## COS 526 - Advanced Computer Graphics |
## Fall 2004 |

Course home | Outline and lecture notes | Assignments |

- As we saw, a single cubic Bézier curve can be defined as
Q(t) = b where the P_{0}(t) P_{0}+ b_{1}(t) P_{1}+ b_{2}(t) P_{2}+ b_{3}(t) P_{3}_{i}are the control points and the b_{i}(t) are the Bernstein polynomials.- 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. - 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.)

- 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
- What is the degree of continuity at an interior point of an
*n*th-order B-spline patch? What is the degree of continuity at a point on the boundary between two such B-spline patches? - 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 Ax^{2}+ Bxy + Cy^{2}+ Dx + Ey + F = 0). Short proofs preferred! - 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.

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

Last update 29-Dec-2010 12:01:30 smr at princeton edu