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. 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.

Submitting

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