Written Exercise 2

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 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.)
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. Describe an algorithm for extracting a discrete approximation of the silhouettes of a surface represented as a signed distance function sampled along a uniform voxel grid. That is, the value at each voxel gives the signed distance to the nearest surface location, negative values occur inside the surface and positive values occur outside the surface. Hint: silhouettes depend on the viewing position and occur at areas on the surface where the dot product of the surface normal and the unit-length vector pointing toward the camera's center of projection are zero. What is the complexity of your algorithm? How does the running time of your algorithm relate to the Marching Cubes algorithm we studied in class?
4. (Optional for extra credit) 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!

Submitting

Please submit the answers to these questions in an email to rpw7e@cs.virginia.edu, with "CS451" 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.