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# COS 526 - Advanced Computer Graphics

### Fall 2003

 Course home Outline and lecture notes Assignments

## Written Exercise 2

Due Wednesday, Oct. 22
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 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.)

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.

Please see the general notes on submitting your assignments, as well as the late policy and the collaboration policy.

Last update 29-Dec-2010 12:02:55
smr@cs.princeton.edu