CS 284: CAGD 
Lecture #16 -- Th 10/19, 2006.

Preparation: -- Review course notes !

Hand back results on Spiral Surface Assignment

Give feedback on Project Proposals

Upcoming Paper Presentations

• The whole, uninterrupted presentation for one paper should last between 20 and 25 minutes.
•  there may be interruptions due to questions, but "the clock will be stopped.".
• Everybody in the class should read the papers carefully before the day that they are presented in class.
• The class should be prepared to help the presenters with constructive and insightful questions and comments.
• The presenters should be prepared to answer such questions.
• The presentations will be judged for:
• Organization of the material
• Clarity of explanations
• Quality of visual materials
• Forcefulness and enthusiasm of oral presentation
• Occasionally I may ask you to answer some questions on-line about the paper before coming to class.
  
• I reserve the right to start a class with a short 10 minute pop quiz about the paper(s) for that day.

How to read a paper ...

• What is the overall goal ?
• What is the high-level approach ?
• What is novel, innovative, reusable in other contexts ?
• What are the results obtained ?
• What else should you remember about this paper ?

Curve Optimization

• What are desirable properties -- and how do we achieve them ?
  
• Curvature-based global functionals: MEC, MVC, Si-MEC, Si-MVC
• The generic optimization step:  A move that makes things better -- how do we find such a step ?
• Perhaps there are many different ways to make things better -- how do we combine them ?
• Gradient descent.
  
• How to reach the global minimum, and what may slow it down ...
  
• What is an adequate curve representation ?  -- e.g., quintic hermite ?  -- or a cubic subdivision curve ?
  
• What are the best "primitives"?
• Does this work in 3D ?

Differential Geometry of Surfaces

Intrinsic Properties of Surfaces (following M. E. Mortenson)

• We are concerned with 2-manifolds p(u,w),
  
• thus need 2 paramters u, w, 
• 2 derivatives, dp/du, dp/dw (= velocity along parameter lines)
• First Fundamental Form: dp * dp = E du du + 2F du dw + G dw dw
• with E=p^u p^u,  F=p^u p^w,  G=p^w p^w;
• describes metric properties of surface.
• Second Fundamental Form: -dp * dn = L du du + 2M du dw + N dw dw
• with L=p^uu n, M=p^uw n, N=p^ww n, where n is the normal;
• describes curving and twisting of surface, assuming a "good" parametrization.
• Descriptive Trihedron: Darboux Frame
• Normal vector
• Tangent plane
• Principal directions
• Normal curvature (curvature of intersection with normal plane)
• Principal curvatures (max. and min. of normal curvature, k₁ and k₂, orthogonal to each other)
• Gaussian curvature: K=k₁*k₂
• K > 0 ==> spherical curvature (dome or bowl);
• K = 0 ==> flat, no curvature (plane, cylinder, or cone);
• K < 0 ==> hyperbolic curvature (saddle points);
• Mean curvature: H=(k₁+k₂)/2
• H > 0 ==> mostly bowl shaped;
• H = 0 ==> a balanced saddle point; minimal surface;
• H < 0 ==> mostly bowl shaped;
• Osculating paraboloid
• best-fitting quadric surface
• corresponds to osculating circle for a curve.
• Dupin indicatrix
• scaled conics obtained from slicing the osculating paraboloid parallel to the tangent plane.
• Curves on a surface
• Geodesic curvature
• Geodesic lines
• Meusnier's sphere (collection of osculating circles of all curves with same tangents through a point)

Reading Assignment:

Homework Assignment:  COMPLETE  QUIZ !  (alone -- no interaction with any other person about it).