CS 284: CAGD 
Lecture #17 -- We 10/22, 2003.

Preparation:

Functional Optimization with Brakke's Surface Evolver

• 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 ?

Feedback on Presentation Format

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Preparing for 5-Minute Project Proposal Presentations 
("Venture Capitalist Rally" -- How to get the "gold".)

• Should last 4-5 minutes (hard 5 minute cutoff).
• Start with an attention-grabber.
• Be enthusiastic, make an impact.
• Your peers will rank-order your performance.

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Differential Geometry of Surfaces, Part I

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)

Start preparing your Project Proposals -->  E-mail outline due by Monday midnight.

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