1. CSC461: Lecture 19 Computer Viewing
Objectives
Introduce computer views and positioning camera
Look at alternate viewing APIs
Introduce the mathematics of projection

2. Computer Viewing
There are three aspects of the viewing process, all of which are implemented in the pipeline,
Positioning the camera
Setting the model-view matrix
Selecting a lens
Setting the projection matrix
Clipping
Setting the view volume

3. The OpenGL Camera
In OpenGL, initially the world and camera frames are the same
Default model-view matrix is an identity
The camera is located at origin and points in the negative z direction
OpenGL also specifies a default view volume that is a cube with sides of length 2 centered at the origin
Default projection matrix is an identity

4. Default Projection
Default projection is orthogonal
clipped out 2
z=0

5. Moving the Camera Frame
If we want to visualize object with both positive and negative z values we can either
Move the camera in the positive z direction
Translate the camera frame
Move the objects in the negative z direction
Translate the world frame
Both of these views are equivalent and are determined by the model-view matrix
A translation: glTranslatef(0.0,0.0,-d);
d > 0

6. Moving Camera back from Origin
default frames
frames after translation by –d
d > 0

7. Moving the Camera
We can move the camera to any desired position by a sequence of rotations and translations
Example: side view
Rotate the camera
Move it away from origin
Model-view matrix
Concatenation of rotation and translation: C = TR
OpenGL code: last transformation specified is first applied
glMatrixMode(GL_MODELVIEW)
glLoadIdentity();
glTranslatef(0.0, 0.0, -d);
glRotatef(90.0, 0.0, 1.0, 0.0);

8. Camera Positioning Where the camera is How the camera is oriented
View-reference point (VRP) in the world system
How the camera is oriented
View-plane normal (VPN): specifying the orientation of the projection plan
View-up vector (VUP): specifying the up direction
With VRP, VPN and VUP, can build a frame
The origin: VRP
Three independent vectors:
VPN, VUP, and
VRN×VUP

9. The LookAt Function Syntax: glLookAt(
The function glLookAt to form the required model-view matrix through a simple interface
Syntax: glLookAt(
eyex,eyey,eyez,
atx,aty,atz,
upx,upy,upz)
VRP = eye, VPN = at-eye, VUP=up
Example: isometric view of cube aligned with axes
glMatrixMode(GL_MODELVIEW):
glLoadIdentity();
gluLookAt(1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0)

10. Constructing LookAt View
Create an isometric view of the cube
Start with default setting
Transformations
Rotate about the y-axis -45 degrees
Rotate about the x-axis degrees
Translate along the z-axis by the distance d
Result:
C = T(0,0,-d)Rx(35.26)Ry(-45)
Equivalent to
gluLookAt(1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0)

11. Other Viewing APIs
The LookAt function is only one possible API for positioning the camera in OpenGL
Others include – not provided by OpenGL
View reference point, view plane normal, view up (PHIGS, GKS-3D)
Yaw, pitch, roll – flight simulation
Elevation, azimuth, twist – rotation about other objects instead of points
Direction angles
All these viewing can be constructed by specifying concatenation of translation and rotations

12. Orthogonal Projections
The default projection in the eye (camera) frame is orthogonal
A special case of parallel projections in which the projectors are perpendicular to the view plane
For points within the default view volume
xp = x, yp = y, zp = 0

13. Homogeneous Coordinate Representation
pp = Mp
xp = x
yp = y
zp = 0
wp = 1
M =
In practice, we can let M = I and set the z term to zero later
Normalization
Most graphics systems use view normalization
All other views are converted to the default view by transformations that determine the projection matrix
Allows use of the same pipeline for all views

14. Simple Perspective Projection
Center of projection at the origin
All projectors pass through the origin
Simple projection: Projection plane z = d, d < 0
General projection: Projection plane can have any orientation w.r.t. the front

15. Simple Perspective Equations
Consider top and side views
xp =
yp =
zp = d

16. Characteristics Nonlinear equations
Non-uniform foreshortening: the images of objects from the COP are reduced in size compared to the images of objects closer to the COP
Can be considered as a transformation of from (x, y, z) to (xp, yp, zp)
Preserves lines but not affine
Irreversible: because all points along a projector project into the same point

17. Homogeneous Coordinate Form
Consider q = Mp where
M transforms p to q
M =
p =
 q =
Represent points (x, y, z) as (wx, wy, wz, w), w≠0
The difference is the coefficient w
When w=1, homogeneous coordinates
Can represent a broader class of transformations

18. Perspective Division
However w  1, so we must divide by w to return to homogeneous coordinates
This perspective division yields
the desired perspective equations
d/z is the foreshortening factor
We will consider the corresponding clipping volume with the OpenGL functions
xp =
yp =
zp = d

19. Projection Pipeline Set the model-view matrix
Apply the projection matrix
Perform a perspective division