Practice problems - set 3
1. You are rendering a number of cubes (some may be intersecting) using the graphics pipeline with
back-face culling turned on. The cubes are triangulated and their triangles are oriented in a way suitable
for back-face culling. Thus, you get the correct image whenever the viewpoint is outside any of the cubes.
Let’s say that you move the viewpoint so that it enters one of the cubes. Describe what happens to the
image.
Now, assume the viewpoint is inside some number of cubes. How does the image look like (in particular,
which cubes are visible and which are not)?
2. Let’s say that the viewpoint is at vand the vertices of the triangle are at p,qand r. Argue that the
triangle is front-facing (its vertices appear in counterclockwise order) if (~pq ×~pr)·~pv > 0 and is back
facing if ( ~pq ×~pr)·~pv < 0. Assume right-handed coordinate system.
3. Let’s say that the viewpoint is at (1,2,3) and you are given three triangles with vertices at:
T1: (0,0,0), (0,1,0), (0,0,1)
T2: (1,2,5), (0,2,6), (0,1,7)
T3: (−1,3,3), (−1,4,3), (0,−1,−1)
Assume that the triangles whose vertices appear in clockwise order after projection are considered back
facing. Which of the three triangles are front facing? (assume vertices are specified in the order given
above)
4. Let’s say you are rendering a cube using graphics pipeline. Assume that the cube appears large on
the screen. Without back-face culling, you are processing Nfragments. With back-face culling (assuming
the vertices are specified in the right order so that the image is correct) you are processing Mfragments.
Come up with a rough approximation of the ratio M:N(explain your answer).
How about the same ratio for a triangulation of another 3D object (assume it’s a closed surface, i.e. one
that bounds a volume in 3D and that the triangles are oriented so that back face culling works).
Hint: think about rays that start at the viewpoint and intersect the volume, number of times they enter/exit
the volume and the relationship to Mand N.
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