1. THREE-DIMENSIONAL VIEWING II12
고려대학교 컴퓨터 학과
김 창 헌

2. Contents Viewing Pipeline Viewing Coordinates
Specifying the Viewing Coordinates
Transformation from WC to VC
View Volumes and Projection Transformations
Clipping
Hardware Implementations
3D Viewing Functions
Summary

3. Viewing Pipeline General 3D Transformation Pipeline Clipping against
Modeling
Coordinates
Modeling
Transformation
World
Coordinates
Viewing
Transformation
Viewing
Coordinates
Clipping against
Viewing volume
Projection
Transformation
Projection
Coordinates
Workstation
Transformation
Device
Coordinates
Project onto
Projection plane

4. The Human Visual System
Cornea (각막) : 렌즈를 보호하는 투명막
Iris (홍채) : 눈에 들어오는 빛의 양을 조절
Lens : 3차원의 물체를 투영하여 2차원의 이미지를 형성함
Retina (망막) : 투영된 이미지가 맺히는 곳
Rods and Cones (시신경의 간상체) : 빛을 감지하는 신경

5. The simplest camera geometric model !
The Pinhole Camera
The simplest camera geometric model !
Side view
Angle of view(field)
The angle made by the largest object that our camera can image on its film plane

6. The Synthetic Camera Model Equivalent views of image formation
Imaging with the
sythetic camera
Imaging system
Projection
Plane
Center of
Projection
Projection
Plane
Equivalent views of image formation

7. Clipping (a) with window in initial position (b) with window shifted
Clipping window
Clipping (a) with window in initial position (b) with window shifted

8. The Programmer’s Interface
Position The camera location usually is given by the position of the center of the lens(the center of projection)
Orientation Place a camera coordinate system with its origin at the center of projection
Focal length The focal length of the lens determines the size of the image on the film plane
Flim plane The back of the camera has a height andq width
Camera specification

9. Viewing Coordinates Camera positioning VRP VPN VUP
Specifying the view plane !
Use A viewing API
Set_view_reference_point(x,y,z);
VRP
Set_view_plane_normal(nx,ny,nz);
VPN
Set_view_up(ux,uy,uz);
VUP

10. Viewing Coordinate System
1. View reference point (VRP)
- The camera position in the World-coordinate system
- The origin of Viewing-coordinate system
VPN(n)
VRP
VUP(v) zw xw yw u
UVN coordinate system
2. View-plane normal (VPN)
- The positive direction for the viewing zv axis
3. View- up vector (VUP)
- Specify what direction is up to the camera
- To establish the positive direction for the
yv axis, VUP is adjusted to a position
perpendicular to the normal vector VPN
View plane
4. Set up the u vector
- Using vectors N and V, determine the right-handed viewing system

11. Camera Positioning in OpenGL
1. Using the Model View Matrices
The Camera Positioning Movement of the world frames
Ex)
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glTranslatef(0.0,0.0,-d);
glRotatef(-90.0,0.0,1.0,0.0);
Movement of the world frames
a) Initial camera position
b) After change in the model view matrix

12. Camera Positioning in OpenGL
2. Using the Look-At Function
gluLookAt(eyex,eyey,eyez,atx,aty,atz,upx,upy,upz);
Look-at positioning

13. Positioning of the Camera
Other Viewing APIs
Roll,pitch and yaw
Elevation and azimuth

14. Transformation from WC to VC
Transformation sequences
1. Translate the view reference point to the origin of the WC system
2. Apply rotations to align the xv, yv, and zv axes with the world axes
General sequence of translate-rotate transformation

15. Transformation from WC to VC (con’t)
Translation
view reference point(x0, y0, z0)
Rotation
rotate around the world xw axis to bring zv into the xwzw plane
rotate around the world yw axis to align the zw and zv axis
final rotation is about the zw axis to align the yw and yv axis

16. Transformation from WC to VC (con’t)
Direct generating the rotation-transformation matrix

17. View Window and View Volume
Definition of a View Volume
View window - How much of the scene is caught on film
- Can be placed any where on the view plane
View volume - Only those objects within it will appear in the display
- The volume size depends on the view window size
- While its shape depends on the type of the projection

18. View Volumes and Projection
View volume boundary
Near plane(Front plane)
Far plane(Back plane)
View window
Parallelpiped Volume
View Frustum
Projection reference point

19. glFrustum(left,right,bottom,top,near,far);
Projections in OpenGL
1. Perspective Projection in OpenGL
glFrustum(left,right,bottom,top,near,far);
View point
near
far
left
right
top
bottom

20. gluPerspective(fovy,aspect,zNear,zFar);
Projections in OpenGL
gluPerspective(fovy,aspect,zNear,zFar);
aspect = w/h  w h
View point
fovy
near
far

21. glOrtho(left,right,bottom,top,near,far);
Projections in OpenGL
2. Parallel Projection in OpenGL
glOrtho(left,right,bottom,top,near,far);
left
top
Toward the
view point
right
bottom
near
far

22. Viewport Transformation
Mapping the Viewing volume to the Viewport
glViewport( x, y, width, height);
gluPerspective(myFovy,10.0, myNear, myFar);
glViewport(0,0,400,400);
gluPerspective(myFovy,10.0, myNear, myFar);
glViewport(0,0,400,200);

23. OpenGL Viewing Summary
Ex)
A Transformed Cube
void CALLBACK myReshape() {
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
glFrustum(-1.0,1.0,-1.0,1.0,1.5,20.0);
glMatrixMode(GL_MODELVIEW);
glViewport(0,0,w,h); }
void main()
auxInitDisplayMode(AUX_SINGLE |AUX_RGBA);
auxInitPosition(0,0,500,500);
auxReshapeFunc(myReshape);
auxMainLoop(display);
#include <GL/glos.h>
#include <GL/gl.h>
#include <GL/glu.h>
#include <GL/glaux.h>
void CALLBACK display () {
glClear(GL_COLOR_BUFFER_BIT);
glColor3f(1.0,1.0,1.0);
glLoadIdentity();
glTranslatef(0.0,0.0,-5.0);
glScalef(1.0,2.0,1.0);
auxWireCube(1.0);
glFlush(); }
Define the
Projection
Define the Viewport
( w,h : Window Size )
Initial Position of
the Window
Positioning/ Scaling
of the Camera

24. View Volume Properties
Changing the shape of the oblique-projection view volume by moving the window
Changing perspective effects by
moving the projection reference
point

25. View Volume Properties
Projected object size depends on
the view plane position relative
to the position of the PRP
Simulate the Camera moving
in Animation
- PRP move with the VRP

26. General Parallel-Projection Transformation
General Parallel-Projection Process
Eliminating
Z coordinate
Oblique projection vector
and view volume
Regular view volume after shearing

27. General Parallel-Projection Transformation
※ Relationship between Vp and
L,, and 

28. General Perspective-Projection Transformation
1. Shear the view voume so that the centerline of the frustum is
perpendicular to the view plane
2. Scale the view volume with a scaling factor that depends on 1/z
Frustum
Centerline zv
(x,y,z)
(x’,y’,z’)
(x”,y”,z”)
View plane
View plane
(xprp, yprp, zprp)
Shearing
View volume
Pyramid
into a Box

29. General Perspective-Projection Transformation
Step 1.
Step 2.

30. Clipping
Clipping
Identify and save all surface segments within the view volume
View-volume clipping boundary : planes

31. Normalized View Volumes
Normalized View Volumes  Why?
View Volume

32. Normalized View Volumes (con’t)
Pipeline

33. Normalized View Volumes (con’t)
Advantages
provides a standard shape for representing any sized view volume
clipping procedures are simplified and standardized with unit cube
depth cueing and visible-surface determination are simplified
Mapping position
within a rectangular view volume to a 3D rectangular viewport
Translation factors
Ratios of the dimensions of
the viewport and view volume

34. 1. Parametric line equation
Viewport Clipping
1. Parametric line equation
2. Calculation of
parameter value
3. Intersection point
Side view of two
line segments that
are to be clipped

35. Clipping in Homogeneous
Clipping in Homogeneous Coordinates

36. Hardware Implementations

37. 3D Viewing Functions
Multiple Views Using Difference Camera Orientations

38. 3D Viewing Functions (con’t)
A Wide-Angle Perspective Display
Same viewing position, but with slight shifts in the viewing direction

39. Summary Modeling Coordinates Modeling Transformation World Coordinates
Viewing
Transformation
Viewing
Coordinates
Projection
Transformation
Projection
Coordinates
Workstation
Transformation
Device
Coordinates