Overall teaching objectives:
Students must understand the differences and overlap between implicit, parametric, and subdivision curves and surfaces. They must understand how to use one or the other form for computing geometric intersections or for rendering. They must know how to create Bezier and B-spline curves and surfaces through subdivision and how to alter these subdivision rules to create other types of curves and surfaces (four-point, Jarek, Js splines). They must be able to implement these as combinations of curve refinement and smoothing steps.
Motivation and relation to other modules: Smooth curves and surfaces are important in modeling and animation. Sometimes it is necessery or more efficient to use a closed form implicit or parametric model of such a shape. At other times, it may be simpler and prefereable to use a piecewise linear approxiamtion that may be smoothened or refined through subdivision to any desired accuracy. Students must be able to select between these three options and use them to solve various problems in graphics.
What students should know:
Examples, advantages, and drawbacks of implicit, parametric, and subdivision cuurves and surfaces and their use in applications.
Bezier subdivision, tangent computation, and convex hull property
Subdivision rules for cubic B-splines, 4-point, and other Js refinements.
Use of curve subdivision to produce surfaces and animations.