Computer Graphics: Homework 1. Due: 9/2/2004
1. Is the perspective projection of a non-obstructed sphere a disk? Prove
your answer. The perspective transformation of a sphere is a surface in the
3D transformed space. Is it a sphere? Is it a quadric (i.e. is it given by a
quadratic equation)?
2. We used 1 −1/z as the depth in class. Two obvious alternatives would be
(a) z
(b) (x2+y2+z2)1/2(the distance between the viewpoint and the point being
projected).
For both of the above, write out a formula that the rasterizer would have
to use to interpolate depth from projected coordinates/depth of vertices of
a a triangle with vertices (x0, y0, z0), (x1, y1, z1), (x2, y2, z2). In other words,
for a fragment at (a, b), how to compute its depth, given the vertices of the
triangle it originated from? (the reason for 1 −1/z is that the interpolation
it requires is linear).
3. Express a 3D rotation (about an arbitrary axis) and a 3D translation
as a composition of symmetries about planes (not necessarily axis-aligned)
(describe how it works and what the planes are in plain English).
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