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COS 526 - Advanced Computer Graphics

Fall 2004

Course home Outline and lecture notes Assignments

Written Exercise 2

Due Tuesday, Oct. 19
  1. As we saw, a single cubic Bézier curve can be defined as
    Q(t) = b0(t) P0 + b1(t) P1 + b2(t) P2 + b3(t) P3
    where the Pi are the control points and the bi(t) are the Bernstein polynomials.
    1. Extend this definition to a bicubic Bézier patch. That is, write down the equation for Q(s,t) in terms of the Bernstein polynomials and the sixteen control points Pij, i=0..3, j=0..3.
    2. How would you go about computing the surface normal at an arbitrary point on a Bézier patch? That is, given some s and t, find the surface normal at Q(s,t). (Explain how you would derive the answer - it is not necessary to write out the full expression explicitly.)
  2. What is the degree of continuity at an interior point of an nth-order B-spline patch? What is the degree of continuity at a point on the boundary between two such B-spline patches?
  3. Show that a quadratic rational Bézier curve in the plane can generate arbitrary conic sections (i.e., arcs of an arbitrary curve of the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0). Short proofs preferred!
  4. The basic operation in hierarchical frustum culling is testing whether some primitive shape lies completely inside, lies completely outside, or crosses the view frustum. Describe how to perform this test for (a) a sphere and (b) an AABB (axis-aligned bounding box). Assume the equations for the planes that make up the view frustum are given.


Please submit the answers to these questions in writing, or in an email to smr@cs.princeton.edu, with "CS526" in the subject line. Plain text email is preferred.

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Last update 29-Dec-2010 12:01:30
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