1. Based on slides created by Edward Angel
Computer Viewing
Mohan Sridharan
Based on slides created by Edward Angel
CS4395: Computer Graphics

2. Objectives Introduce the mathematics of projection.
Introduce OpenGL viewing functions.
Look at alternate viewing APIs.
CS4395: Computer Graphics

3. Computer Viewing
There are three aspects of the viewing process, all implemented in the pipeline:
Positioning the camera:
Setting the model-view matrix.
Selecting a lens:
Setting the projection matrix.
Clipping:
Setting the view volume.
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4. The OpenGL Camera Initially, object and camera frames are the same:
Default model-view matrix is identity.
The camera located at origin and points in the negative z direction.
OpenGL also specifies default view volume that is a cube with side of length 2 centered at the origin.
Default projection matrix is identity.
CS4395: Computer Graphics

5. Default Projection Default projection is orthogonal: clipped out 2 z=0
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6. Moving the Camera Frame
To visualize objects with both positive and negative z values:
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 views are equivalent and determined by the model-view matrix:
Need a translation: glTranslatef(0.0,0.0,-d)
d > 0
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7. Moving Camera back from Origin
frames after translation by –d
d > 0
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8. Moving the Camera
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.
Modelview matrix: C = TR
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9. OpenGL code
Remember that last transformation specified is first to be applied!
glMatrixMode(GL_MODELVIEW)
glLoadIdentity();
glTranslatef(0.0, 0.0, -d);
glRotatef(90.0, 0.0, 1.0, 0.0);
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10. The LookAt Function
GLU library contains the function (gluLookAt) to form the modelview matrix through a simple interface.
Need to set up a direction and initialize:
Concatenate with modeling transformations.
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., );
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11. Function: gluLookAt()
glLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz)
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12. Other Viewing APIs
The LookAt function is only one possible API for positioning the camera.
Others include (Sections 5.3.2, 5.3.4):
View reference point, view plane normal, view up (PHIGS, GKS-3D).
Yaw, pitch, roll.
Elevation, azimuth, twist.
Direction angles.
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13. Projections and Normalization
The default projection in the eye (camera) frame is orthogonal.
For points within the default view volume:
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.
xp x
yp y
zp = 0
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14. Homogeneous Coordinate Representation
Default orthographic projection:
In practice, can let M=I and set z term to zero later.
pp = Mp
xp = x yp = y zp = 0 wp = 1
M =
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15. Perspective Projection
Preserves lines but not affine.
Irreversible: all points along a projector project to the same point.
Later: invertible variant of projection transformation that preserves distances (hidden-surface removal).
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16. Simple Perspective Projection
Center of projection at the origin.
Projection plane z = d, d < 0
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17. Simple Perspective Equations
Consider top and side views:
xp =
yp =
zp = d
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18. Simple Perspective Projection
Representation in 4D for a 3D point:
Recover 3D point by dividing first three components by the fourth component.
Points in 3D become lines through the origin in 4D.
Large class of transformations using 4x4 matrices with the last row altered.
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19. Perspective Projection Example
consider q = Mp where
M =
 q =
p =
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20. Perspective Division
However w  1, so divide by w to return from homogeneous coordinates to 3D space.
This perspective division yields:
In homogeneous coordinates:
Apply projection matrix after model-view matrix, then perform perspective division.
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21. OpenGL Orthogonal Viewing
glOrtho(left,right,bottom,top,near,far)
near and far measured from camera
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22. OpenGL Perspective glFrustum(left,right,bottom,top,near,far)
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23. Using Field of View
With glFrustum it is often difficult to get the desired view.
gluPerpective(fovy, aspect, near, far) often provides a better interface.
front plane
aspect = w/h
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24. Other Issues Hidden surface removal: Section 5.6.
Interactive mesh displays: Section 5.7.
CS4395: Computer Graphics