Final Exam Study Topics

Math 155B - Winter 2000

This is an outline of possible topics for the final exam.  The final may cover any topic covered in lecture up through the Monday, March 13 lecture.   I have tried to make list of topics complete, but it is possible a few topics have been omitted.   The exam will not cover forward or inverse kinematics.

1. Basics of Splines.  B-splines.  Blending functions and their properties..  The Cox - de Boor definition. Piecewise polynomial curves.  Local control.  Continuous derivatives.  Knot vectors and non-uniformly spaced  knots.  How repeated knots affect continuiuty properties.   Converting between Bézier curves and B-splines.  Böhm knot insertion.   (I will supply you with the Cox - de Boor definition --- you do not need to memorise the formula for knot insertion.)

2. Rational Splines & NURBS.  Equivalence with weighting.  Uses with points at infinity.  Conic sections, especially circular arcs, defined with degree 2 rational B-splines or rational Bézier curves.

3. Interpolation with splines.  Catmull-Rom interpolation.  Interpolating many points with a B-spline curve.

4. Recursive ray tracing.  Reflection, refraction.   Angles of reflection and refraction.  Index of refraction. Local lighting model and new term for transmission.  Shadow feelers.  Global light calculations.  What can and cannot be handled well with recursive ray tracing.

5. Distributed ray tracing. From viewpoint: antialising, depth of field, motion blur.  Soft shadows and penumbra.  Jitter.  Soft reflections.  Wavelength dependent refraction/reflection angles.

6. Backwards ray tracing.

7. Cheats to avoid raytracing.  Environment maps, accumulation buffer, mipmapping, transparencies with alpha value, shadows via perspective transformations.

8. Intersection testing.  Intersecting a ray with convex polygons, convex polytopes, spheres, conic sections.

9. Radiosity. Lambertian reflection.  Form factor calculations.  Hardware speedup of form factor calculations. Jacobi iteration.   Gauss-Seidel method.  Gathering method.  Shooting method.  Proof of convergence.

10. Quaternions.  Definitions and algebraic properties.  Rotations via quaternions.  Spherical linear interpolation with quaternions.  Catmull-Rom style splines with quaternion values for generating orientation curves.

11. Animation.  Keyframing and inbetweening.   Motion capture.  Use of splines.