#
CS 284: CAGD

Lecture #17 -- Tu 10/24, 2006.

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**COLLECT QUIZ !**

# Differential Geometry of Surfaces (cont.)

- Descriptive Trihedron:
**Darboux Frame**
- Normal vector
- Tangent plane
- Principal directions

**
Normal curvature** (curvature of intersection with normal plane)-
P
**rincipal 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 in some direction (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 (soap film);
- H < 0 ==> mostly hill 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 (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

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

**
**

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

**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 D**_{Lh} 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 D_{Lh} 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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