D Viewing V view up vector P0 = (x0, y0, z0) view point yw
N viewplane normal
Viewplane is at point zvp in negative zv direction
V is perpendicular to N xw yw zw xv yv zv V P0 N

6. 3D Viewing xv yv zv is a right-handed viewing coordinate system yv xvu n

7. 3D Viewing
Transformation from World to Viewing Coordinates:
1. Translate viewing coordinate origin to the origin of world coordinate system
2. Apply rotations to align xv yv zv axes with xw yw zw axes respectively
MWC,VC = R . T yv xw yw zw xv zv P0 xw yw zw yv xv zv

8. 3D Viewing Transformation PipelineMC
Modeling Transformation WC
Viewing Transformation VC
Projection Transformation PC
Normalization Transformation and Clipping NC
Viewport Transformation DC

9. Projections isometric axonometric orthogonal Parallel cavalier oblique
cabinet
Perspective

10. Projections view plane view plane Perspective projection
Parallel projection

11. Orthogonal Projection

12. Orthogonal Projection
Axonometric: displays more than one face of an object
Isometric: projection plane intersects each coordinate axis at the sane distance from the origin
Isometric

13. 3D Viewing Transformation PipelineMC
Modeling Transformation WC
Viewing Transformation VC
Projection Transformation PC
Normalization Transformation and Clipping NC
Viewport Transformation DC

14. Orthogonal Projection
Clipping
view volume
clipping
window xv yv zv
far plane
near plane

15. Orthogonal Projection
Normalization
Mortho,norm . R . T
ynorm
znorm
(1,1,1) yv
xnorm
(-1,-1,-1) xv zv

16. Oblique Projection
Oblique projection: projection path is not perpendicular to the view plane
view plane

17. Oblique Projection yv
(xp,yp,zvp) xv zv a f
(x,y,zvp)
(x,y,z)

18. Oblique Projection xp = x + L1 (zvp – z) cos f
yp = y + L1 (zvp – z) sin f
shearing transformation L1 f
view plane

19. Oblique Projection Cavalier Projection Cabinet Projection f = 450

20. Oblique Projection window a f Vp view volume Vp yv xv zv (xp,yp,zvp)
(x,y,zvp)
view
volume Vp
(x,y,z)

21. Oblique Projection MWC,VC . Morth,norm . Mobl Mobl,norm shear Vp Vp
window
window
shear Vp Vp
orthogonal
coordinates
MWC,VC . Morth,norm . Mobl
Mobl,norm

22. Parallel Projection
Viewer’s
position
View plane

23. Perspective Projection
Viewer’s
position
View plane

24. Perspective Projectionyv
projection
reference
point (prp) xv zv
view
plane

25. Perspective Projectionyv
(xp,yp,zvp)
(xprp,yprp,zprp) xv zv
(x,y,z)

26. Perspective Projection
At any point (xp, yp, zp) along the projection line:
zp = zvp
xp = x (zprp-zvp) / (zprp-z) + xprp (zvp-z) / (zprp-z)
yp = y (zprp-zvp) / (zprp-z) + yprp (zvp-z) / (zprp-z)

27. Perspective Projection
Vanishing point y x z
One-point perspective projection
x-axis
vanishing point
z-axis
vanishing point
Two-point perspective projection

28. Perspective Projection
Far plane
Near plane
Viewer’s
position

29. Perspective Projection
Far plane
Near plane
Viewer’s
position

30. Perspective Projection
Clipping
window
Rectangular
Frustum
view volume q
projection
reference
point (prp)
Far
clipping
plane
Near
clipping
plane

31. Perspective Projection
P=(x, y, z, 1)
Ph=(xh, yh, zh, h)
xp = x (zprp-zvp) / (zprp-z) + xprp (zvp-z) / (zprp-z)
yp = y (zprp-zvp) / (zprp-z) + yprp (zvp-z) / (zprp-z)
h = zprp-z
xh = x (zprp-zvp) + xprp (zvp-z)
yp = y (zprp-zvp) + yprp (zvp-z)
xp = xh /h
yp = yh /h

32. Perspective Projection
Ph= Mpers . P
Mpers =
zprp-zvp xprp xprpzprp
0 zprp-zvp -yprp yprpzprp
sz tz
zprp

33. Perspective Projection
Symmetric Frustum
Frustum
center line
Far plane
view volume
Clipping
window
Near plane q
view plane
(xprp, yprp, zvp)
(xprp, yprp, zprp)

34. Perspective Projection
200
600

35. Perspective Projection
Parallelepiped
view volume
Symmetric
Frustum
Perspective Mapping
Then apply normalization transformation

36. Perspective Projection
Parallelepiped
view volume
Oblique
Frustum
Perspective Mapping
(xprp, yprp, zprp)=(0,0,0)
1. Transform to asymmetric frustum (z-axis shear)
2. Normalize viewvolume

37. Perspective Projection
shzx= -(xwmin+xwmax)/2.znear
Mshear = shzy= -(ywmin+ywmax)/2.znear
assuming that viewplane is at the
position of the near plane
Mpers =
Moblpers= Mpers . Mshear
shzx 0
shzy 0
znear
0 -znear
sz tz

38. Perspective Projection
Normalization
ynorm
(xwmax, ywmaz, zfar)
transformed frustum
znorm
(1,1,1)
xnorm yv xv zv
PRP
(-1,-1,-1)
(xwmin, ywmin, znear)

39. Perspective Projection
Mnormpers = Mxyscale . Moblpers =
-znearsx sx(xwmin+xwmax)/2 0
znearsy sy(ywmin+ywmax)/2 0
sz tz
-2znearsx/(xwmax-xwmin) (xwmin+xwmax)/(xwmax-xwmin) 0
znearsy/(ywmax-ywmin) (ywmin+ywmax)/(ywmax-ywmin) 0
(znear+zfar)/(znear-zfar) -2.znear.zfar/(znear-zfar)

40. Perspective Projection
For symmetric frustum with field-of-view angle q:
Mnormsymmpers =
Mnormsymmpers . R . T
cot(q/2)/aspect
0 cot(q/2) 0 0
(znear+zfar)/(znear-zfar) -2.znear.zfar/(znear-zfar)

41. 3D Viewing Transformation PipelineMC
Modeling Transformation WC
Viewing Transformation VC
Projection Transformation PC
Normalization Transformation and Clipping NC
Viewport Transformation DC

42. Viewport Transformation
ynorm
(xvmax, yvmax, 0)
znorm
viewport
screen
(1,1,1)
xnorm
(-1,-1,-1)
(xvmin, yvmin, 0)
Mnormviewvol,3Dscreen =
(xvmax-xvmin)/ (xvmin+xvmax)/2
(yvmax-yvmin)/2 0 (yvmin+yvmax)/2
0 0 1/2 1/2

43. 3D Clipping Inside if: -h≤xh≤h, -h≤yh≤h, -h≤zh≤h
Inside if:
-h≤xh≤h, -h≤yh≤h, -h≤zh≤h
Use sign bits of h±xh, h±yh, h±zh to set bit1-bit6
far near top bottom right left
ynorm
znorm
(1,1,1)
xnorm
(-1,-1,-1)

44. Point Clipping Test sign bits of h±xh, h±yh, h±zh
If region code is then inside
otherwise eliminate

45. Line Clipping 1. Test region codes
if for both endpoints => inside
if RC1 V RC2 = => trivially accept
if RC1 Λ RC2 ≠ => trivially reject
2. If a line fails the above tests, use line equation to determine whether there is an intersection

46. Polygon Clipping 1. Check coordinate limits of the object
if all limits are inside all boundaries => save the entire object
if all limits are outside any one of the boundaries => eliminate the entire object
2. Otherwise process the vertices of the polygons
Apply 2D polygon clipping
Clip edges to obtain new vertex list.
Update polygon tables to add new surfaces
If the object consists of triangle polygons, use Sutherland-Hodgman algorithm.

47. OpenGL Orthogonal projection glMatrixMode (GL_PROJECTION)
gluOrtho (xwmin, xwmax, ywmin, ywmax, dnear, dfar)
Orthogonal projection
gluPerspective (theta, aspect, dnear, dfar)
theta: field-of-view angle (00 – 1800)
aspect: aspect ratio of the clipping window (width/height)
dnear, dfar: positions of the near and far planes (must have positive values)
glFrustum (xwmin, xwmax, ywmin, ywmax, dnear, dfar)
if xwmin = -xwmax and ywmin = -ywmax then symmetric view volume

48. OpenGL glMatrixMode (GL_MODELVIEW)
gluLookAt (x0, y0, z0, xref, yref, zref, Vx, Vy, Vz)
Designates the origin of the viewing reference frame as,
the world coordinate position P0=(x0, y0, z0)
the reference position Pref=(xref, yref, zref) and
the viewup vector V=(Vx, Vy, Vz) xw yw zw xv yv zv V P0

49. OpenGL float eye_x, eye_y, eye_z; void init (void) {
glClearColor (1.0, 1.0, 1.0, 0.0); // Set display-window color to white.
glMatrixMode (GL_PROJECTION); // Set projection parameters.
glLoadIdentity();
gluPerspective(80.0, 1.0, 50.0, 500.0); // degree, aspect, near, far
glMatrixMode (GL_MODELVIEW);
eye_x = eye_y = eye_z = 60;
gluLookAt(eye_x, eye_y, eye_z, 0,0,0, 0,1,0); // eye, look-at, view-up }
void main (int argc, char** argv) {
glutInit (&argc, argv); // Initialize GLUT.
glutInitDisplayMode (GLUT_SINGLE | GLUT_RGB); // Set display mode.
glutInitWindowPosition (50, 500); // Set top-left display-window position.
glutInitWindowSize (400, 300); // Set display-window width and height.
glutCreateWindow ("An Example OpenGL Program"); // Create display window.
glViewport (0, 0, 400, 300);
init ( ); // Execute initialization procedure.
glutDisplayFunc (my_objects); // Send graphics to display window.
glutReshapeFunc(reshape);
glutKeyboardFunc(keybrd);
glutMainLoop ( ); // Display everything and wait.