Computational Mathematics
Object Space
This is where the object is defined in its own coordinate system
World space
Create clones of the objects in the object spaces; these clones are transformed to the
position, orientation and the size required in the world space
Camera space
The world space, aligned so that the camera is at the origin looking down the z-axis
Screen space
Project the camera space onto the screen using perspective projection
Clipped screen space
Clip the objects to the boundaries of the viewing frustum and apply hidden surface
determination
Raster
The screen space which is defined using vector graphics is approximated using a raster
(x', y', z')
(-fx/z, -fy/z, -f)
Projection matrix P
(1 0 0 0 )
(0 1 0 0 )
(0 0 1 0 )
(0 0 1/f 0)
Why are 4x4 matrices used?
4x4 matrices are used because the use of homogenous coordinates allows translation
transformations to be represented by matrices so that transformations are handled
consistently and any allowable transformation can be represented by a matrix.
T
A translation matrix that will translate the objects in the space so that the rotation axis
goes through the origin. This can be achieved by translating the vector -c.
Back-face culling (applied to remove hidden surfaces in screen space)
Any polygons whose normal vector is pointing away from the viewer is removed from
the scene.
If v • n > 0, then the polygon is back facing where v is the viewing vector and n is the
normal vector to the polygon.
Painter's algorithm (applied to remove hidden surfaces in screen space)
