COS 526 - Advanced Computer Graphics

Fall 2003

Written Exercise 2

1. B-Splines and continuity:
   1. Why do computer graphics applications use piecewise polynomial curves of degree 3 rather than single curves of higher order?
   2. What is C² continuity? How does it differ from C¹?
   3. What is the degree of continuity at an interior point of a B-spline patch? What is the degree of continuity at a point on the boundary between two B-spline patches?

2. 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 P_i are the control points and the b_i(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 P_ij, 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.)

3. 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 sphere and an AABB (axis-aligned bounding box). Assume the equations for the planes that make up the view frustum are given.

Submitting

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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