**Final Exam Study Topics**

Math 155B - Spring 2001

This is an outline of possible topics for the final exam. The final may cover any topic covered in lecture throughout the entire quarter. I have tried to make list of topics complete, but it is possible a few topics have been omitted. The exam will not cover any technical details about 3D Studio Max, nor will it cover the command syntax of OpenGL. The emphasis is on the Math 155B topics listed below, but of course you are expected to know the Math 155A material too.

**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 continuity properties. Converting between Bézier curves and B-splines. Böhm knot insertion. (You do not need to memorise the formula for knot insertion.) You are also expected to know the Bézier curve material from last quarter.**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.**Interpolation with splines.**Catmull-Rom interpolation. Overhauser splines. Interpolating many points with a B-spline curve.**Color.**RGB and CMY primaries. Additive versus subtractive primaries. The trichromatic theory of light and the opponent theory of light. Conversion from RGB to HSL.**Recursive ray tracing.**Reflection, refraction. Angles of reflection and refraction. Index of refraction. Local lighting model and new terms for reflection and transmission. Shadow feelers. Global light calculations. What can and cannot be handled well with recursive ray tracing.**Distributed ray tracing.**From viewpoint: antialising, depth of field, motion blur. Soft shadows and penumbra. Jitter. Soft reflections. Wavelength dependent refraction/reflection angles.**Backwards ray tracing.****Cheats to avoid raytracing.**Environment maps, accumulation buffer, mipmapping, transparencies with alpha value, shadows via perspective transformations.**Intersection testing.**Ray versus sphere. Ray versus triangle. Ray versus convex polygon (or convex polytope). Bounding spheres. Bounding boxes.**Animation.**Forward/inverse kinematics. Forward/inverse dynamics. Key framing. Motion capture.**Orientations and quaternions.**Yaw-pitch-roll. Euler angles. Rotation matrices. Quaternions. Definition of quarternions. Multiplication and inverses of quaternions. Distributive and associative properties. Quaternion representation of orientation. Spherical linear interpolation (slerp-ing).**Forward and Inverse kinematics.**Rotational joints and rigid links. Forward kinematics gives positions as functions of the joint angles. Partial derivatives give a Jacobian matrix that drives inverse kinematics.**Radiosity.**Global diffuse light transport. Retangular patches. Form factors. Hemicube method. Jacobi iteration. Gauss-Seidel iteration. The shooting method.

**Final exam date: Monday, June 11, 11:30**. Usual room. No
notes or textbook permitted. Calculators are permitted but not particularly
required.

**Review session**, **Friday evening, June 8, 5:00pm**, in **APM
7421 (NOTE TIME CHANGE!)**