CS216 Spring 2010Practice Midterm

To study for the midterm I would suggest you: (1) Know how to do (by heart) all the practice problems. (2) Go over your notes at least three times. Second and third time try to see how much you can remember from the first time. (3) Go over the homework problems. (4) Try to create your own problems similar to the ones I have given and solve them. (5) Skim the relevant sections from the book. (6) If you want to study in groups, at this point you are ready to quiz each other. The practice midterm is below. Here are some facts about the actual midterm: (a) It is closed book, closed notes. Nothing will be permitted on your desk except your pen (pencil) and test. (b) You should bring photo ID. (c) There will be more than one version of the test. Each version will be of comparable difficulty. (d) One problem (less typos) on the actual test will be from the practice test. (e) One problem on the actual test will be about one of my HW solutions.

1. Write out completely (including the blending functions) the degree three Bézier curve with control point `(0,0)`, `(1,1)`, `(2, -1)`, `(3, 0)`. Use de Casteljau method, to compute `\vec q(1/5)`.

2. Explain how recursive subdivision could be used to determine if a line segment intersects with a Bézier curve.

3. Convert the Bézier curve of Problem 1 to a Hermite representation.

4. What is variation diminishing property? Do Bézier curves have it? Do B-splines have it?

5. Give the control points of a degree four Bézier curve which is identical to that of Problem 1.

6. Briefly describe what each of the following OpenGL functions does: glMap2f, glMapGrid2f, glEvalMesh

7. What are TCB splines? What are they used for? How are the control points `D\vec q_i^(\pm)` defined?

8. Suppose we have the knot vector `[0, 2, 4, 6]`. Use the Cox de Boor formula to write out the blending functions `N_(i,k)` for `k = 1, 2`.

9. Describe the algorithm for converting from a B-spline curve to a piecewise Bézier curve.

10. Briefly describe the following approaches to geometric modeling: Cell Decomposition, Volume Modeling, Euler Operations Representation.