# CS 284: CAGD  Lecture #17 -- Tu 10/24, 2006.

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# Differential Geometry of Surfaces (cont.)

• 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 in some direction (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 (soap film);
• H < 0 ==> mostly hill shaped;
• Osculating paraboloid
• 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 (curvature on projection of tangent plane)
• Geodesic lines have zero geodesic curvature everywhere (bend with the surface, but do not curve in the surface).
See: "Intrinsic Properties of a Surface" by M. E. Mortenson  (handout)
Differential Geometry of Surfaces -- Selected Formulas by Jordan Smith

# Introduction and Motivation to Minimum Variation Surfaces

Discussion of the salient points of the papers: Data structures, algorithms, difficulties ...

## Reading Assignment: BR'92: "The Surface Evolver" K. Brakke

- skip Sections  3.4, 3.5, 4.2, 4.8, 5.4, 5.5, 6.2, 6.7, 9, 10.

## Definitive Project Description (Phase 2):

Before Thursday 10/26/2006, midnight: E-mail me:  a 0.5 to 1.0 page description of your chosen project. Specify:
• What you plan to accomplish.
• The approach you will take.
• The deliverables and demos you hope to provide.
• Milestones: What you plan to have done before Thanksgiving

## Homework Assignment:  Make a genus-L Surface with DLh Symmetry (Phase 1)

Generate a rough polyhedral surface of genus-L, where L is the number of letters in your first name.
The surface should have DLh symmetry; i.e. L-fold rotational symmetry around the z-axis, as well as
mirror symmetry across the x-y-plane and across L planes that comprise the z-axis.
Build only the fundamental domain of this surface; i.e.  1 / 4L  of the total object.
Keep it as simple as possible; i.e. use only 3-5 quadrilaterals or about twice as many triangles.
This piece of polyhedral surface then get's suitably mirrored and replicated to form the whole closed, watertight genus-L surface.
Start from the basic set-up in  GenusL_SymmDLh.slf
All you have to do in this file is: replace the two place-holder quadrilaterals and adjust the replication number in the final surface assembly.
In the next phase of the assignment we will then convert this surface piece into the format of Brakke's Surface Evolver
and experiment with it.
The goal is to find the surface of lowest bending energy while maintaining the specified genus and symmetries.

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