CS 284: CAGD
Lecture #16  Th 10/19, 2006.
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Preparation:  Review course notes !

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 online 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 ...
You probably don't have the time to read every paragraph of every assigned
paper,
but at least you should try to find answers to questions like these:

What is the overall goal ?

What is the highlevel 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 ?
 Curvaturebased global functionals: MEC, MVC, SiMEC, SiMVC
 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)
See: "Intrinsic Properties of a Surface" by M. E. Mortenson (handout)
Differential
Geometry of Surfaces  Selected Formulas by Jordan Smith

We are concerned with 2manifolds 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, k_{1}
and k_{2}, orthogonal to each other)

Gaussian curvature: K=k_{1}*k_{2}
 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=(k_{1}+k_{2})/2
 H > 0 ==> mostly bowl shaped;
 H = 0 ==> a balanced saddle point; minimal surface;
 H < 0 ==> mostly bowl shaped;

Osculating paraboloid
 bestfitting 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:
FO'92: "Functional
Optimization for Fair Surface Design" by H. P. Moreton and C. H. Sequin.
Homework Assignment: COMPLETE QUIZ ! (alone  no interaction with any other person about it).
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